NUMBER SYSTEM

The mysterious world of numbers…

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data-sgid="s85" data-fillcolor = "16777215" style="z-index:4;left:79px;top:0px;width:640px;height:540px"> <img id="s85_i" src="8ABC37C2.png" class="sg" style="z-index:1;left:0px;top:0px;width:640.85px;height:540.48px"></div> <div id="s91" class="sg" data-sgid="s91" data-fillcolor = "16777215" style="z-index:5;left:73px;top:-3px;width:12px;height:546px"> <img id="s91_i" src="2086EAC3.png" class="sg" style="z-index:1;left:0px;top:0px;width:12.95px;height:546.71px"></div> <div id="s102" class="sg" data-sgid="s102" data-pointer-events = "none" style="z-index:1;left:641px;top:0px;width:78px;height:540px"> <img id="s102_i" src="2821B18.png" class="sg" style="z-index:1;left:0px;top:0px;width:78.81px;height:540.48px"></div> <div id="s172" class="sg" data-sgid="Rectangle 1" data-fillcolor = "16777215" style="z-index:1;left:79px;top:0px;width:640px;height:540px"> <img id="s172_i" src="8ABC37C2.png" class="sg" style="z-index:1;left:0px;top:0px;width:640.71px;height:540.48px"></div> <div id="s173" class="sg" data-sgid="Rectangle 2" data-fillcolor = "16777215" style="z-index:2;left:73px;top:-3px;width:12px;height:546px"> <img id="s173_i" src="2086EAC3.png" class="sg" style="z-index:1;left:0px;top:0px;width:12.95px;height:546.71px"></div> <div id="s180" class="sg" data-sgid="s180" data-fillcolor = "16777215" style="z-index:1;left:38px;top:467px;width:389px;height:73px"> <img id="s180_i" src="FCD12B6A.png" class="sg" style="z-index:1;left:0px;top:0px;width:389.80px;height:73.53px"></div> <div id="s181" class="sg" data-sgid="s181" data-fillcolor = "0" style="z-index:2;left:37px;top:466px;width:291px;height:74px"> <img id="s181_i" src="EF820D70.png" class="sg" style="z-index:1;left:0px;top:0px;width:291.47px;height:74.35px"></div> <div id="s182" class="sg" data-sgid="s182" data-pointer-events = "none" style="z-index:3;left:0px;top:455px;width:268px;height:85px"> <img id="s182_i" src="B785516D.png" class="sg" style="z-index:1;left:0px;top:0px;width:268.56px;height:85.96px"></div> <div id="s183" class="sg" data-sgid="s183" data-pointer-events = "none" style="z-index:4;left:-1px;top:454px;width:269px;height:87px"> <img id="s183_i" src="EC21C7F1.png" class="sg" style="z-index:1;left:0px;top:0px;width:269.86px;height:86.32px"></div> <div id="s217" class="sg" data-sgid="s217" data-fillcolor = "16777215" style="z-index:1;left:0px;top:96px;width:720px;height:26px"> <img id="s217_i" src="41C2B0B0.png" class="sg" style="z-index:1;left:0px;top:0px;width:720.48px;height:26.00px"></div> <div id="s218" class="sg" data-sgid="s218" data-fillcolor = "16777215" style="z-index:2;left:0px;top:100px;width:42px;height:18px"> <img id="s218_i" src="1699214F.png" class="sg" style="z-index:1;left:0px;top:0px;width:42.83px;height:18.95px"></div> <div id="s219" class="sg" data-sgid="s219" data-fillcolor = "16777215" style="z-index:3;left:45px;top:100px;width:674px;height:18px"> <img id="s219_i" src="F5358E43.png" class="sg" style="z-index:1;left:0px;top:0px;width:674.05px;height:18.95px"></div> <div id="s243" class="sg" data-sgid="s243" data-fillcolor = "16777215" style="z-index:1;left:0px;top:28px;width:720px;height:7px"> <img id="s243_i" src="AA45D626.png" class="sg" style="z-index:1;left:0px;top:0px;width:720.48px;height:7.56px"></div> <div id="s244" class="sg" data-sgid="s244" data-fillcolor = "16777215" style="z-index:2;left:0px;top:0px;width:720px;height:25px"> <img id="s244_i" src="A73D7E45.png" class="sg" style="z-index:1;left:0px;top:0px;width:720.48px;height:25.29px"></div> <div id="s245" class="sg" data-sgid="s245" data-fillcolor = "16777215" style="z-index:3;left:0px;top:23px;width:720px;height:8px"> <img id="s245_i" src="9CC4243.png" class="sg" style="z-index:1;left:0px;top:0px;width:720.48px;height:8.16px"></div> <div id="s246" class="sg" data-sgid="s246" data-fillcolor = "16777215" style="z-index:4;left:425px;top:27px;width:294px;height:8px"> <img id="s246_i" src="40495839.png" class="sg" style="z-index:1;left:0px;top:0px;width:294.70px;height:8.04px"></div> <div id="s247" class="sg" data-sgid="s247" data-fillcolor = "16777215" style="z-index:5;left:425px;top:34px;width:294px;height:15px"> <img id="s247_i" src="81A9B6F0.png" class="sg" style="z-index:1;left:0px;top:0px;width:294.70px;height:15.13px"></div> <div id="s248" class="sg" data-sgid="s248" data-pointer-events = "none" style="z-index:6;left:425px;top:38px;width:242px;height:3px"> <img id="s248_i" src="238D3004.png" class="sg" style="z-index:1;left:0px;top:0px;width:242.04px;height:3.12px"></div> <div id="s249" class="sg" data-sgid="s249" data-pointer-events = "none" style="z-index:7;left:579px;top:45px;width:126px;height:3px"> <img id="s249_i" src="F36FFEB.png" class="sg" style="z-index:1;left:0px;top:0px;width:126.88px;height:3.85px"></div> <div id="s250" class="sg" data-sgid="s250" data-fillcolor = "16777215" style="z-index:8;left:714px;top:0px;width:5px;height:49px"> <img id="s250_i" src="8AEC82E9.png" class="sg" style="z-index:1;left:0px;top:0px;width:5.41px;height:49.65px"></div> <div id="s251" class="sg" data-sgid="s251" data-fillcolor = "16777215" style="z-index:9;left:711px;top:0px;width:3px;height:49px"> <img id="s251_i" src="208CD3A8.png" class="sg" style="z-index:1;left:0px;top:0px;width:3.12px;height:49.65px"></div> <div id="s252" class="sg" data-sgid="s252" data-fillcolor = "16777215" style="z-index:10;left:709px;top:0px;width:1px;height:49px"> <img id="s252_i" src="890DC690.png" class="sg" style="z-index:1;left:0px;top:0px;width:1.56px;height:49.65px"></div> <div id="s253" class="sg" data-sgid="s253" data-fillcolor = "16777215" style="z-index:11;left:705px;top:0px;width:3px;height:49px"> <img id="s253_i" src="BA37F0BB.png" class="sg" style="z-index:1;left:0px;top:0px;width:3.00px;height:49.65px"></div> <div id="s254" class="sg" data-sgid="s254" data-fillcolor = "16777215" style="z-index:12;left:701px;top:0px;width:5px;height:47px"> <img id="s254_i" src="8085A798.png" class="sg" style="z-index:1;left:0px;top:0px;width:5.27px;height:47.02px"></div> <div id="s255" class="sg" data-sgid="s255" data-fillcolor = "16777215" style="z-index:13;left:697px;top:0px;width:1px;height:47px"> <img id="s255_i" src="25DB63F8.png" class="sg" style="z-index:1;left:0px;top:0px;width:1.44px;height:47.02px"></div> <div id="s298" class="sg" data-sgid="s298" data-fillcolor = "16777215" style="z-index:1;left:38px;top:467px;width:389px;height:73px"> <img id="s298_i" src="23D3402C.png" class="sg" style="z-index:1;left:0px;top:0px;width:389.80px;height:73.53px"></div> <div id="s299" class="sg" data-sgid="s299" data-fillcolor = "0" style="z-index:2;left:37px;top:466px;width:291px;height:74px"> <img id="s299_i" src="EF820D70.png" class="sg" style="z-index:1;left:0px;top:0px;width:291.47px;height:74.35px"></div> <div id="s300" class="sg" data-sgid="s300" data-pointer-events = "none" style="z-index:3;left:0px;top:455px;width:268px;height:85px"> <img id="s300_i" src="EECC858F.png" class="sg" style="z-index:1;left:0px;top:0px;width:268.56px;height:85.96px"></div> <div id="s301" class="sg" data-sgid="s301" data-pointer-events = "none" style="z-index:4;left:-1px;top:454px;width:269px;height:87px"> <img id="s301_i" src="DBD25D2D.png" class="sg" style="z-index:1;left:0px;top:0px;width:269.86px;height:86.32px"></div> <div id="s329" class="sg" data-sgid="Rectangle 3" data-fillcolor = "16777215" style="z-index:1;left:0px;top:119px;width:720px;height:90px"> <img id="s329_i" src="C3125175.png" class="sg" style="z-index:1;left:0px;top:0px;width:720.48px;height:90.78px"></div> <div id="s330" class="sg" data-sgid="Rectangle 4" data-fillcolor = "16777215" style="z-index:2;left:0px;top:125px;width:102px;height:78px"> <img id="s330_i" src="28927E52.png" class="sg" style="z-index:1;left:0px;top:0px;width:102.78px;height:78.77px"></div> <div id="s331" class="sg" data-sgid="Rectangle 5" data-fillcolor = "16777215" style="z-index:3;left:107px;top:125px;width:612px;height:78px"> <img id="s331_i" src="1BDBB10.png" class="sg" style="z-index:1;left:0px;top:0px;width:612.52px;height:78.77px"></div> <div id="s34" class="sg sgTitle" data-sgid="Title 1" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:35px;top:54px;width:648px;height:90px"> <div id="s34_t" class="sg" style="z-index:2;left:0px;top:0px;width:649px;height:123px" ><div><p para="{al:2}"><span data-textwidth="196.56" class="sg nowrap f3" style="z-index:2;color:#4F271C;font-size:300%; left:226.56px;top:31.37px;">Contents</span></p></div></div></div> <div id="s35" class="sg" data-sgid="Content Placeholder 4" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:44px;top:207px;width:318px;height:242px"> <div id="s35_t" class="sg" style="z-index:2;left:-2px;top:-6px;width:322px;height:249px" ><div><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="120.00" class="sg nowrap f4" style="z-index:2;color:#0BD0D9;font-size:106%; left:10.59px;top:6.67px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:10.68px;">Acknowledgment</span></span></p><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="92.76" class="sg nowrap f4" style="z-index:4;color:#0BD0D9;font-size:106%; left:10.59px;top:35.47px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:11.04px;">Introduction</span></span></p><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="212.64" class="sg nowrap f4" style="z-index:6;color:#0BD0D9;font-size:106%; left:10.59px;top:64.27px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:11.04px;">Brief Introduction about numbers</span></span></p><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="174.36" class="sg nowrap f4" style="z-index:8;color:#0BD0D9;font-size:106%; left:10.59px;top:93.07px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:11.04px;">History of Number System</span></span></p><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="241.92" class="sg nowrap f4" style="z-index:10;color:#0BD0D9;font-size:106%; left:10.59px;top:121.87px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:11.04px;">Number System according to different </span></span><span data-textwidth="66.24" class="sg nowrap f5" style="z-index:12;font-size:111%; left:32.07px;top:147.43px;">civilizations</span></p><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="126.48" class="sg nowrap f4" style="z-index:13;color:#0BD0D9;font-size:106%; left:10.59px;top:174.67px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:11.04px;">Types of Numbers</span></span></p></div></div></div> <div id="s36" class="sg" data-sgid="Content Placeholder 5" data-valign = "top" data-pointer-events = "none" style="z-index:3;left:364px;top:201px;width:318px;height:248px"> <div id="s36_t" class="sg" style="z-index:2;left:-2px;top:-6px;width:322px;height:254px" ><div><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="245.16" class="sg nowrap f4" style="z-index:2;color:#0BD0D9;font-size:106%; left:10.74px;top:6.78px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:11.04px;">Decimal Expansion Of Number System</span></span></p><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="225.72" class="sg nowrap f4" style="z-index:4;color:#0BD0D9;font-size:106%; left:10.74px;top:35.58px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:11.04px;">Scientists related to Number System</span></span></p><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="157.44" class="sg nowrap f4" style="z-index:6;color:#0BD0D9;font-size:106%; left:10.74px;top:64.38px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:14.88px;">What is a number line?</span></span></p><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="275.40" class="sg nowrap f4" style="z-index:8;color:#0BD0D9;font-size:106%; left:10.74px;top:93.18px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:10.92px;">What is the difference between numeral and </span></span><span data-textwidth="49.56" class="sg nowrap f5" style="z-index:10;font-size:111%; left:32.22px;top:118.74px;">number?</span></p><p para="{bl:{ch:61591},spbf:0.20}"><span data-textwidth="124.92" class="sg nowrap f4" style="z-index:11;color:#0BD0D9;font-size:106%; left:10.74px;top:145.98px;"><span class="f5" style="font-size:105%; color:#000000;margin-left:10.92px;">Word Alternatives</span></span></p></div></div></div> <div id="s37" class="sg" data-sgid="Picture 5" style="z-index:4;left:33px;top:38px;width:176px;height:159px"> <img id="s37_i" src="A1036926.png" class="sg" style="z-index:1;left:0px;top:0px;width:176.32px;height:159.97px"></div> <div id="s54" class="sg sgTitle" data-sgid="Title 1" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:44px;top:44px;width:588px;height:90px"> <div id="s54_t" class="sg" style="z-index:2;left:0px;top:0px;width:588px;height:116px" ><div><p para="{al:2}"><span data-textwidth="452.76" class="sg nowrap f6" style="z-index:2;color:#4E5B6F;font-size:267%; left:67.94px;top:34.74px;">A<span class="f6" style="font-size:79%; ">CKNOWLEDGEMENT</span></span></p></div></div></div> <div id="s55" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:35px;top:156px;width:666px;height:293px"> <div id="s55_t" class="sg" style="z-index:2;left:0px;top:-3px;width:677px;height:297px" ><div><p para="{spbf:6.00}"><span data-textwidth="610.80" class="sg nowrap f2" style="z-index:2;color:#003300;font-size:133%; left:29.52px;top:9.90px;">We would like to thank Teema madam for giving us an </span><span data-textwidth="566.64" class="sg nowrap f2" style="z-index:3;color:#003300;font-size:133%; left:29.52px;top:38.70px;">opportunity to express ourselves via mathematical </span><span data-textwidth="584.76" class="sg nowrap f2" style="z-index:4;color:#003300;font-size:133%; left:29.52px;top:67.50px;">projects. We are also thankful to Bharti madam, our </span><span data-textwidth="633.36" class="sg nowrap f2" style="z-index:5;color:#003300;font-size:133%; left:29.52px;top:96.30px;">computer teacher for letting us use the school computers </span><span data-textwidth="586.44" class="sg nowrap f2" style="z-index:6;color:#003300;font-size:133%; left:29.52px;top:125.10px;">for presentation and providing us with an e-mail ID. </span><span data-textwidth="540.96" class="sg nowrap f2" style="z-index:9;color:#003300;font-size:133%; left:29.40px;top:153.90px;">Also, we thank our friends for their ideas and co-</span><span data-textwidth="611.88" class="sg nowrap f2" style="z-index:11;color:#003300;font-size:133%; left:29.52px;top:182.70px;">operation they provided to us. We are grateful to all of </span><span data-textwidth="64.08" class="sg nowrap f2" style="z-index:12;color:#003300;font-size:133%; left:29.52px;top:211.50px;">them.