<div dir="ltr" style="text-align: left;" trbidi="on"> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Hence </span><span style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; position: relative; top: 5.5pt;"><v:shapetype coordsize="21600,21600" filled="f" id="_x0000_t75" o:preferrelative="t" o:spt="75" path=" m@4@5 l@4@11@9@11@9@5 xe" stroked="f"><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0 "></v:f><v:f eqn="sum @0 1 0 "></v:f><v:f eqn="sum 0 0 @1 "></v:f><v:f eqn="prod @2 1 2 "></v:f><v:f eqn="prod @3 21600 pixelWidth "></v:f><v:f eqn="prod @3 21600 pixelHeight "></v:f><v:f eqn="sum @0 0 1 "></v:f><v:f eqn="prod @6 1 2 "></v:f><v:f eqn="prod @7 21600 pixelWidth "></v:f><v:f eqn="sum @8 21600 0 "></v:f><v:f eqn="prod @7 21600 pixelHeight "></v:f><v:f eqn="sum @10 21600 0 "></v:f></v:formulas><v:path gradientshapeok="t" o:connecttype="rect" o:extrusionok="f"></v:path><o:lock aspectratio="t" v:ext="edit"></o:lock></v:shapetype><v:shape coordsize="21600,21600" id="_x0000_i1025" style="height: 18.75pt; width: 15.75pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="kkkkkkk_files/image001.png"></v:imagedata></v:shape></span><span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;"> is not a rational number, it is irrational. <o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">2. Prove that 3+2</span><span style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; position: relative; top: 5.5pt;"><v:shape coordsize="21600,21600" id="_x0000_i1025" style="height: 18.75pt; width: 15.75pt;" type="#_x0000_t75"> <v:imagedata chromakey="white" o:title="" src="kkkkkkk_files/image001.png"></v:imagedata></v:shape></span><span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;"> is irrational<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <b><span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Answer</span></b><span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;"><o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Let take that 3+2</span><span style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; position: relative; top: 5.5pt;"><v:shape coordsize="21600,21600" id="_x0000_i1025" style="height: 19.5pt; width: 15.75pt;" type="#_x0000_t75"> <v:imagedata chromakey="white" o:title="" src="kkkkkkk_files/image003.png"></v:imagedata></v:shape></span><span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;"> is a rational number.<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">So we can write this number as<o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: 0pt; text-align: justify;"> <br /></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">3+2</span><span style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; position: relative; top: 5.5pt;"><v:shape coordsize="21600,21600" id="_x0000_i1025" style="height: 18.75pt; width: 15.75pt;" type="#_x0000_t75"> <v:imagedata chromakey="white" o:title="" src="kkkkkkk_files/image001.png"></v:imagedata></v:shape></span><span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">=a/b<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Here a and b are two co prime number and b is not equal to 0 <o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Subtract 3 both side we get <o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">2</span><span style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; position: relative; top: 5.5pt;"><v:shape coordsize="21600,21600" id="_x0000_i1025" style="height: 18.75pt; width: 15.75pt;" type="#_x0000_t75"> <v:imagedata chromakey="white" o:title="" src="kkkkkkk_files/image001.png"></v:imagedata></v:shape></span><span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;"> = a/b -3<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">2</span><span style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; position: relative; top: 5.5pt;"><v:shape coordsize="21600,21600" id="_x0000_i1025" style="height: 19.5pt; width: 15.75pt;" type="#_x0000_t75"> <v:imagedata chromakey="white" o:title="" src="kkkkkkk_files/image003.png"></v:imagedata></v:shape></span><span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;"> = (a-<span class="SpellE">3b</span>)/b<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Now divide by 2, we get <o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> </div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> <span style="font-family: "calibri" , "sans-serif"; font-size: 11pt; line-height: 115%; position: relative; top: 5.5pt;"><v:shape coordsize="21600,21600" id="_x0000_i1025" style="height: 19.5pt; width: 18.75pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="kkkkkkk_files/image005.png"></v:imagedata></v:shape></span><span style="font-size: 13pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">= (a-<span class="SpellE">3b</span>)/<span class="SpellE">2b</span><o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">3. Prove that the following are irrationals: <o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">(i) 1</span><!--[if gte msEquation 12]><m:oMath><span style='font-size:13.0pt;font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>√2</m:r></span></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 115%; position: relative; top: 5.5pt;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:15.75pt; height:18.75pt'> <v:imagedata src="file:///C:\Users\win7\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png" o:title="" chromakey="white"/> </v:shape><![endif]--><!--[if !vml]--><img height="25" src="file:///C:/Users/win7/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" v:shapes="_x0000_i1025" width="21" /><!--[endif]--></span><!--[endif]--><span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> (ii) 7</span><!--[if gte msEquation 12]><m:oMath><span style='font-size:13.0pt;font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>√5</m:r></span></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 115%; position: relative; top: 5.5pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:15.75pt;height:18.75pt'> <v:imagedata src="file:///C:\Users\win7\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png" o:title="" chromakey="white"/> </v:shape><![endif]--><!