</span></p></div></div></div> <div id="s56" class="sg" data-sgid="AutoShape 5" data-pointer-events = "none" style="z-index:3;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s57" class="sg" data-sgid="AutoShape 7" data-pointer-events = "none" style="z-index:4;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s58" class="sg" data-sgid="Picture 11" style="z-index:5;left:16px;top:0px;width:98px;height:158px"> <img id="s58_i" src="8377CF2B.png" class="sg" style="z-index:1;left:0px;top:0px;width:98.22px;height:158.30px"></div> <div id="s75" class="sg" data-sgid="Picture 10" style="z-index:1;left:530px;top:0px;width:189px;height:151px"> <img id="s75_i" src="A5A07C81.png" class="sg" style="z-index:1;left:0px;top:0px;width:189.04px;height:151.45px"></div> <div id="s76" class="sg sgTitle" data-sgid="Rectangle 2" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:106px;top:38px;width:591px;height:90px"> <div id="s76_t" class="sg" style="z-index:2;left:0px;top:-4px;width:591px;height:118px" ><div><p para="{al:2}"><span data-textwidth="287.64" class="sg nowrap f6" style="z-index:2;color:#E75C01;font-size:300%; left:151.98px;top:22.99px;"><span style="text-shadow:1.48px 1.48px 2.10px rgba(192,192,192,1.00)">Introduction</span></span></p></div></div></div> <div id="s77" class="sg" data-sgid="Rectangle 3" data-shape="pict" data-pointer-events = "none" data-valign = "top" style="z-index:3;left:89px;top:151px;width:613px;height:308px"> <div id="s77_t" class="sg" style="z-index:2;left:0px;top:-9px;width:620px;height:324px" ><div ><p para="{spbf:5.00,spaf:5.00}"><span data-textwidth="536.04" class="sg nowrap f7" style="z-index:2;color:#8A7007;font-size:133%; left:36.76px;top:9.88px;">A number system defines a set of values used to </span><span data-textwidth="568.92" class="sg nowrap f7" style="z-index:3;color:#8A7007;font-size:133%; left:36.88px;top:32.92px;">represent a quantity. We talk about the number of </span><span data-textwidth="561.96" class="sg nowrap f7" style="z-index:4;color:#8A7007;font-size:133%; left:36.88px;top:55.96px;">people attending school, number of modules taken </span><span data-textwidth="179.76" class="sg nowrap f7" style="z-index:5;color:#8A7007;font-size:133%; left:36.88px;top:79.00px;">per student etc. </span></p><p para="{spbf:5.00,spaf:5.00}"><span data-textwidth="542.76" class="sg nowrap f7" style="z-index:6;color:#8A7007;font-size:133%; left:36.88px;top:112.00px;">Quantifying items and values in relation to each </span><span data-textwidth="478.32" class="sg nowrap f7" style="z-index:7;color:#8A7007;font-size:133%; left:36.88px;top:135.04px;">other is helpful for us to make sense of our </span><span data-textwidth="154.92" class="sg nowrap f7" style="z-index:8;color:#8A7007;font-size:133%; left:36.88px;top:158.08px;">environment. </span></p><p para="{spbf:5.00,spaf:5.00}"><span data-textwidth="483.36" class="sg nowrap f7" style="z-index:9;color:#8A7007;font-size:133%; left:36.88px;top:191.20px;">The study of numbers is not only related to </span><span data-textwidth="501.12" class="sg nowrap f7" style="z-index:10;color:#8A7007;font-size:133%; left:36.88px;top:214.24px;">computers. We apply numbers everyday, and </span><span data-textwidth="492.72" class="sg nowrap f7" style="z-index:11;color:#8A7007;font-size:133%; left:36.88px;top:237.28px;">knowing how numbers work, will give us an </span><span data-textwidth="529.08" class="sg nowrap f7" style="z-index:12;color:#8A7007;font-size:133%; left:36.88px;top:260.32px;">insight of how computers manipulate and store </span><span data-textwidth="111.00" class="sg nowrap f7" style="z-index:13;color:#8A7007;font-size:133%; left:36.88px;top:283.36px;">numbers. </span></p></div></div></div> <div id="s78" class="sg" data-sgid="Picture 4" style="z-index:4;left:111px;top:33px;width:114px;height:113px"> <img id="s78_i" src="459359726.gif" class="sg" style="z-index:1;left:0px;top:0px;width:114.42px;height:113.32px"></div> <div id="s79" class="sg" data-sgid="AutoShape 6" data-pointer-events = "none" style="z-index:5;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s80" class="sg" data-sgid="AutoShape 8" data-pointer-events = "none" style="z-index:6;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s81" class="sg" data-sgid="AutoShape 8 (2)" data-pointer-events = "none" style="z-index:7;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s97" class="sg sgTitle" data-sgid="Title 3" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:50px;top:27px;width:570px;height:90px"> <img class="sg" src="966886E5.png" id="s97_tt" style="left:0.00px;top:-14.53px;width:570.54px;height:105.42px"></div> <div id="s98" class="sg" data-sgid="Content Placeholder 4" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:21px;top:128px;width:572px;height:265px"> <div id="s98_t" class="sg" style="z-index:2;left:0px;top:-3px;width:588px;height:289px" ><div><p para="{spbf:6.00}"><span data-textwidth="474.48" class="sg nowrap f8" style="z-index:2;font-size:133%; left:29.65px;top:9.53px;">A <span class="f9" style="font-weight:bold; ">number</span> is a mathematical object used in </span><span data-textwidth="502.92" class="sg nowrap f8" style="z-index:5;color:#E36406;font-size:133%; left:29.65px;top:38.33px;">counting<span style="color:#000000;"> and measuring. It is used in counting </span></span><span data-textwidth="482.16" class="sg nowrap f8" style="z-index:7;font-size:133%; left:29.65px;top:67.13px;">and measuring. Numerals are often used for </span><span data-textwidth="542.04" class="sg nowrap f8" style="z-index:8;font-size:133%; left:29.65px;top:95.93px;">labels, for ordering serial numbers, and for codes </span><span data-textwidth="485.28" class="sg nowrap f8" style="z-index:9;font-size:133%; left:29.65px;top:124.73px;">like ISBNs. In mathematics, the definition of </span><span data-textwidth="491.04" class="sg nowrap f8" style="z-index:10;font-size:133%; left:29.65px;top:153.53px;">number has been extended over the years to </span><span data-textwidth="542.16" class="sg nowrap f8" style="z-index:11;font-size:133%; left:29.65px;top:182.33px;">include such numbers as zero, negative numbers<span class="f8" style="font-size:117%; ">, </span></span><span data-textwidth="461.16" class="sg nowrap f8" style="z-index:13;font-size:133%; left:29.65px;top:215.93px;">rational numbers, irrational numbers, and </span><span data-textwidth="198.48" class="sg nowrap f8" style="z-index:14;font-size:133%; left:29.65px;top:244.73px;">complex numbers<span class="f8" style="font-size:108%; ">.</span></span></p></div></div></div> <div id="s99" class="sg" data-sgid="AutoShape 6" data-pointer-events = "none" style="z-index:3;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s100" class="sg" data-sgid="AutoShape 12" data-pointer-events = "none" style="z-index:4;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s101" class="sg" data-sgid="Picture 14" style="z-index:5;left:286px;top:370px;width:197px;height:144px"> <img id="s101_i" src="1217A0B9.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:197.31px;height:144.74px"></div> <div id="s113" class="sg" data-sgid="Picture 8" data-shape="pict" style="z-index:1;left:0px;top:16px;width:140px;height:154px"> <img id="s113_i" src="98CC8CFC.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:141px;height:155px"></div> <div id="s114" class="sg sgTitle" data-sgid="Title 3" data-shape="pict" data-pointer-events = "none" data-valign = "bottom" style="z-index:2;left:72px;top:50px;width:570px;height:90px"> <img class="sg" src="res/201409/35c9CF/DD87CCE7.png" id="s114_tt" style="left:0.00px;top:-14.53px;width:570.68px;height:105.31px"></div> <div id="s115" class="sg" data-sgid="Group 6" style="z-index:3;left:449px;top:376px;width:160px;height:147px"> <img id="s115_i" src="6190E250.png" class="sg" style="z-index:1;left:0px;top:0px;width:160.25px;height:147.51px"></div> <div id="s116" class="sg" data-sgid="Content Placeholder 2" data-shape="pict" data-pointer-events = "none" data-valign = "top" style="z-index:4;left:0px;top:128px;width:600px;height:351px"> <div id="s116_t" class="sg" style="z-index:2;left:0px;top:0px;width:600px;height:352px" ><div ><p para="{spbf:6.00}"></p><p para="{spbf:6.00}"><span data-textwidth="550.80" class="sg nowrap f8" style="z-index:2;font-size:133%; left:29.53px;top:41.10px;">Numbers were probably first used many thousands </span><span data-textwidth="552.36" class="sg nowrap f8" style="z-index:3;font-size:133%; left:29.53px;top:69.90px;">of years ago in commerce, and initially only whole </span><span data-textwidth="488.52" class="sg nowrap f8" style="z-index:4;font-size:133%; left:29.53px;top:98.70px;">numbers and perhaps rational numbers were </span><span data-textwidth="555.72" class="sg nowrap f8" style="z-index:5;font-size:133%; left:29.53px;top:127.50px;">needed. But already in Babylonian times, practical </span><span data-textwidth="509.52" class="sg nowrap f8" style="z-index:6;font-size:133%; left:29.53px;top:156.30px;">problems of geometry began to require square </span><span data-textwidth="70.44" class="sg nowrap f8" style="z-index:7;font-size:133%; left:29.53px;top:185.10px;">roots. </span></p><p para="{spbf:6.00}"><span data-textwidth="476.04" class="sg nowrap f8" style="z-index:8;font-size:133%; left:29.53px;top:219.90px;">Certain procedures which take one or more </span><span data-textwidth="550.08" class="sg nowrap f8" style="z-index:9;font-size:133%; left:29.53px;top:248.70px;">numbers as input and produce a number as output </span><span data-textwidth="343.68" class="sg nowrap f8" style="z-index:10;font-size:133%; left:29.53px;top:277.50px;">are called <span class="f10" style="font-style:italic; color:#E36406;">numerical operation.</span></span></p></div></div></div> <div id="Erijc3" class="sg" data-sgid="Line Erijc3" data-shape="line" style="z-index:5;left:360px;top:390px;width:120px;height:40px"> <canvas xmlns="canvas" id="Erijc3_i" class="sg" width="130" height="50" data-path="|F|S0,0,0,1|W1|Y0,1|M0,40|L120,0" style="z-index:1;left:-5px;top:-5px" data-offset = "5"></canvas></div> <div id="s117" class="sg sgTitle" data-sgid="Title 1" data-shape="pict" data-pointer-events = "none" data-valign = "bottom" style="z-index:1;left:33px;top:173px;width:648px;height:90px"> <div id="s117_t" class="sg" style="z-index:2;left:0px;top:-97px;width:649px;height:188px" ><div ><p para="{al:2}"><span data-textwidth="514.20" class="sg nowrap f6" style="z-index:2;color:#4F271C;font-size:267%; left:73.70px;top:20.38px;">The History Of Number </span><span data-textwidth="150.24" class="sg nowrap f6" style="z-index:3;color:#4F271C;font-size:267%; left:249.74px;top:77.98px;">System</span></p></div></div></div> <div id="s118" class="sg" data-sgid="Subtitle 2" data-shape="pict" data-pointer-events = "none" data-valign = "top" style="z-index:2;left:196px;top:241px;width:457px;height:265px"> <div id="s118_t" class="sg" style="z-index:2;left:0px;top:-3px;width:457px;height:269px" ><div style="cursor: auto;"><p id="s0NkcU" para="{al:4,spbf:0.20}"><span data-textwidth="420.60" class="sg nowrap f11" style="z-index: 2; font-size: 133%; left: 30px; top: 10px;">The<span style="margin-left:12.84px;">number</span><span style="margin-left:12.96px;">system</span><span style="margin-left:12.96px;">with</span><span style="margin-left:12.96px;">which</span><span style="margin-left:12.96px;">we</span><span style="margin-left:12.96px;">are</span></span><span data-textwidth="420.60" class="sg nowrap f11" style="z-index: 9; font-size: 133%; left: 30px; top: 39px;">most<span style="margin-left:17.40px;">familiar</span><span style="margin-left:17.52px;">is</span><span style="margin-left:17.52px;">the</span><span style="color:#0070C0;margin-left:17.40px;">decimal</span><span style="color:#0070C0;margin-left:17.52px;">(base</span><span style="color:#0070C0;">-10)</span></span><span data-textwidth="420.60" class="sg nowrap f11" style="z-index: 18; font-size: 133%; left: 30px; top: 68px;">system<span style="margin-left:9.84px;">,</span><span style="margin-left:9.96px;">but</span><span style="margin-left:9.96px;">over</span><span style="margin-left:9.96px;">time</span><span style="margin-left:9.96px;">our</span><span style="margin-left:9.96px;">ancestor</span><span style="margin-left:9.96px;">have</span></span><span data-textwidth="420.60" class="sg nowrap f11" style="z-index: 26; font-size: 133%; left: 30px; top: 97px;">experimented<span style="margin-left:35.28px;">with</span><span style="margin-left:17.76px;">a</span><span style="margin-left:17.76px;">wide</span><span style="margin-left:17.76px;">range</span><span style="margin-left:17.76px;">of</span></span><span data-textwidth="420.60" class="sg nowrap f11" style="z-index: 32; font-size: 133%; left: 30px; top: 125px;">alternatives,<span style="margin-left:10.92px;">including</span><span style="color:#0070C0;margin-left:11.16px;">duo</span><span style="color:#0070C0;">-decimal</span><span style="color:#0070C0;margin-left:11.04px;">(base</span><span style="color:#0070C0;">-</span></span><span data-textwidth="420.60" class="sg nowrap f11" style="z-index: 39; color: rgb(0, 112, 192); font-size: 133%; left: 30px; top: 154px;">12)<span style="color:#000000;">,</span><span style="margin-left:10.92px;">vigesimal</span><span style="margin-left:10.92px;">(base</span>-20)<span style="color:#000000;">,</span><span style="color:#000000;margin-left:10.92px;">and</span><span style="margin-left:10.92px;">sexagesimal</span></span><span data-textwidth="112.80" class="sg nowrap f11" style="z-index: 50; color: rgb(0, 112, 192); font-size: 133%; left: 30px; top: 183px;">(base-60)<span style="color:#000000;">…</span></span></p></div></div></div> <div id="s119" class="sg" data-sgid="Picture 5" style="z-index:3;left:16px;top:230px;width:198px;height:198px"> <img id="s119_i" src="2ADD4BF3.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:198.87px;height:198.69px"></div> <div id="45F8C1" class="sg" data-sgid="Line 45F8C1" data-shape="line" style="z-index:4;left:530px;top:430px;width:160px;height:10px"> <canvas xmlns="canvas" id="45F8C1_i" class="sg" width="170" height="20" data-path="|F|S0,0,0,1|W1|M0,10|L160,0" style="z-index:1;left:-5px;top:-5px" data-offset = "5"></canvas></div> <div id="6dVQSh" class="sg" data-sgid="Line 6dVQSh" data-shape="line" style="z-index:5;left:100px;top:400px;width:110px;height:50px"> <canvas xmlns="canvas" id="6dVQSh_i" class="sg" width="120" height="60" data-path="|F|S0,0,0,1|W1|Y0,1|M0,50|L110,0" style="z-index:1;left:-5px;top:-5px" data-offset = "5"></canvas></div> <div id="s125" class="sg sgTitle" data-sgid="Title 1" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:44px;top:21px;width:648px;height:90px"> <div id="s125_t" class="sg" style="z-index:2;left:0px;top:0px;width:649px;height:120px" ><div><p para="{al:2}"><span data-textwidth="468.84" class="sg nowrap f6" style="z-index:2;color:#424456;font-size:267%; left:90.38px;top:38.82px;">The Ancient Egyptians</span></p></div></div></div> <div id="s126" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:21px;top:123px;width:648px;height:301px"> <div id="s126_t" class="sg" style="z-index:2;left:0px;top:-3px;width:655px;height:305px" ><div><p para="{spbf:0.20}"><span data-textwidth="554.28" class="sg nowrap f12" style="z-index:2;font-size:133%; left:29.65px;top:10.32px;">The Ancient Egyptians experimented with <span style="color:#006600;">duo-decimal </span></span><span data-textwidth="611.40" class="sg nowrap f12" style="z-index:6;color:#006600;font-size:133%; left:29.65px;top:39.12px;">(base-12)<span style="color:#000000;"> system in which they counted finger-joints instead of </span></span><span data-textwidth="604.56" class="sg nowrap f12" style="z-index:12;font-size:133%; left:29.65px;top:67.92px;">finger . Each of our finger has three joints. In addition to their </span><span data-textwidth="582.36" class="sg nowrap f12" style="z-index:13;font-size:133%; left:29.65px;top:96.72px;">base-twelve system, the Egyptians also experimented with a </span><span data-textwidth="610.80" class="sg nowrap f12" style="z-index:16;font-size:133%; left:29.65px;top:125.52px;">sort –of-base-ten system. In this system , the number 1 through </span><span data-textwidth="585.48" class="sg nowrap f12" style="z-index:23;font-size:133%; left:29.65px;top:154.32px;">9 were drawn using the appropriate number of vertical lines.</span></p></div></div></div> <div id="s127" class="sg" data-sgid="AutoShape 7" data-pointer-events = "none" style="z-index:3;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s128" class="sg" data-sgid="AutoShape 9" data-pointer-events = "none" style="z-index:4;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s129" class="sg" data-sgid="Picture 11" style="z-index:5;left:359px;top:303px;width:165px;height:224px"> <img id="s129_i" src="926CF7B9.png" class="sg" style="z-index:1;left:0px;top:0px;width:165.01px;height:224.24px"></div> <div id="s130" class="sg" data-sgid="Straight Arrow Connector 8" data-pointer-events = "none" style="z-index:6;left:281px;top:376px;width:159px;height:98px"> <img id="s130_i" src="289729DE.png" class="sg" style="z-index:1;left:0px;top:0px;width:159.89px;height:98.45px"></div> <div id="s131" class="sg" data-sgid="Straight Arrow Connector 9" data-pointer-events = "none" style="z-index:7;left:297px;top:360px;width:126px;height:81px"> <img id="s131_i" src="836B7F57.png" class="sg" style="z-index:1;left:0px;top:0px;width:126.40px;height:81.55px"></div> <div id="s132" class="sg" data-sgid="Straight Arrow Connector 10" data-pointer-events = "none" style="z-index:8;left:264px;top:337px;width:160px;height:98px"> <img id="s132_i" src="A3D9070D.png" class="sg" style="z-index:1;left:0px;top:0px;width:160.00px;height:98.34px"></div> <div id="s133" class="sg" data-sgid="TextBox 12" data-valign = "top" style="z-index:9;left:21px;top:415px;width:259px;height:66px"> <div id="s133_t" class="sg" style="z-index:2;left:-2px;top:-1px;width:261px;height:73px" ><div><p><span data-textwidth="239.52" class="sg nowrap f13" style="z-index:2;font-style:italic; font-size:89%; left:9.97px;top:8.10px;">A human hand palm was the way </span><span data-textwidth="170.04" class="sg nowrap f13" style="z-index:3;font-style:italic; font-size:89%; left:10.21px;top:27.30px;">of counting used by the </span><span data-textwidth="86.28" class="sg nowrap f13" style="z-index:4;font-style:italic; font-size:89%; left:10.21px;top:46.50px;">Egyptians…</span></p></div></div></div> <div id="s134" class="sg" data-sgid="Picture 11" style="z-index:1;left:33px;top:21px;width:113px;height:132px"> <img id="s134_i" src="A70663EF.png" class="sg" style="z-index:1;left:0px;top:0px;width:113.34px;height:132.87px"></div> <div id="s135" class="sg sgTitle" data-sgid="Title 1" data-valign = "bottom" data-pointer-events = "none" style="z-index:2;left:89px;top:44px;width:648px;height:90px"> <div id="s135_t" class="sg" style="z-index:2;left:0px;top:0px;width:649px;height:120px" ><div><p para="{al:2}"><span data-textwidth="515.52" class="sg nowrap f6" style="z-index:2;color:#424456;font-size:267%; left:67.12px;top:38.71px;">The Ancient Babylonians</span></p></div></div></div> <div id="s136" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:3;left:35px;top:151px;width:645px;height:326px"> <div id="s136_t" class="sg" style="z-index:2;left:0px;top:-3px;width:655px;height:339px" ><div><p para="{spbf:0.20}"><span data-textwidth="593.88" class="sg nowrap f11" style="z-index:2;font-size:133%; left:29.52px;top:10.09px;">Babylonians, were famous for their astrological observations </span><span data-textwidth="607.32" class="sg nowrap f11" style="z-index:3;font-size:133%; left:29.52px;top:38.89px;">and calculations, and used a <span style="color:#006600;">sexagesimal (base-60)</span> numbering </span><span data-textwidth="606.24" class="sg nowrap f11" style="z-index:8;font-size:133%; left:29.52px;top:67.69px;">system. In addition to using base sixty, the babylonians also </span><span data-textwidth="543.00" class="sg nowrap f11" style="z-index:9;font-size:133%; left:29.52px;top:96.49px;">made use of six and ten as sub-bases. The babylonians </span><span data-textwidth="611.76" class="sg nowrap f11" style="z-index:12;font-size:133%; left:29.52px;top:125.29px;">sexagesimal system which first appeared around 1900 to 1800 </span><span data-textwidth="603.72" class="sg nowrap f11" style="z-index:13;font-size:133%; left:29.52px;top:154.09px;">BC, is also credited with being the first known place-value of </span><span data-textwidth="556.68" class="sg nowrap f11" style="z-index:16;font-size:133%; left:29.52px;top:182.89px;">a particular digit depends on both the digit itself and its </span><span data-textwidth="605.40" class="sg nowrap f11" style="z-index:17;font-size:133%; left:29.52px;top:211.69px;">position within the number . This as an extremely important </span><span data-textwidth="591.84" class="sg nowrap f11" style="z-index:18;font-size:133%; left:29.52px;top:240.49px;">development, because – prior to place-value system – people </span><span data-textwidth="580.08" class="sg nowrap f11" style="z-index:25;font-size:133%; left:29.52px;top:269.29px;">were obliged to use different symbol to represent different </span><span data-textwidth="159.72" class="sg nowrap f11" style="z-index:26;font-size:133%; left:29.52px;top:298.09px;">power of a base.</span></p></div></div></div> <div id="s137" class="sg" data-sgid="AutoShape 5" data-pointer-events = "none" style="z-index:4;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s138" class="sg" data-sgid="AutoShape 7" data-pointer-events = "none" style="z-index:5;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s139" class="sg" data-sgid="AutoShape 9" data-pointer-events = "none" style="z-index:6;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s140" class="sg" data-sgid="Picture 11" style="z-index:1;left:504px;top:375px;width:193px;height:147px"> <img id="s140_i" src="8F311386.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:193.60px;height:147.85px"></div> <div id="s141" class="sg sgTitle" data-sgid="Title 1" data-valign = "bottom" data-pointer-events = "none" style="z-index:2;left:27px;top:72px;width:648px;height:90px"> <div id="s141_t" class="sg" style="z-index:2;left:0px;top:-39px;width:659px;height:159px" ><div><p para="{al:2}"><span data-textwidth="600.00" class="sg nowrap f6" style="z-index:2;color:#424456;font-size:267%; left:30.74px;top:20.35px;">Aztecs, Eskimos, And Indian </span><span data-textwidth="227.40" class="sg nowrap f6" style="z-index:3;color:#424456;font-size:267%; left:211.10px;top:77.95px;">Merchants.</span></p></div></div></div> <div id="s142" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:3;left:33px;top:151px;width:648px;height:341px"> <div id="s142_t" class="sg" style="z-index:2;left:0px;top:-3px;width:661px;height:345px" ><div><p para="{spbf:0.20}"><span data-textwidth="597.36" class="sg nowrap f11" style="z-index:2;font-size:133%; left:29.54px;top:10.09px;">Other cultures such as the Aztecs, developed <span style="color:#006600;">vigesimal (base-</span></span><span data-textwidth="600.48" class="sg nowrap f11" style="z-index:5;color:#006600;font-size:133%; left:29.54px;top:38.89px;">20)<span style="color:#000000;"> systems because they counted using both finger and toes. </span></span><span data-textwidth="606.60" class="sg nowrap f11" style="z-index:7;font-size:133%; left:29.54px;top:67.69px;">The Ainu of Japan and the Eskimos of Greenland are two of </span><span data-textwidth="578.88" class="sg nowrap f11" style="z-index:8;font-size:133%; left:29.54px;top:96.49px;">the peoples who make use of vigesimal systems of present </span><span data-textwidth="595.08" class="sg nowrap f11" style="z-index:9;font-size:133%; left:29.54px;top:125.29px;">day . Another system that is relatively easy to understand is </span><span data-textwidth="589.92" class="sg nowrap f11" style="z-index:10;font-size:133%; left:29.54px;top:154.09px;">quinary (base-5), which uses five digit : 0, 1, 2, 3, and 4. The </span><span data-textwidth="575.28" class="sg nowrap f11" style="z-index:13;font-size:133%; left:29.54px;top:182.89px;">system is particularly interesting , in that a quinary finger-</span><span data-textwidth="617.28" class="sg nowrap f11" style="z-index:15;font-size:133%; left:29.54px;top:211.69px;">counting scheme is still in use today by Indian merchant near </span><span data-textwidth="564.84" class="sg nowrap f11" style="z-index:16;font-size:133%; left:29.54px;top:240.49px;">Bombay . This allow them to perform calculations on one </span><span data-textwidth="506.88" class="sg nowrap f11" style="z-index:17;font-size:133%; left:29.54px;top:269.29px;">hand while serving their customers with the other.</span></p></div></div></div> <div id="s143" class="sg" data-sgid="AutoShape 7" data-pointer-events = "none" style="z-index:4;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s144" class="sg" data-sgid="AutoShape 9" data-pointer-events = "none" style="z-index:5;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s145" class="sg" data-sgid="TextBox 7" data-valign = "top" style="z-index:6;left:297px;top:482px;width:208px;height:46px"> <div id="s145_t" class="sg" style="z-index:2;left:-2px;top:-1px;width:211px;height:53px" ><div><p><span data-textwidth="165.48" class="sg nowrap f13" style="z-index:2;font-style:italic; font-size:89%; left:10.11px;top:8.04px;">Aztecs were the ethnic </span><span data-textwidth="113.88" class="sg nowrap f13" style="z-index:3;font-style:italic; font-size:89%; left:10.35px;top:27.24px;">group of Mexico</span></p></div></div></div> <div id="s146" class="sg" data-sgid="Picture 4" style="z-index:1;left:162px;top:196px;width:420px;height:315px"> <img id="s146_i" src="1BF0422E.