--[if !vml]--><img height="25" src="file:///C:/Users/win7/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" v:shapes="_x0000_i1025" width="21" /><!--[endif]--></span><!--[endif]--><span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> (iii) 6+</span><!--[if gte msEquation 12]><m:oMath><span style='font-size:13.0pt;font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>√2</m:r></span></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 115%; position: relative; top: 5.5pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:15.75pt;height:18.75pt'> <v:imagedata src="file:///C:\Users\win7\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png" o:title="" chromakey="white"/> </v:shape><![endif]--><!--[if !vml]--><img height="25" src="file:///C:/Users/win7/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" v:shapes="_x0000_i1025" width="21" /><!--[endif]--></span><!--[endif]--><span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><o:p></o:p></span></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <br /></div> <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <b><span style="font-size: 13.0pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Answer<o:p></o:p></span></b></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <br /></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">(i) Let take that 1</span><!--[if gte msEquation 12]><m:oMath><span style='font-size:13.0pt;font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>√2</m:r></span></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 115%; position: relative; top: 5.5pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:15.75pt;height:18.75pt'> <v:imagedata src="file:///C:\Users\win7\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png" o:title="" chromakey="white"/> </v:shape><![endif]--><!--[if !vml]--><img height="25" src="file:///C:/Users/win7/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" v:shapes="_x0000_i1025" width="21" /><!--[endif]--></span><!--[endif]--><span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> is a rational number. <o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So we can write this number as<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">1</span><!--[if gte msEquation 12]><m:oMath><span style='font-size:13.0pt;font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>√2</m:r></span></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 115%; position: relative; top: 5.5pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:15.75pt;height:18.75pt'> <v:imagedata src="file:///C:\Users\win7\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png" o:title="" chromakey="white"/> </v:shape><![endif]--><!--[if !vml]--><img height="25" src="file:///C:/Users/win7/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" v:shapes="_x0000_i1025" width="21" /><!--[endif]--></span><!--[endif]--><span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">=a/b<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Here a and b are two co prime number and b is not equal to 0<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Multiply by </span><!--[if gte msEquation 12]><m:oMath><span style='font-size:13.0pt;font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>√2</m:r></span></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 115%; position: relative; top: 5.5pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:15.75pt;height:18.75pt'> <v:imagedata src="file:///C:\Users\win7\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png" o:title="" chromakey="white"/> </v:shape><![endif]--><!--[if !vml]--><img height="25" src="file:///C:/Users/win7/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" v:shapes="_x0000_i1025" width="21" /><!--[endif]--></span><!--[endif]--><span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> both sides we get<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">1=(a</span><!--[if gte msEquation 12]><m:oMath><span style='font-size:13.0pt;font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>√2</m:r></span></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 115%; position: relative; top: 5.5pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:15.75pt;height:18.75pt'> <v:imagedata src="file:///C:\Users\win7\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png" o:title="" chromakey="white"/> </v:shape><![endif]--><!--[if !vml]--><img height="25" src="file:///C:/Users/win7/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" v:shapes="_x0000_i1025" width="21" /><!--[endif]--></span><!--[endif]--><span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">)/b<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Now multiply by b<o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">b=a</span><!--[if gte msEquation 12]><m:oMath><span style='font-size:13.0pt;font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>√2</m:r></span></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 115%; position: relative; top: 5.5pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:15.75pt;height:18.75pt'> <v:imagedata src="file:///C:\Users\win7\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png" o:title="" chromakey="white"/> </v:shape><![endif]--><!--[if !vml]--><img height="25" src="file:///C:/Users/win7/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" v:shapes="_x0000_i1025" width="21" /><!--[endif]--></span><!--[endif]--><span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><o:p></o:p></span></div> <div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: justify;"> </div> <div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;"> <span style="font-size: 13.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">divide by a we get<o:p></o:p></span></div> </div>

Hence  is not a rational number, it is irrational. 2. Prove that 3+2  is irrational Answer Let take that 3+2  is a rational num...

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