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:420.74px;height:315.73px"></div> <div id="s147" class="sg sgTitle" data-sgid="Title 3" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:106px;top:100px;width:591px;height:90px"> <div id="s147_t" class="sg" style="z-index:2;left:-22px;top:-27px;width:613px;height:163px" ><div><p><span data-textwidth="549.36" class="sg nowrap f6" style="z-index:2;color:#545E70;font-size:267%; left:30.04px;top:21.21px;"><span style="text-shadow:1.48px 1.48px 2.10px rgba(192,192,192,1.00)">Number System according </span></span><span data-textwidth="533.28" class="sg nowrap f6" style="z-index:3;color:#545E70;font-size:267%; left:30.04px;top:78.81px;"><span style="text-shadow:1.48px 1.48px 2.10px rgba(192,192,192,1.00)">to different civilizations… </span></span></p></div></div></div> <div id="s152" class="sg sgTitle" data-sgid="Title 1" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:-62px;top:61px;width:588px;height:90px"> <div id="s152_t" class="sg" style="z-index:2;left:0px;top:-21px;width:588px;height:132px" ><div><p para="{al:2}"><span data-textwidth="410.40" class="sg nowrap f6" style="z-index:2;color:#4E5B6F;font-size:222%; left:99.10px;top:16.70px;">T<span class="f6" style="font-size:80%; ">HE</span> D<span class="f6" style="font-size:80%; ">ECIMAL</span> N<span class="f6" style="font-size:80%; ">UMBER</span></span><span data-textwidth="140.16" class="sg nowrap f6" style="z-index:8;color:#4E5B6F;font-size:222%; left:224.38px;top:64.70px;">S<span class="f6" style="font-size:80%; ">YSTEM</span></span></p></div></div></div> <div id="s153" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:27px;top:168px;width:588px;height:257px"> <div id="s153_t" class="sg" style="z-index:2;left:0px;top:-3px;width:589px;height:339px" ><div><p para="{spbf:6.00}"><span data-textwidth="535.56" class="sg nowrap f2" style="z-index:2;font-size:133%; left:29.54px;top:9.78px;">The number system we use on day-to-day basis </span><span data-textwidth="499.44" class="sg nowrap f2" style="z-index:7;font-size:133%; left:29.54px;top:38.58px;">in the decimal system , which is based on <span class="f2" style="font-weight:bold; color:#006600;">ten</span></span><span data-textwidth="545.76" class="sg nowrap f2" style="z-index:9;font-size:133%; left:29.54px;top:67.38px;">digits: zero through nine. As the decimal system </span><span data-textwidth="534.84" class="sg nowrap f2" style="z-index:10;font-size:133%; left:29.54px;top:96.18px;">is based on ten digits, it is said to be base -10 or </span><span data-textwidth="511.56" class="sg nowrap f2" style="z-index:13;font-size:133%; left:29.54px;top:124.98px;">radix-10. Outside of specialized requirement </span><span data-textwidth="529.20" class="sg nowrap f2" style="z-index:16;font-size:133%; left:29.54px;top:153.78px;">such as computing , base-10 numbering system </span><span data-textwidth="478.56" class="sg nowrap f2" style="z-index:19;font-size:133%; left:29.54px;top:182.58px;">have been adopted almost universally. The </span><span data-textwidth="494.88" class="sg nowrap f2" style="z-index:20;font-size:133%; left:29.54px;top:211.38px;">decimal system with which we are fated is a </span><span data-textwidth="537.96" class="sg nowrap f2" style="z-index:21;font-size:133%; left:29.54px;top:240.18px;">place-value system, which means that the value </span><span data-textwidth="513.84" class="sg nowrap f2" style="z-index:24;font-size:133%; left:29.54px;top:268.98px;">of a particular digit depends both on the itself </span><span data-textwidth="423.84" class="sg nowrap f2" style="z-index:25;font-size:133%; left:29.54px;top:297.78px;">and on its position within the number.</span></p></div></div></div> <div id="s154" class="sg" data-sgid="Picture 5" style="z-index:3;left:466px;top:21px;width:215px;height:132px"> <img id="s154_i" src="6DAD028A.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:215.30px;height:132.87px"></div> <div id="s155" class="sg sgTitle" data-sgid="Rectangle 2" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:35px;top:21px;width:588px;height:90px"> <div id="s155_t" class="sg" style="z-index:2;left:0px;top:0px;width:588px;height:111px" ><div><p para="{al:2}"><span data-textwidth="536.88" class="sg nowrap f6" style="z-index:2;color:#4E5B6F;font-weight:bold; font-size:222%; left:25.94px;top:43.10px;">MAYAN NUMBER SYSTEM</span></p></div></div></div> <div id="s156" class="sg" data-sgid="Rectangle 3" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:35px;top:125px;width:588px;height:384px"> <div id="s156_t" class="sg" style="z-index:2;left:0px;top:-6px;width:600px;height:391px" ><div><p para="{spbf:15.00}"><span data-textwidth="502.80" class="sg nowrap f2" style="z-index:2;font-size:133%; left:29.52px;top:9.76px;">This system is unique to our current decimal </span><span data-textwidth="544.56" class="sg nowrap f2" style="z-index:3;font-size:133%; left:29.52px;top:35.68px;">system, as our current decimal system uses base </span><span data-textwidth="521.76" class="sg nowrap f2" style="z-index:4;font-size:133%; left:29.52px;top:61.60px;">-10 whereas, the Mayan Number System uses </span><span data-textwidth="97.68" class="sg nowrap f2" style="z-index:6;color:#006600;font-size:133%; left:29.52px;top:87.52px;">base- 20.</span></p><p para="{spbf:15.00}"><span data-textwidth="511.68" class="sg nowrap f2" style="z-index:9;font-size:133%; left:29.52px;top:128.44px;">The Mayan system used a combination of two </span><span data-textwidth="556.80" class="sg nowrap f2" style="z-index:10;font-size:133%; left:29.52px;top:154.36px;">symbols. A dot (.) was used to represent the units </span><span data-textwidth="509.88" class="sg nowrap f2" style="z-index:11;font-size:133%; left:29.52px;top:180.28px;">and a dash (-) was used to represent five. The </span><span data-textwidth="475.92" class="sg nowrap f2" style="z-index:14;font-size:133%; left:29.52px;top:206.20px;">Mayan's wrote their numbers vertically as </span><span data-textwidth="439.80" class="sg nowrap f2" style="z-index:15;font-size:133%; left:29.52px;top:232.12px;">opposed to horizontally with the lowest </span><span data-textwidth="326.04" class="sg nowrap f2" style="z-index:16;font-size:133%; left:29.52px;top:258.04px;">denomination on the bottom. </span></p></div></div></div> <div id="s157" class="sg" data-sgid="Picture 3" style="z-index:3;left:168px;top:398px;width:309px;height:48px"> <img id="s157_i" src="F920A920.png" class="sg" style="z-index:1;left:0px;top:0px;width:309.69px;height:48.58px"></div> <div id="s158" class="sg" data-sgid="TextBox 4" data-valign = "top" style="z-index:4;left:325px;top:443px;width:343px;height:51px"> <div id="s158_t" class="sg" style="z-index:2;left:-3px;top:-1px;width:347px;height:60px" ><div><p><span data-textwidth="303.00" class="sg nowrap f13" style="z-index:2;font-style:italic; left:11.55px;top:8.62px;">Several numbers according to Mayan </span><span data-textwidth="128.88" class="sg nowrap f13" style="z-index:3;font-style:italic; left:11.55px;top:30.22px;">Number System</span></p></div></div></div> <div id="s159" class="sg sgTitle" data-sgid="Rectangle 2" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:72px;top:50px;width:590px;height:90px"> <div id="s159_t" class="sg" style="z-index:2;left:0px;top:-43px;width:590px;height:159px" ><div><p para="{al:2}"><span data-textwidth="437.64" class="sg nowrap f6" style="z-index:2;color:#495A74;font-size:267%; left:82.97px;top:20.45px;">BINARY NUMBER </span><span data-textwidth="203.28" class="sg nowrap f6" style="z-index:3;color:#495A74;font-size:267%; left:194.09px;top:78.05px;">SYSTEM</span></p></div></div></div> <div id="s160" class="sg" data-sgid="Rectangle 3" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:21px;top:168px;width:664px;height:325px"> <div id="s160_t" class="sg" style="z-index:2;left:0px;top:-5px;width:666px;height:331px" ><div><p para="{spbf:15.00}"><span data-textwidth="567.60" class="sg nowrap f14" style="z-index:2;font-size:133%; left:29.65px;top:10.98px;">The <span class="f14" style="font-weight:bold; ">binary numeral system</span>, or <span class="f14" style="font-weight:bold; ">base-2 number system</span>, represents </span><span data-textwidth="622.32" class="sg nowrap f14" style="z-index:9;font-size:133%; left:29.65px;top:36.90px;">numeric values using two symbols, 0 and 1. More specifically, the usual </span><span data-textwidth="605.52" class="sg nowrap f14" style="z-index:10;font-size:133%; left:29.65px;top:62.82px;">base-2 system is a positional notation with a radix of 2. Owing to its </span><span data-textwidth="589.68" class="sg nowrap f14" style="z-index:13;font-size:133%; left:29.65px;top:88.74px;">straight forward implementation in digital electronic circuitry using </span><span data-textwidth="537.00" class="sg nowrap f14" style="z-index:14;font-size:133%; left:29.65px;top:114.66px;">logic gates, the binary system is used internally by all modern </span><span data-textwidth="573.48" class="sg nowrap f14" style="z-index:15;font-size:133%; left:29.65px;top:140.58px;">computers. Counting in binary is similar to counting in any other </span><span data-textwidth="562.44" class="sg nowrap f14" style="z-index:16;font-size:133%; left:29.65px;top:166.50px;">number system. Beginning with a single digit, counting proceeds </span><span data-textwidth="596.40" class="sg nowrap f14" style="z-index:17;font-size:133%; left:29.65px;top:192.42px;">through each symbol, in increasing order. Decimal counting uses the </span><span data-textwidth="569.40" class="sg nowrap f14" style="z-index:18;font-size:133%; left:29.65px;top:218.34px;">symbols <span class="f14" style="font-weight:bold; ">0</span> through <span class="f14" style="font-weight:bold; ">9</span>, while binary only uses the symbols <span class="f14" style="font-weight:bold; ">0</span> and <span class="f14" style="font-weight:bold; ">1</span>.</span></p></div></div></div> <div id="s161" class="sg sgTitle" data-sgid="Rectangle 2" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:27px;top:66px;width:642px;height:78px"> <div id="s161_t" class="sg" style="z-index:2;left:0px;top:-33px;width:642px;height:133px" ><div><p para="{al:2}"><span data-textwidth="568.20" class="sg nowrap f6" style="z-index:2;color:#4E5B6F;font-size:222%; left:42.37px;top:16.71px;">FRACTIONS AND ANCIENT </span><span data-textwidth="140.28" class="sg nowrap f6" style="z-index:3;color:#4E5B6F;font-size:222%; left:251.41px;top:64.71px;">EGYPT</span></p></div></div></div> <div id="s162" class="sg" data-sgid="Rectangle 3" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:38px;top:179px;width:642px;height:296px"> <div id="s162_t" class="sg" style="z-index:2;left:0px;top:-5px;width:654px;height:302px" ><div><p para="{spbf:15.00}"><span data-textwidth="590.04" class="sg nowrap f14" style="z-index:2;font-size:133%; left:49.58px;top:10.98px;">Ancient Egyptians had an understanding of fractions, however they </span><span data-textwidth="493.32" class="sg nowrap f14" style="z-index:3;font-size:133%; left:51.26px;top:36.90px;">did not write simple fractions as 3/5 or 4/9 because of </span><span data-textwidth="570.36" class="sg nowrap f14" style="z-index:4;font-size:133%; left:51.26px;top:62.82px;">restrictions in notation. The Egyptian scribe wrote fractions with </span><span data-textwidth="580.32" class="sg nowrap f14" style="z-index:5;font-size:133%; left:51.26px;top:88.74px;">the numerator of 1. They used the hieroglyph “an open mouth" </span><span data-textwidth="581.76" class="sg nowrap f14" style="z-index:6;font-size:133%; left:51.26px;top:114.66px;">above the number to indicate its reciprocal. The number 5, written </span><span data-textwidth="514.92" class="sg nowrap f14" style="z-index:7;font-size:133%; left:51.26px;top:140.58px;">, as a fraction 1/5 would be written as . . . There are some </span><span data-textwidth="545.76" class="sg nowrap f14" style="z-index:8;font-size:133%; left:51.26px;top:166.50px;">exceptions. There was a special hieroglyph for 2/3, , and some </span><span data-textwidth="584.04" class="sg nowrap f14" style="z-index:9;font-size:133%; left:51.26px;top:192.42px;">evidence that 3/4 also had a special hieroglyph. All other fractions </span><span data-textwidth="554.52" class="sg nowrap f14" style="z-index:10;font-size:133%; left:51.26px;top:218.34px;">were written as the sum of unit fractions. For example 3/8 was </span><span data-textwidth="201.36" class="sg nowrap f14" style="z-index:11;font-size:133%; left:51.26px;top:244.26px;">written as 1/4 + 1/8. </span></p></div></div></div> <div id="s163" class="sg" data-sgid="Picture 3" style="z-index:3;left:488px;top:263px;width:23px;height:17px"> <img id="s163_i" src="7D800ABC.png" class="sg" style="z-index:1;left:0px;top:0px;width:23.37px;height:17.76px"></div> <div id="s164" class="sg" data-sgid="Picture 4" style="z-index:4;left:426px;top:308px;width:29px;height:30px"> <img id="s164_i" src="76E1DEFC.png" class="sg" style="z-index:1;left:0px;top:0px;width:29.04px;height:30.45px"></div> <div id="s165" class="sg sgTitle" data-sgid="Title 3" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:55px;top:100px;width:588px;height:90px"> <div id="s165_t" class="sg" style="z-index:2;left:0px;top:-9px;width:589px;height:139px" ><div><p para="{al:2}"><span data-textwidth="501.36" class="sg nowrap f6" style="z-index:2;color:#4F271C;font-size:367%; left:44.30px;top:27.37px;">Types of numbers</span></p></div></div></div> <div id="s166" class="sg" data-sgid="Picture 20" style="z-index:2;left:162px;top:207px;width:355px;height:294px"> <img id="s166_i" src="5C04A86D.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:355.13px;height:294.63px"></div> <div id="s171" class="sg" data-sgid="Picture 3" style="z-index:1;left:0px;top:0px;width:720px;height:540px"> <img id="s171_i" src="D1B6CB0.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:720.48px;height:540.48px"></div> <div id="s177" class="sg" data-sgid="Subtitle 2" data-valign = "top" data-pointer-events = "none" style="z-index:1;left:27px;top:95px;width:648px;height:272px"> <div id="s177_t" class="sg" style="z-index:2;left:0px;top:-4px;width:650px;height:277px" ><div><p para="{spbf:4.00}"><span data-textwidth="582.60" class="sg nowrap f14" style="z-index:2;font-size:156%; left:36.86px;top:12.38px;">The real numbers include all of the measuring numbers . </span><span data-textwidth="592.80" class="sg nowrap f14" style="z-index:3;font-size:156%; left:36.86px;top:45.98px;">Real numbers are usually written using decimal numerals , </span><span data-textwidth="596.16" class="sg nowrap f14" style="z-index:4;font-size:156%; left:36.86px;top:79.58px;">in which a decimal point is placed to the right of the digit </span><span data-textwidth="210.00" class="sg nowrap f14" style="z-index:5;font-size:156%; left:36.86px;top:113.18px;">with place value one.</span></p><p para="{bl:{ch:61565},spbf:4.00}"><span data-textwidth="593.64" class="sg nowrap f15" style="z-index:6;color:#7FD13B;font-size:106%; left:16.70px;top:146.86px;"><span class="f14" style="font-size:147%; color:#000000;position:relative;bottom:-4px;margin-left:10.08px;">It includes all types of numbers such as Integers, Whole </span></span><span data-textwidth="571.92" class="sg nowrap f14" style="z-index:8;font-size:156%; left:36.86px;top:184.46px;">numbers, Natural numbers, Rational number, Irrational </span><span data-textwidth="194.52" class="sg nowrap f14" style="z-index:9;font-size:156%; left:36.86px;top:218.06px;">numbers and etc…</span><span data-textwidth="274.80" class="sg nowrap f14" style="z-index:10;font-size:156%; left:302.90px;top:218.06px;">Let us see them in detail…</span></p></div></div></div> <div id="s178" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:50px;top:21px;width:648px;height:90px"> <img class="sg" src="AD9277D3.png" id="s178_tt" style="left:0.00px;top:0.00px;width:648.50px;height:90.89px"></div> <div id="s179" class="sg" data-sgid="Picture 5" style="z-index:3;left:239px;top:342px;width:431px;height:182px"> <img id="s179_i" src="5951AE2.png" class="sg" style="z-index:1;left:0px;top:0px;width:431.05px;height:182.04px"></div> <div id="s194" class="sg" data-sgid="Subtitle 2" data-valign = "top" data-pointer-events = "none" style="z-index:1;left:235px;top:151px;width:450px;height:221px"> <div id="s194_t" class="sg" style="z-index:2;left:0px;top:-4px;width:462px;height:328px" ><div><p para="{spbf:4.00}"><span data-textwidth="406.32" class="sg nowrap f14" style="z-index:2;font-size:156%; left:37.07px;top:12.40px;">A rational number is a number that can </span><span data-textwidth="344.40" class="sg nowrap f14" style="z-index:3;font-size:156%; left:37.07px;top:46.00px;">be expressed as a fraction with an </span><span data-textwidth="347.28" class="sg nowrap f14" style="z-index:4;font-size:156%; left:37.07px;top:79.60px;">integer numerator and a non-zero </span><span data-textwidth="350.76" class="sg nowrap f14" style="z-index:7;font-size:156%; left:37.07px;top:113.20px;">natural number denominator. The </span><span data-textwidth="408.24" class="sg nowrap f14" style="z-index:8;font-size:156%; left:37.07px;top:146.80px;">symbol of the rational number is ‘Q’. It </span><span data-textwidth="364.44" class="sg nowrap f14" style="z-index:9;font-size:156%; left:37.07px;top:180.40px;">includes all types of numbers other </span><span data-textwidth="397.80" class="sg nowrap f14" style="z-index:10;font-size:156%; left:37.07px;top:214.00px;">than irrational numbers, i.e. it includes </span><span data-textwidth="324.48" class="sg nowrap f14" style="z-index:11;font-size:156%; left:37.07px;top:247.60px;">integers, whole number, natural </span><span data-textwidth="150.60" class="sg nowrap f14" style="z-index:12;font-size:156%; left:37.07px;top:281.20px;">numbers etc…</span></p></div></div></div> <div id="s195" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:33px;top:50px;width:648px;height:90px"> <img class="sg" src="5AE51D7C.png" id="s195_tt" style="left:0.00px;top:0.00px;width:648.50px;height:90.78px"></div> <div id="s196" class="sg" data-sgid="Picture 5" style="z-index:3;left:21px;top:145px;width:237px;height:303px"> <img id="s196_i" src="72597FFC.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:237.00px;height:303.30px"></div> <div id="s197" class="sg" data-sgid="Content Placeholder 1" data-valign = "top" data-pointer-events = "none" style="z-index:1;left:44px;top:151px;width:648px;height:357px"> <div id="s197_t" class="sg" style="z-index:2;left:0px;top:-3px;width:653px;height:361px" ><div><p para="{spbf:4.00}"><span data-textwidth="574.68" class="sg nowrap f16" style="z-index:2;font-size:133%; left:36.86px;top:10.41px;">This is a type of a rational number. Fractions are </span><span data-textwidth="560.76" class="sg nowrap f16" style="z-index:3;font-size:133%; left:36.86px;top:39.21px;">written as two numbers, the numerator and the </span><span data-textwidth="561.84" class="sg nowrap f16" style="z-index:4;font-size:133%; left:36.86px;top:68.01px;">denominator ,with a dividing bar between them.</span></p><p para="{bl:{ch:61565},spbf:4.00}"><span data-textwidth="571.56" class="sg nowrap f15" style="z-index:5;color:#7FD13B;font-size:89%; left:16.70px;top:96.89px;"><span class="f16" style="font-size:150%; color:#000000;position:relative;bottom:-4px;margin-left:11.52px;">In the fraction m/n ‘m’ represents equal parts, </span></span><span data-textwidth="551.64" class="sg nowrap f16" style="z-index:7;font-size:133%; left:36.86px;top:129.69px;">where ‘n’ equal parts of that size make up one </span><span data-textwidth="75.96" class="sg nowrap f16" style="z-index:8;font-size:133%; left:36.86px;top:158.49px;">whole.</span></p><p para="{bl:{ch:61565},spbf:4.00}"><span data-textwidth="593.04" class="sg nowrap f15" style="z-index:9;color:#7FD13B;font-size:89%; left:16.70px;top:187.25px;"><span class="f16" style="font-size:150%; color:#000000;position:relative;bottom:-4px;margin-left:11.52px;">If the absolute value of m is greater than n ,then </span></span><span data-textwidth="572.16" class="sg nowrap f16" style="z-index:11;font-size:133%; left:36.86px;top:220.05px;">the absolute value of the fraction is greater than </span><span data-textwidth="602.52" class="sg nowrap f16" style="z-index:12;font-size:133%; left:36.86px;top:248.85px;">1.Fractions can be greater than ,less than ,or equal </span><span data-textwidth="555.12" class="sg nowrap f16" style="z-index:13;font-size:133%; left:36.86px;top:277.65px;">to1 and can also be positive ,negative , or zero.</span></p></div></div></div> <div id="s198" class="sg sgTitle" data-sgid="Title 2" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:35px;top:21px;width:648px;height:90px"> <img class="sg" src="4D48FED7.png" id="s198_tt" style="left:0.00px;top:0.00px;width:648.53px;height:90.78px"></div> <div id="s199" class="sg" data-sgid="Picture 5" style="z-index:3;left:21px;top:0px;width:194px;height:147px"> <img id="s199_i" src="8D212E9D.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:194.31px;height:147.74px"></div> <div id="s200" class="sg" data-sgid="Subtitle 2" data-valign = "top" data-pointer-events = "none" style="z-index:1;left:33px;top:140px;width:648px;height:357px"> <div id="s200_t" class="sg" style="z-index:2;left:0px;top:-3px;width:649px;height:361px" ><div><p para="{spbf:4.00}"><span data-textwidth="588.24" class="sg nowrap f16" style="z-index:2;font-size:133%; left:36.86px;top:10.40px;">If a real number cannot be written as a fraction of </span><span data-textwidth="530.28" class="sg nowrap f16" style="z-index:3;font-size:133%; left:36.86px;top:39.20px;">two integers, i.e. it is not rational, it is called </span><span data-textwidth="588.36" class="sg nowrap f16" style="z-index:4;font-size:133%; left:36.86px;top:68.00px;">irrational numbers . A decimal that can be written </span><span data-textwidth="544.56" class="sg nowrap f16" style="z-index:5;font-size:133%; left:36.86px;top:96.80px;">as a fraction either ends(terminates)or forever </span><span data-textwidth="567.00" class="sg nowrap f16" style="z-index:6;font-size:133%; left:36.86px;top:125.60px;">repeats about which we will see in detail further.</span></p><p para="{spbf:4.00}"></p><p para="{spbf:4.00}"><span data-textwidth="540.00" class="sg nowrap f16" style="z-index:7;font-size:133%; left:36.86px;top:191.24px;">Real number pi (π) is an example of irrational.</span></p><p para="{spbf:4.00}"><span data-textwidth="553.32" class="sg nowrap f16" style="z-index:8;font-size:133%; left:36.86px;top:224.00px;">π=3.14159365358979……the number neither </span><span data-textwidth="566.16" class="sg nowrap f16" style="z-index:9;font-size:133%; left:36.86px;top:252.80px;">start repeating themselves or come in a specific </span><span data-textwidth="91.92" class="sg nowrap f16" style="z-index:10;font-size:133%; left:36.86px;top:281.60px;">pattern.</span></p></div></div></div> <div id="s201" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:35px;top:21px;width:648px;height:90px"> <img class="sg" src="850A90A0.png" id="s201_tt" style="left:0.00px;top:0.00px;width:648.53px;height:90.78px"></div> <div id="s202" class="sg" data-sgid="Picture 5" style="z-index:3;left:21px;top:21px;width:102px;height:98px"> <img id="s202_i" src="8BCC3A8E.png" class="sg" style="z-index:1;left:0px;top:0px;width:102.07px;height:98.94px"></div> <div id="s203" class="sg" data-sgid="Picture 7" style="z-index:1;left:21px;top:0px;width:264px;height:190px"> <img id="s203_i" src="F53CFC5A.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:264.82px;height:190.42px"></div> <div id="s204" class="sg" data-sgid="Subtitle 2" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:35px;top:116px;width:648px;height:357px"> <div id="s204_t" class="sg" style="z-index:2;left:0px;top:-4px;width:649px;height:385px" ><div><p para="{bl:{ch:61565},spbf:4.00}"><span data-textwidth="546.24" class="sg nowrap f15" style="z-index:2;color:#7FD13B;left:16.68px;top:6.62px;"><span class="f16" style="font-size:150%; color:#000000;position:relative;bottom:-5px;margin-left:10.44px;">Integers are the number which includes </span></span><span data-textwidth="410.76" class="sg nowrap f16" style="z-index:4;font-size:150%; left:36.84px;top:44.02px;">positive and negative numbers.</span></p><p para="{bl:{ch:61565},spbf:4.00}"><span data-textwidth="608.04" class="sg nowrap f15" style="z-index:5;color:#7FD13B;left:16.68px;top:75.38px;"><span class="f16" style="font-size:150%; color:#000000;position:relative;bottom:-5px;margin-left:10.44px;">Negative numbers are numbers that are less </span></span><span data-textwidth="530.88" class="sg nowrap f16" style="z-index:7;font-size:150%; left:36.84px;top:112.78px;">than zero. They are opposite of positive </span><span data-textwidth="536.76" class="sg nowrap f16" style="z-index:8;font-size:150%; left:36.84px;top:145.18px;">numbers . Negative numbers are usually </span><span data-textwidth="539.64" class="sg nowrap f16" style="z-index:9;font-size:150%; left:36.84px;top:177.58px;">written with a negative sign(also called a </span><span data-textwidth="567.96" class="sg nowrap f16" style="z-index:10;font-size:150%; left:36.84px;top:209.98px;">minus sign)in front of the number they are </span><span data-textwidth="497.64" class="sg nowrap f16" style="z-index:11;font-size:150%; left:36.84px;top:242.38px;">opposite of .When the set of negative </span><span data-textwidth="506.64" class="sg nowrap f16" style="z-index:12;font-size:150%; left:36.84px;top:274.78px;">numbers is combined with the natural </span><span data-textwidth="596.16" class="sg nowrap f16" style="z-index:13;font-size:150%; left:36.84px;top:307.18px;">numbers zero, the result is the set of integer </span><span data-textwidth="337.68" class="sg nowrap f16" style="z-index:14;font-size:150%; left:36.84px;top:339.58px;">numbers , also called ‘Z’. </span></p></div></div></div> <div id="s205" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:3;left:35px;top:21px;width:648px;height:90px"> <img class="sg" src="9F4676C3.png" id="s205_tt" style="left:0.00px;top:0.00px;width:648.53px;height:90.78px"></div> <div id="s206" class="sg" data-sgid="Subtitle 2" data-valign = "top" data-pointer-events = "none" style="z-index:1;left:35px;top:116px;width:648px;height:357px"> <div id="s206_t" class="sg" style="z-index:2;left:0px;top:-4px;width:649px;height:361px" ><div><p para="{bl:{ch:61565},spbf:4.00}"><span data-textwidth="583.68" class="sg nowrap f15" style="z-index:2;color:#7FD13B;left:16.68px;top:6.62px;"><span class="f16" style="font-size:150%; color:#000000;position:relative;bottom:-5px;margin-left:10.44px;">The most familiar numbers are the natural </span></span><span data-textwidth="557.88" class="sg nowrap f16" style="z-index:4;font-size:150%; left:36.84px;top:44.02px;">numbers or counting numbers: One, Two, </span><span data-textwidth="248.28" class="sg nowrap f16" style="z-index:5;font-size:150%; left:36.84px;top:76.42px;">Three and so on….</span></p><p para="{bl:{ch:61565},spbf:4.00}"><span data-textwidth="518.16" class="sg nowrap f15" style="z-index:6;color:#7FD13B;left:16.68px;top:107.78px;"><span class="f16" style="font-size:150%; color:#000000;position:relative;bottom:-5px;margin-left:10.44px;">Traditionally, the sequence of natural </span></span><span data-textwidth="569.88" class="sg nowrap f16" style="z-index:8;font-size:150%; left:36.84px;top:145.18px;">numbers started with 1.However in the 19<span class="f16" style="font-size:67%; position:relative;bottom:10px;">th</span></span><span data-textwidth="585.84" class="sg nowrap f16" style="z-index:10;font-size:150%; left:36.84px;top:177.58px;">century, mathematicians started including 0 </span><span data-textwidth="390.60" class="sg nowrap f16" style="z-index:11;font-size:150%; left:36.84px;top:209.98px;">in the set of natural numbers.</span></p><p para="{bl:{ch:61565},spbf:4.00}"><span data-textwidth="583.08" class="sg nowrap f15" style="z-index:12;color:#7FD13B;left:16.68px;top:241.34px;"><span class="f16" style="font-size:150%; color:#000000;position:relative;bottom:-5px;margin-left:10.44px;">The mathematical symbol for the set of all </span></span><span data-textwidth="308.64" class="sg nowrap f16" style="z-index:14;font-size:150%; left:36.84px;top:278.74px;">natural numbers is ‘N’. </span></p></div></div></div> <div id="s207" class="sg sgTitle" data-sgid="Title 3" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:106px;top:33px;width:591px;height:90px"> <img class="sg" src="3CB6D42D.png" id="s207_tt" style="left:0.00px;top:0.00px;width:591.19px;height:90.78px"></div> <div id="s208" class="sg" data-sgid="Rectangle 4" data-pointer-events = "none" style="z-index:3;left:162px;top:232px;width:15px;height:73px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s209" class="sg" data-sgid="Subtitle 2" data-valign = "top" data-pointer-events = "none" style="z-index:1;left:35px;top:116px;width:648px;height:357px"> <div id="s209_t" class="sg" style="z-index:2;left:0px;top:-3px;width:660px;height:360px" ><div><p para="{spbf:4.00}"><span data-textwidth="576.48" class="sg nowrap f11" style="z-index:2;font-size:133%; left:36.84px;top:9.96px;">Moving to a greater level of abstraction, the real numbers </span><span data-textwidth="531.84" class="sg nowrap f11" style="z-index:3;font-size:133%; left:36.84px;top:38.76px;">can be extended to the complex numbers. This set of </span><span data-textwidth="606.36" class="sg nowrap f11" style="z-index:4;font-size:133%; left:36.84px;top:67.56px;">number arose historically, from trying to find closed formulas </span><span data-textwidth="594.00" class="sg nowrap f11" style="z-index:5;font-size:133%; left:36.84px;top:96.36px;">for the roots of cubic and quadratic polynomials. This led to </span><span data-textwidth="596.16" class="sg nowrap f11" style="z-index:6;font-size:133%; left:36.84px;top:125.16px;">expressions involving the square roots of negative numbers, </span><span data-textwidth="602.52" class="sg nowrap f11" style="z-index:7;font-size:133%; left:36.84px;top:153.96px;">eventually to the definition of a new number: the square root </span><span data-textwidth="609.24" class="sg nowrap f11" style="z-index:8;font-size:133%; left:36.84px;top:182.76px;">of negative one denoted by “I”. The complex numbers consist </span><span data-textwidth="576.60" class="sg nowrap f11" style="z-index:9;font-size:133%; left:36.84px;top:211.56px;">of all numbers of the form (a+bi) ; Where a and b are real </span><span data-textwidth="95.64" class="sg nowrap f11" style="z-index:10;font-size:133%; left:36.84px;top:240.36px;">numbers. </span></p></div></div></div> <div id="s210" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:35px;top:21px;width:648px;height:90px"> <img class="sg" src="6DF0AC21.png" id="s210_tt" style="left:0.00px;top:0.00px;width:648.53px;height:90.78px"></div> <div id="s211" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:1;left:66px;top:10px;width:591px;height:90px"> <div id="s211_t" class="sg" style="z-index:2;left:0px;top:0px;width:591px;height:103px" ><div><p para="{al:2}"><span data-textwidth="258.00" class="sg nowrap f6" style="z-index:2;color:#D9253E;font-size:267%; left:167.07px;top:22.25px;">Other Types</span></p></div></div></div> <div id="s212" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:33px;top:190px;width:664px;height:259px"> <div id="s212_t" class="sg" style="z-index:2;left:-7px;top:-3px;width:671px;height:263px" ><div><p para="{spbf:7.00}"><span data-textwidth="603.96" class="sg nowrap f17" style="z-index:2;font-size:133%; left:14.89px;top:10.72px;">There are different kind of other numbers too. It includes </span></p><p para="{bl:{ch:118},spbf:7.00}"><span data-textwidth="240.00" class="sg nowrap f18" style="z-index:3;color:#60B5CC;font-size:78%; left:14.89px;top:46.48px;"><span class="f17" style="font-size:171%; color:#000000;margin-left:12.36px;">hyper</span><span class="f17" style="font-size:171%; color:#000000;">-real numbers,</span></span></p><p para="{bl:{ch:118},spbf:7.00}"><span data-textwidth="289.44" class="sg nowrap f18" style="z-index:7;color:#60B5CC;font-size:78%; left:14.89px;top:82.24px;"><span class="f17" style="font-size:171%; color:#000000;margin-left:12.36px;">hyper</span><span class="f17" style="font-size:171%; color:#000000;">-complex numbers,</span></span></p><p para="{bl:{ch:118},spbf:7.00}"><span data-textwidth="197.28" class="sg nowrap f18" style="z-index:11;color:#60B5CC;font-size:78%; left:14.89px;top:118.12px;"><span class="f17" style="font-size:171%; color:#000000;margin-left:12.36px;">p</span><span class="f17" style="font-size:171%; color:#000000;">-adic numbers,</span></span></p><p para="{bl:{ch:118},spbf:7.00}"><span data-textwidth="244.08" class="sg nowrap f18" style="z-index:15;color:#60B5CC;font-size:78%; left:14.89px;top:153.88px;"><span class="f17" style="font-size:171%; color:#000000;margin-left:12.36px;">surreal numbers etc.</span></span></p><p para="{spbf:7.00}"><span data-textwidth="571.08" class="sg nowrap f17" style="z-index:17;font-size:133%; left:46.33px;top:189.64px;">These numbers are rarely used in our day-to-day life. </span><span data-textwidth="533.76" class="sg nowrap f17" style="z-index:22;font-size:133%; left:40.09px;top:218.44px;">Therefore, we need not know about them in detail.</span></p></div></div></div> <div id="s226" class="sg sgTitle" data-sgid="Title 1" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:55px;top:50px;width:570px;height:90px"> <div id="s226_t" class="sg" style="z-index:2;left:0px;top:-29px;width:570px;height:142px" ><div><p para="{al:2}"><span data-textwidth="410.76" class="sg nowrap f6" style="z-index:2;color:#424456;font-size:239%; left:85.34px;top:18.65px;">Decimal Expansion of </span><span data-textwidth="169.08" class="sg nowrap f6" style="z-index:3;color:#424456;font-size:239%; left:200.90px;top:70.25px;">Numbers</span></p></div></div></div> <div id="s227" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:27px;top:151px;width:570px;height:295px"> <div id="s227_t" class="sg" style="z-index:2;left:-7px;top:-3px;width:579px;height:299px" ><div><p para="{spbf:6.00}"><span data-textwidth="528.60" class="sg nowrap f2" style="z-index:2;font-size:133%; left:36.24px;top:9.89px;">A decimal expansion of a number can be either, </span></p><p para="{bl:{ch:113},spbf:6.00}"><span data-textwidth="159.12" class="sg nowrap f18" style="z-index:3;color:#53548A;font-size:94%; left:14.88px;top:44.69px;"><span class="f2" style="font-size:141%; color:#000000;"> Terminating</span></span></p><p para="{bl:{ch:113},spbf:6.00}"><span data-textwidth="374.28" class="sg nowrap f18" style="z-index:5;color:#53548A;font-size:94%; left:14.88px;top:79.49px;"><span class="f2" style="font-size:141%; color:#000000;"> Non-terminating, non recurring</span></span></p><p para="{bl:{ch:113},spbf:6.00}"><span data-textwidth="331.32" class="sg nowrap f18" style="z-index:9;color:#53548A;font-size:94%; left:14.88px;top:114.29px;"><span class="f2" style="font-size:141%; color:#000000;"> Non terminating, recurring </span></span></p><p para="{spbf:6.00}"></p><p para="{spbf:6.00}"></p><p para="{spbf:6.00}"><span data-textwidth="345.60" class="sg nowrap f2" style="z-index:11;font-size:133%; left:14.88px;top:218.69px;">Let us see each of the following</span></p><p para="{spbf:6.00}"><span data-textwidth="96.00" class="sg nowrap f2" style="z-index:12;font-size:133%; left:21.48px;top:253.49px;">briefly…</span></p></div></div></div> <div id="s228" class="sg" data-sgid="AutoShape 5" data-pointer-events = "none" style="z-index:3;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s229" class="sg" data-sgid="Picture 7" style="z-index:4;left:398px;top:297px;width:225px;height:170px"> <img id="s229_i" src="C5FD8BE.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:225.75px;height:170.17px"></div> <div id="s230" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:1;left:35px;top:89px;width:648px;height:84px"> <div id="s230_t" class="sg" style="z-index:2;left:0px;top:-1px;width:649px;height:101px" ><div><p para="{al:2}"><span data-textwidth="425.16" class="sg nowrap f6" style="z-index:2;color:#575F6D;font-size:267%; left:112.20px;top:20.50px;">Terminating decimal</span></p></div></div></div> <div id="s231" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:35px;top:176px;width:648px;height:341px"> <div id="s231_t" class="sg" style="z-index:2;left:0px;top:-3px;width:649px;height:345px" ><div><p para="{spbf:3.00}"><span data-textwidth="575.64" class="sg nowrap f19" style="z-index:2;font-size:133%; left:36.36px;top:11.00px;">A decimal expansion in which the remainder becomes </span><span data-textwidth="311.04" class="sg nowrap f19" style="z-index:3;font-size:133%; left:36.84px;top:39.80px;">zero. For example, 54<span style="margin-left:42.60px;">9 = </span></span></p><p para="{spbf:3.00}"></p><p para="{spbf:3.00}"></p><p para="{spbf:3.00}"></p><p para="{spbf:3.00}"></p><p para="{spbf:3.00}"></p><p para="{spbf:3.00}"><span data-textwidth="596.76" class="sg nowrap f19" style="z-index:5;font-size:133%; left:36.84px;top:230.60px;">Terminating decimal is always a rational number. It can </span><span data-textwidth="242.76" class="sg nowrap f19" style="z-index:6;font-size:133%; left:36.60px;top:259.40px;">be written in p/q form.</span></p></div></div></div> <div id="s232" class="sg" data-sgid="Division 3" data-pointer-events = "none" style="z-index:3;left:305px;top:220px;width:23px;height:19px"> <img id="s232_i" src="4744E27A.png" class="sg" style="z-index:1;left:0px;top:0px;width:23.26px;height:19.54px"></div> <div id="s233" class="sg" data-sgid="Straight Connector 9" data-pointer-events = "none" style="z-index:4;left:224px;top:274px;width:63px;height:1px"> <img id="s233_i" src="D30BFEEB.png" class="sg" style="z-index:1;left:0px;top:0px;width:63.23px;height:1.00px"></div> <div id="s234" class="sg" data-sgid="Freeform 11" data-pointer-events = "none" style="z-index:5;left:213px;top:275px;width:18px;height:33px"> <img id="s234_i" src="B5401489.png" class="sg" style="z-index:1;left:0px;top:0px;width:18.58px;height:33.20px"></div> <div id="s235" class="sg" data-sgid="TextBox 12" data-valign = "top" style="z-index:6;left:230px;top:274px;width:57px;height:42px"> <div id="s235_t" class="sg" style="z-index:2;left:-9px;top:-4px;width:66px;height:59px" ><div><p><span data-textwidth="31.20" class="sg nowrap f17" style="z-index:2;font-size:156%; left:17.39px;top:13.33px;">54</span></p></div></div></div> <div id="s236" class="sg" data-sgid="TextBox 15" data-valign = "top" style="z-index:7;left:201px;top:274px;width:23px;height:37px"> <div id="s236_t" class="sg" style="z-index:2;left:-7px;top:-3px;width:43px;height:51px" ><div><p><span data-textwidth="13.32" class="sg nowrap f17" style="z-index:2;font-size:133%; left:15.09px;top:10.67px;">9</span></p></div></div></div> <div id="s237" class="sg" data-sgid="TextBox 16" data-valign = "top" style="z-index:8;left:235px;top:235px;width:45px;height:42px"> <div id="s237_t" class="sg" style="z-index:2;left:-9px;top:-4px;width:55px;height:59px" ><div><p><span data-textwidth="15.60" class="sg nowrap f17" style="z-index:2;font-size:156%; left:17.50px;top:13.31px;">6</span></p></div></div></div> <div id="s238" class="sg" data-sgid="TextBox 17" data-valign = "top" style="z-index:9;left:230px;top:308px;width:57px;height:63px"> <div id="s238_t" class="sg" style="z-index:2;left:-9px;top:-4px;width:66px;height:68px" ><div><p><span data-textwidth="31.20" class="sg nowrap f17" style="z-index:2;font-size:156%; left:17.39px;top:13.37px;">54</span></p></div></div></div> <div id="s239" class="sg" data-sgid="Straight Connector 21" data-pointer-events = "none" style="z-index:10;left:229px;top:347px;width:57px;height:1px"> <img id="s239_i" src="110A52A2.png" class="sg" style="z-index:1;left:0px;top:0px;width:57.59px;height:1.00px"></div> <div id="s240" class="sg" data-sgid="TextBox 22" data-valign = "top" style="z-index:11;left:241px;top:348px;width:62px;height:37px"> <div id="s240_t" class="sg" style="z-index:2;left:-7px;top:-3px;width:69px;height:51px" ><div><p><span data-textwidth="13.32" class="sg nowrap f17" style="z-index:2;font-size:133%; left:15.13px;top:10.71px;">0</span></p></div></div></div> <div id="s241" class="sg" data-sgid="Straight Arrow Connector 24" data-pointer-events = "none" style="z-index:12;left:274px;top:324px;width:69px;height:46px"> <img id="s241_i" src="48915DD8.png" class="sg" style="z-index:1;left:0px;top:0px;width:69.32px;height:46.99px"></div> <div id="s242" class="sg" data-sgid="TextBox 27" data-valign = "top" style="z-index:13;left:342px;top:286px;width:248px;height:51px"> <div id="s242_t" class="sg" style="z-index:2;left:-3px;top:-1px;width:260px;height:60px" ><div><p><span data-textwidth="237.96" class="sg nowrap f13" style="z-index:2;font-style:italic; left:11.08px;top:8.49px;">As the remainder is zero, this </span><span data-textwidth="194.04" class="sg nowrap f13" style="z-index:3;font-style:italic; left:11.44px;top:30.09px;">is a terminating decimal </span></p></div></div></div> <div id="s266" class="sg" data-sgid="Picture 5" style="z-index:1;left:38px;top:123px;width:264px;height:60px"> <img id="s266_i" src="70609A3A.png" class="sg" style="z-index:1;left:0px;top:0px;width:264.70px;height:60.81px"></div> <div id="s267" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:33px;top:78px;width:570px;height:90px"> <div id="s267_t" class="sg" style="z-index:2;left:0px;top:-27px;width:571px;height:159px" ><div><p para="{al:2}"><span data-textwidth="443.28" class="sg nowrap f6" style="z-index:2;color:#5A6378;font-size:267%; left:70.22px;top:20.46px;">Non terminating non </span><span data-textwidth="185.16" class="sg nowrap f6" style="z-index:3;color:#5A6378;font-size:267%; left:193.34px;top:78.06px;">recurring</span></p></div></div></div> <div id="s268" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:3;left:38px;top:201px;width:658px;height:304px"> <div id="s268_t" class="sg" style="z-index:2;left:0px;top:-3px;width:659px;height:316px" ><div><p para="{spbf:3.00}"><span data-textwidth="576.48" class="sg nowrap f19" style="z-index:2;font-size:133%; left:36.86px;top:11.12px;">“Recurring” means “repeating”. In this form, when we </span><span data-textwidth="584.40" class="sg nowrap f19" style="z-index:3;font-size:133%; left:36.86px;top:39.92px;">divide a number by another, remainder never becomes </span><span data-textwidth="599.64" class="sg nowrap f19" style="z-index:4;font-size:133%; left:36.86px;top:68.72px;">zero, and also the number does not repeat themselves in </span><span data-textwidth="597.00" class="sg nowrap f19" style="z-index:5;font-size:133%; left:36.86px;top:97.52px;">any specific pattern. If a number is non terminating and </span><span data-textwidth="565.32" class="sg nowrap f19" style="z-index:6;font-size:133%; left:36.86px;top:126.32px;">non repeating, they are always classified as irrational </span><span data-textwidth="234.84" class="sg nowrap f19" style="z-index:7;font-size:133%; left:36.86px;top:155.12px;">number. For example,</span></p><p para="{spbf:3.00}"></p><p para="{spbf:3.00}"><span data-textwidth="584.04" class="sg nowrap f19" style="z-index:8;font-size:133%; left:36.86px;top:218.72px;">0.10100100010000100000100.... does have a pattern, </span><span data-textwidth="542.64" class="sg nowrap f19" style="z-index:9;font-size:133%; left:36.62px;top:247.52px;">but it is not a fixed-length recurring pattern, so the </span><span data-textwidth="225.36" class="sg nowrap f19" style="z-index:12;font-size:133%; left:36.86px;top:276.32px;">number is irrational. </span></p></div></div></div> <div id="s269" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:1;left:35px;top:89px;width:648px;height:84px"> <div id="s269_t" class="sg" style="z-index:2;left:0px;top:-1px;width:649px;height:101px" ><div><p para="{al:2}"><span data-textwidth="550.80" class="sg nowrap f6" style="z-index:2;color:#04617B;font-size:267%; left:49.32px;top:20.50px;">Non terminating, recurring</span></p></div></div></div> <div id="s270" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:44px;top:168px;width:619px;height:301px"> <div id="s270_t" class="sg" style="z-index:2;left:0px;top:-3px;width:625px;height:305px" ><div><p para="{spbf:3.00}"><span data-textwidth="554.28" class="sg nowrap f20" style="z-index:2;font-size:133%; left:36.86px;top:11.42px;">In this form, when a number is divided by the other, the </span><span data-textwidth="546.00" class="sg nowrap f20" style="z-index:3;font-size:133%; left:36.86px;top:40.22px;">remainder never becomes zero, instead the numbers of </span><span data-textwidth="573.96" class="sg nowrap f20" style="z-index:4;font-size:133%; left:36.86px;top:69.02px;">the quotient start repeating themselves. Such numbers are </span><span data-textwidth="427.08" class="sg nowrap f20" style="z-index:5;font-size:133%; left:36.86px;top:97.82px;">classified as rational numbers. For example,</span></p><p para="{spbf:3.00}"></p><p para="{spbf:3.00}"><span data-textwidth="254.40" class="sg nowrap f20" style="z-index:6;font-size:133%; left:34.70px;top:161.42px;">3.7250725072507250…</span></p><p para="{spbf:3.00}"><span data-textwidth="524.04" class="sg nowrap f20" style="z-index:7;font-size:133%; left:36.86px;top:193.22px;">In this example, “7250” have started repeating itself. </span><span data-textwidth="563.64" class="sg nowrap f20" style="z-index:8;font-size:133%; left:36.86px;top:222.02px;">Hence, it is a rational number. It can be expressed in p/q </span><span data-textwidth="53.64" class="sg nowrap f20" style="z-index:9;font-size:133%; left:36.86px;top:250.82px;">form.</span></p></div></div></div> <div id="s271" class="sg" data-sgid="Picture 5" style="z-index:1;left:522px;top:269px;width:197px;height:233px"> <img id="s271_i" src="FE9A61FE.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:197.31px;height:233.99px"></div> <div id="s272" class="sg sgTitle" data-sgid="Rectangle 2" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:50px;top:55px;width:588px;height:90px"> <div id="s272_t" class="sg" style="z-index:2;left:0px;top:-14px;width:589px;height:132px" ><div><p para="{al:2}"><span data-textwidth="447.36" class="sg nowrap f6" style="z-index:2;font-size:222%; left:75.99px;top:16.58px;">Mathematicians related to </span><span data-textwidth="276.24" class="sg nowrap f6" style="z-index:3;font-size:222%; left:156.63px;top:64.58px;">Number System</span></p></div></div></div> <div id="s273" class="sg" data-sgid="Rectangle 3" data-valign = "top" data-pointer-events = "none" style="z-index:3;left:21px;top:156px;width:588px;height:330px"> <div id="s273_t" class="sg" style="z-index:2;left:0px;top:-6px;width:589px;height:336px" ><div><p para="{spbf:5.75}"><span data-textwidth="108.96" class="sg nowrap f21" style="z-index:2;color:#006600;font-style:italic; font-size:178%; left:29.05px;top:13.87px;">Euclid <span class="f19" style="font-style:normal; ">:</span></span></p><p para="{spbf:5.75}"><span data-textwidth="458.52" class="sg nowrap f19" style="z-index:4;color:#1C1911;font-size:133%; left:28.45px;top:61.75px;">Euclid was an ancient mathematician from </span><span data-textwidth="536.88" class="sg nowrap f19" style="z-index:5;color:#1C1911;font-size:133%; left:29.17px;top:91.51px;">Alexandria, who is best known for his major work, </span><span data-textwidth="477.36" class="sg nowrap f19" style="z-index:6;color:#1C1911;font-size:133%; left:29.65px;top:120.31px;">Elements. He told about the division lemma, </span><span data-textwidth="212.52" class="sg nowrap f19" style="z-index:7;color:#1C1911;font-size:133%; left:29.65px;top:149.11px;">according to which, </span></p><p para="{spbf:5.75}"><span data-textwidth="497.76" class="sg nowrap f21" style="z-index:8;color:#1C1911;font-style:italic; font-size:133%; left:23.53px;top:183.91px;">A prime number that divides a product of two </span><span data-textwidth="464.28" class="sg nowrap f21" style="z-index:9;color:#1C1911;font-style:italic; font-size:133%; left:29.65px;top:212.71px;">integers must divide one of the two integer.</span></p></div></div></div> <div id="s274" class="sg" data-sgid="TextBox 4" data-valign = "top" style="z-index:4;left:258px;top:443px;width:265px;height:27px"> <div id="s274_t" class="sg" style="z-index:2;left:-2px;top:-1px;width:267px;height:34px" ><div><p><span data-textwidth="223.80" class="sg nowrap f13" style="z-index:2;font-style:italic; font-size:89%; left:10.34px;top:8.00px;">Euclid – The father of geometry</span></p></div></div></div> <div id="s275" class="sg" data-sgid="Picture 9" style="z-index:1;left:567px;top:397px;width:96px;height:120px"> <img id="s275_i" src="9625373F.png" class="sg" style="z-index:1;left:0px;top:0px;width:96.43px;height:120.15px"></div> <div id="s276" class="sg sgTitle" data-sgid="Rectangle 2" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:50px;top:66px;width:644px;height:83px"> <div id="s276_t" class="sg" style="z-index:2;left:0px;top:-18px;width:651px;height:132px" ><div><p para="{al:2}"><span data-textwidth="598.68" class="sg nowrap f6" style="z-index:2;font-size:222%; left:28.59px;top:16.61px;">Mathematicians related to Number </span><span data-textwidth="125.04" class="sg nowrap f6" style="z-index:3;font-size:222%; left:260.43px;top:64.61px;">System</span></p></div></div></div> <div id="s277" class="sg" data-sgid="Rectangle 3" data-valign = "top" data-pointer-events = "none" style="z-index:3;left:38px;top:173px;width:616px;height:298px"> <div id="s277_t" class="sg" style="z-index:2;left:0px;top:-4px;width:619px;height:304px" ><div><p para="{spbf:3.00}"><span data-textwidth="358.20" class="sg nowrap f21" style="z-index:2;color:#006600;font-style:italic; font-size:156%; left:41.18px;top:12.68px;">R. Dedekind And G. Cantor :</span></p><p para="{spbf:3.00}"><span data-textwidth="541.08" class="sg nowrap f19" style="z-index:3;color:#534733;font-size:133%; left:41.54px;top:52.76px;">In 1870s two German mathematicians; Cantor and </span><span data-textwidth="257.16" class="sg nowrap f19" style="z-index:4;color:#534733;font-size:133%; left:36.86px;top:82.52px;">Dedekind, showed that :</span></p><p para="{spbf:3.00}"><span data-textwidth="572.76" class="sg nowrap f21" style="z-index:5;color:#534733;font-style:italic; font-size:133%; left:31.82px;top:114.32px;">Corresponding to every real number, there is a point </span><span data-textwidth="517.32" class="sg nowrap f21" style="z-index:6;color:#534733;font-style:italic; font-size:133%; left:36.86px;top:143.12px;">on the number line, and corresponding to every </span><span data-textwidth="550.68" class="sg nowrap f21" style="z-index:7;color:#534733;font-style:italic; font-size:133%; left:35.06px;top:171.92px;">point on the number line, there exists a unique real </span><span data-textwidth="96.96" class="sg nowrap f21" style="z-index:8;color:#534733;font-style:italic; font-size:133%; left:36.86px;top:200.72px;">number. </span></p></div></div></div> <div id="s278" class="sg" data-sgid="AutoShape 5" data-pointer-events = "none" style="z-index:4;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s279" class="sg" data-sgid="AutoShape 7" data-pointer-events = "none" style="z-index:5;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s280" class="sg" data-sgid="TextBox 6" data-valign = "top" style="z-index:6;left:454px;top:471px;width:180px;height:27px"> <div id="s280_t" class="sg" style="z-index:2;left:-2px;top:-1px;width:183px;height:34px" ><div><p><span data-textwidth="88.08" class="sg nowrap f13" style="z-index:2;font-style:italic; font-size:89%; left:10.43px;top:8.16px;">R. Dedekind</span></p></div></div></div> <div id="s281" class="sg" data-sgid="TextBox 8" data-valign = "top" style="z-index:7;left:140px;top:471px;width:158px;height:27px"> <div id="s281_t" class="sg" style="z-index:2;left:-2px;top:-1px;width:160px;height:34px" ><div><p><span data-textwidth="69.36" class="sg nowrap f13" style="z-index:2;font-style:italic; font-size:89%; left:10.28px;top:8.16px;">G. Cantor</span></p></div></div></div> <div id="s282" class="sg" data-sgid="Picture 13" style="z-index:8;left:33px;top:392px;width:101px;height:126px"> <img id="s282_i" src="457871C.png" class="sg" style="z-index:1;left:0px;top:0px;width:101.33px;height:126.41px"></div> <div id="s283" class="sg" data-sgid="Rectangle 3" data-valign = "top" data-pointer-events = "none" style="z-index:1;left:38px;top:183px;width:648px;height:233px"> <div id="s283_t" class="sg" style="z-index:2;left:0px;top:-6px;width:657px;height:239px" ><div><p para="{spbf:3.00}"><span data-textwidth="188.88" class="sg nowrap f21" style="z-index:2;color:#006600;font-style:italic; font-size:178%; left:34.34px;top:13.76px;">Archimedes<span class="f19" style="font-style:normal; color:#7030A0;margin-left:10.08px;">:</span></span></p><p para="{spbf:3.00}"></p><p para="{spbf:3.00}"><span data-textwidth="540.36" class="sg nowrap f19" style="z-index:4;color:#534733;font-size:133%; left:33.98px;top:96.20px;">He was a Greek mathematician. He was the first to </span><span data-textwidth="606.24" class="sg nowrap f19" style="z-index:5;color:#534733;font-size:133%; left:36.86px;top:125.00px;">compute the digits in the decimal expansion of π (pi). He </span><span data-textwidth="143.52" class="sg nowrap f19" style="z-index:6;color:#534733;font-size:133%; left:36.86px;top:153.80px;">showed that -</span></p><p para="{spbf:3.00}"><span data-textwidth="343.32" class="sg nowrap f19" style="z-index:8;color:#534733;font-size:133%; left:152.06px;top:185.60px;">3.140845 &lt; π &lt; 3.142857</span></p></div></div></div> <div id="s284" class="sg sgTitle" data-sgid="Rectangle 2" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:44px;top:55px;width:648px;height:90px"> <div id="s284_t" class="sg" style="z-index:2;left:0px;top:-14px;width:653px;height:132px" ><div><p para="{al:2}"><span data-textwidth="598.68" class="sg nowrap f6" style="z-index:2;font-size:222%; left:30.38px;top:16.58px;">Mathematicians related to Number </span><span data-textwidth="125.04" class="sg nowrap f6" style="z-index:3;font-size:222%; left:262.22px;top:64.58px;">System</span></p></div></div></div> <div id="s285" class="sg" data-sgid="Picture 5" style="z-index:3;left:27px;top:359px;width:135px;height:162px"> <img id="s285_i" src="EF6344BF.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:135.89px;height:162.13px"></div> <div id="s286" class="sg" data-sgid="TextBox 4" data-valign = "top" style="z-index:4;left:173px;top:460px;width:276px;height:30px"> <div id="s286_t" class="sg" style="z-index:2;left:-3px;top:-1px;width:279px;height:38px" ><div><p><span data-textwidth="94.68" class="sg nowrap f13" style="z-index:2;font-style:italic; left:11.00px;top:8.63px;">Archimedes</span></p></div></div></div> <div id="s287" class="sg sgTitle" data-sgid="Rectangle 2" data-valign = "bottom" data-pointer-events = "none" style="z-index:1;left:35px;top:24px;width:570px;height:90px"> <img class="sg" src="BB7CC527.png" id="s287_tt" style="left:0.00px;top:0.00px;width:570.57px;height:90.89px"></div> <div id="s288" class="sg" data-sgid="Rectangle 3" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:27px;top:162px;width:570px;height:355px"> <div id="s288_t" class="sg" style="z-index:2;left:0px;top:-3px;width:571px;height:359px" ><div><p para="{spbf:6.00}"><span data-textwidth="474.96" class="sg nowrap f11" style="z-index:2;font-size:133%; left:29.54px;top:9.98px;">A number line is a line with marks on it that are </span><span data-textwidth="468.84" class="sg nowrap f11" style="z-index:3;font-size:133%; left:29.54px;top:38.78px;">placed at equal distance apart. One mark on the </span><span data-textwidth="483.84" class="sg nowrap f11" style="z-index:4;font-size:133%; left:29.54px;top:67.58px;">number line is usually labeled zero and then each </span><span data-textwidth="525.60" class="sg nowrap f11" style="z-index:5;font-size:133%; left:29.54px;top:96.38px;">successive mark to the left or to the write of the zero </span><span data-textwidth="500.76" class="sg nowrap f11" style="z-index:6;font-size:133%; left:29.54px;top:125.18px;">represents a particular unit such as 1, or 0.5. It is a </span><span data-textwidth="245.76" class="sg nowrap f11" style="z-index:7;font-size:133%; left:29.54px;top:153.98px;">picture of a straight line.</span></p></div></div></div> <div id="s289" class="sg" data-sgid="Picture 3" style="z-index:3;left:106px;top:364px;width:450px;height:118px"> <img id="s289_i" src="AD2141F6.png" class="sg" style="z-index:1;left:0px;top:0px;width:450.74px;height:118.82px"></div> <div id="s290" class="sg" data-sgid="TextBox 5" data-valign = "top" style="z-index:4;left:387px;top:471px;width:203px;height:29px"> <div id="s290_t" class="sg" style="z-index:2;left:-3px;top:-1px;width:206px;height:38px" ><div><p><span data-textwidth="110.16" class="sg nowrap f13" style="z-index:2;font-style:italic; left:11.12px;top:8.64px;">A number line</span></p></div></div></div> <div id="s291" class="sg" data-sgid="Rectangle 3" data-valign = "top" data-pointer-events = "none" style="z-index:1;left:35px;top:179px;width:588px;height:242px"> <div id="s291_t" class="sg" style="z-index:2;left:0px;top:-1px;width:589px;height:244px" ><div><p para="{spbf:4.00}"><span data-textwidth="479.16" class="sg nowrap f16" style="z-index:2;font-size:133%; left:36.84px;top:10.42px;">No, number and numerals are not same. </span><span data-textwidth="516.96" class="sg nowrap f16" style="z-index:3;font-size:133%; left:36.84px;top:40.18px;">Numerals are used to make numbers. It is a </span><span data-textwidth="439.20" class="sg nowrap f16" style="z-index:4;font-size:133%; left:36.84px;top:68.98px;">symbol used to represent a number. </span></p><p para="{bl:{ch:61565},spbf:4.00}"></p><p para="{spbf:4.00}"><span data-textwidth="521.64" class="sg nowrap f16" style="z-index:5;font-size:133%; left:36.84px;top:134.62px;">For example, the <span class="f16" style="font-style:italic; ">NUMERAL </span>4 is the name of </span><span data-textwidth="167.88" class="sg nowrap f16" style="z-index:8;font-style:italic; font-size:133%; left:36.84px;top:163.42px;">NUMBER <span class="f16" style="font-style:normal; ">four.</span></span></p></div></div></div> <div id="s292" class="sg sgTitle" data-sgid="Rectangle 2" data-valign = "middle" data-pointer-events = "none" style="z-index:2;left:44px;top:61px;width:588px;height:90px"> <img class="sg" src="56CBD909.png" id="s292_tt" style="left:0.00px;top:-4.81px;width:588.53px;height:95.71px"></div> <div id="s293" class="sg" data-sgid="Picture 5" style="z-index:3;left:449px;top:345px;width:146px;height:194px"> <img id="s293_i" src="D5269530.png" class="sg" style="z-index:1;left:0px;top:0px;width:146.32px;height:194.98px"></div> <div id="s294" class="sg" data-sgid="Straight Arrow Connector 5" data-pointer-events = "none" style="z-index:4;left:325px;top:398px;width:164px;height:52px"> <img id="s294_i" src="2DFDBB5D.png" class="sg" style="z-index:1;left:0px;top:0px;width:164.82px;height:52.54px"></div> <div id="s295" class="sg" data-sgid="Straight Arrow Connector 6" data-pointer-events = "none" style="z-index:5;left:544px;top:448px;width:108px;height:52px"> <img id="s295_i" src="6E92B362.png" class="sg" style="z-index:1;left:0px;top:0px;width:108.56px;height:52.54px"></div> <div id="s296" class="sg" data-sgid="TextBox 8" data-valign = "top" style="z-index:6;left:156px;top:449px;width:158px;height:30px"> <div id="s296_t" class="sg" style="z-index:2;left:-3px;top:-1px;width:161px;height:38px" ><div><p><span data-textwidth="94.68" class="sg nowrap f13" style="z-index:2;font-style:italic; left:11.47px;top:8.62px;">Numeral “7”</span></p></div></div></div> <div id="s297" class="sg" data-sgid="TextBox 9" data-valign = "top" style="z-index:7;left:589px;top:426px;width:113px;height:27px"> <div id="s297_t" class="sg" style="z-index:2;left:-2px;top:-1px;width:115px;height:34px" ><div><p><span data-textwidth="70.20" class="sg nowrap f13" style="z-index:2;font-style:italic; font-size:89%; left:10.50px;top:8.11px;">Number 7</span></p></div></div></div> <div id="s312" class="sg sgTitle" data-sgid="Title 1" data-valign = "middle" data-pointer-events = "none" style="z-index:1;left:35px;top:44px;width:648px;height:90px"> <div id="s312_t" class="sg" style="z-index:2;left:0px;top:0px;width:648px;height:107px" ><div><p para="{al:2}"><span data-textwidth="381.96" class="sg nowrap f6" style="z-index:2;color:#3E8EA9;font-size:267%; left:133.46px;top:22.14px;"><span style="text-shadow:1.48px 1.48px 2.10px rgba(192,192,192,1.00)">Word Alternatives</span></span></p></div></div></div> <div id="s313" class="sg" data-sgid="Content Placeholder 2" data-valign = "top" data-pointer-events = "none" style="z-index:2;left:89px;top:183px;width:648px;height:357px"> <div id="s313_t" class="sg" style="z-index:2;left:0px;top:-3px;width:662px;height:360px" ><div><p para="{spbf:6.00}"><span data-textwidth="611.16" class="sg nowrap f17" style="z-index:2;font-size:133%; left:36.88px;top:10.60px;">Some numbers traditionally have words to express them, </span><span data-textwidth="241.44" class="sg nowrap f17" style="z-index:3;font-size:133%; left:36.88px;top:39.40px;">including the following:</span></p><p para="{bl:{ch:167},spbf:6.00}"><span data-textwidth="234.36" class="sg nowrap f18" style="z-index:4;color:#CEB966;font-size:106%; left:14.56px;top:74.20px;"><span class="f17" style="font-size:126%; color:#000000;margin-left:13.56px;">Pair, couple, brace: </span></span><span data-textwidth="13.32" class="sg nowrap f17" style="z-index:6;font-size:133%; left:368.08px;top:74.20px;">2</span></p><p para="{bl:{ch:167},spbf:6.00}"><span data-textwidth="98.16" class="sg nowrap f18" style="z-index:7;color:#CEB966;font-size:106%; left:14.56px;top:109.00px;"><span class="f17" style="font-size:126%; color:#000000;margin-left:13.56px;">Dozen:</span></span><span data-textwidth="26.64" class="sg nowrap f17" style="z-index:9;font-size:133%; left:296.08px;top:109.00px;">12</span></p><p para="{bl:{ch:167},spbf:6.00}"><span data-textwidth="175.56" class="sg nowrap f18" style="z-index:10;color:#CEB966;font-size:106%; left:14.56px;top:143.80px;"><span class="f17" style="font-size:126%; color:#000000;margin-left:13.56px;">Bakers dozen:</span></span><span data-textwidth="26.64" class="sg nowrap f17" style="z-index:12;font-size:133%; left:296.08px;top:143.80px;">13</span></p><p para="{bl:{ch:167},spbf:6.00}"><span data-textwidth="91.56" class="sg nowrap f18" style="z-index:13;color:#CEB966;font-size:106%; left:14.56px;top:178.60px;"><span class="f17" style="font-size:126%; color:#000000;margin-left:13.56px;">Score:</span></span><span data-textwidth="33.36" class="sg nowrap f17" style="z-index:15;font-size:133%; left:296.08px;top:178.60px;">20 </span></p><p para="{bl:{ch:167},spbf:6.00}"><span data-textwidth="93.00" class="sg nowrap f18" style="z-index:16;color:#CEB966;font-size:106%; left:14.56px;top:213.40px;"><span class="f17" style="font-size:126%; color:#000000;margin-left:13.56px;">Gross:</span></span><span data-textwidth="39.96" class="sg nowrap f17" style="z-index:18;font-size:133%; left:296.08px;top:213.40px;">144</span></p><p para="{bl:{ch:167},spbf:6.00}"><span data-textwidth="321.36" class="sg nowrap f18" style="z-index:19;color:#CEB966;font-size:106%; left:14.56px;top:248.20px;"><span class="f17" style="font-size:126%; color:#000000;margin-left:13.56px;">Ream(new measure):</span><span class="f17" style="font-size:126%; color:#000000;margin-left:28.32px;">500</span></span></p><p para="{bl:{ch:167},spbf:6.00}"><span data-textwidth="154.20" class="sg nowrap f18" style="z-index:22;color:#CEB966;font-size:106%; left:14.56px;top:283.00px;"><span class="f17" style="font-size:126%; color:#000000;margin-left:13.56px;">Great gross:</span></span><span data-textwidth="53.28" class="sg nowrap f17" style="z-index:24;font-size:133%; left:296.08px;top:283.00px;">1728</span></p></div></div></div> <div id="s314" class="sg" data-sgid="Rectangle 6" data-pointer-events = "none" style="z-index:1;left:0px;top:376px;width:731px;height:147px"> <img id="s314_i" src="D9FC0BD2.png" class="sg" style="z-index:1;left:0px;top:0px;width:731.64px;height:147.51px"></div> <div id="s315" class="sg" data-sgid="Picture 6" style="z-index:2;left:78px;top:27px;width:467px;height:366px"> <img id="s315_i" src="556123ED.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:467.54px;height:366.34px"></div> <div id="s326" class="sg" data-sgid="Text Placeholder 7" data-valign = "top" data-pointer-events = "none" style="z-index:1;left:33px;top:66px;width:648px;height:409px"> <div id="s326_t" class="sg" style="z-index:2;left:-1px;top:-10px;width:664px;height:512px" ><div><p para="{al:2,spbf:7.00}"><span data-textwidth="606.36" class="sg nowrap f13" style="z-index:2;color:#775F55;font-style:italic; font-size:267%; left:29.41px;top:21.40px;">Project made and Compiled </span><span data-textwidth="92.16" class="sg nowrap f13" style="z-index:3;color:#775F55;font-style:italic; font-size:267%; left:279.61px;top:79.00px;">by ~</span></p><p para="{spbf:7.00}"></p><p para="{spbf:7.00}"></p><p para="{al:3,bl:{ch:252},spbf:7.00}"><span data-textwidth="226.44" class="sg nowrap f18" style="z-index:4;color:#0D0D0D;font-size:94%; left:415.69px;top:222.72px;"><span class="f17" style="font-size:165%; color:#775F55;">Samarth Agrawal</span></span></p><p para="{al:3,bl:{ch:252},spbf:7.00}"><span data-textwidth="186.60" class="sg nowrap f18" style="z-index:6;color:#0D0D0D;font-size:94%; left:455.53px;top:263.28px;"><span class="f17" style="font-size:165%; color:#775F55;">Yogesh Surve</span></span></p><p para="{al:3,bl:{ch:252},spbf:7.00}"><span data-textwidth="145.56" class="sg nowrap f18" style="z-index:8;color:#0D0D0D;font-size:94%; left:496.57px;top:303.96px;"><span class="f17" style="font-size:165%; color:#775F55;">Arnab Das</span></span></p><p para="{al:3,bl:{ch:252},spbf:7.00}"><span data-textwidth="173.40" class="sg nowrap f18" style="z-index:10;color:#0D0D0D;font-size:94%; left:468.73px;top:344.52px;"><span class="f17" style="font-size:165%; color:#775F55;">Arijit Sharma</span></span></p><p para="{al:3,bl:{ch:252},spbf:7.00}"><span data-textwidth="170.40" class="sg nowrap f18" style="z-index:12;color:#0D0D0D;font-size:94%; left:471.73px;top:385.08px;"><span class="f17" style="font-size:165%; color:#775F55;">Ankita Sinha</span></span></p><p para="{al:3,bl:{ch:252},spbf:7.00}"><span data-textwidth="148.32" class="sg nowrap f18" style="z-index:14;color:#0D0D0D;font-size:94%; left:494.17px;top:425.76px;"><span class="f17" style="font-size:165%; color:#775F55;">Ayushi Sur</span></span></p><p para="{al:3,bl:{ch:252},spbf:7.00}"><span data-textwidth="193.80" class="sg nowrap f18" style="z-index:16;color:#0D0D0D;font-size:94%; left:448.33px;top:466.32px;"><span class="f17" style="font-size:165%; color:#775F55;">Nimisha Singh</span></span></p></div></div></div> <div id="s327" class="sg" data-sgid="AutoShape 5" data-pointer-events = "none" style="z-index:2;left:11px;top:-11px;width:24px;height:24px"> <div style="width:100%;height:100%;background:rgba(0,0,0,0)"></div></div> <div id="s328" class="sg" data-sgid="Picture 7" style="z-index:3;left:83px;top:213px;width:259px;height:299px"> <img id="s328_i" src="33559416.jpg" class="sg" style="z-index:1;left:0px;top:0px;width:259.58px;height:299.07px"></div>