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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
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Sia ? 6 years, 4 months ago
{tex}\mathrm { HCF } \left( \frac { 8 } { 9 } , \frac { 10 } { 27 } , \frac { 16 } { 81 } \right){/tex} {tex}= \frac { \mathrm { HCF } \text { of the numerators } } { \mathrm { LCM } \text { of the denominators } }{/tex} {tex}= \frac { H C F ( 8,10,16 ) } { L C M ( 9,27,81 ) }{/tex}
and {tex}\operatorname { LCM } \left( \frac { 8 } { 9 } , \frac { 10 } { 27 } , \frac { 16 } { 81 } \right){/tex} {tex}= \frac { L C M \text { of the numerators } } { \text { HCF of the denominators } }{/tex} {tex}= \frac { L C M ( 8,10,16 ) } { H C F ( 9,27,81 ) }{/tex}
Consider,
8 = 2 {tex}\times{/tex} 2 {tex}\times{/tex} 2 = 23
10 = 2 {tex}\times{/tex}5
16 = 2 {tex}\times{/tex} 2 {tex}\times{/tex} 2{tex}\times{/tex} 2 = 24
So, the HCF{tex} (8, 10, 16){/tex} = 2 and LCM {tex}(8, 10, 16){/tex} {tex}= 2 ^ { 4 } \times 5 = 80{/tex}
9 = 3 {tex}\times{/tex}3 = 32
27 =3 {tex}\times{/tex} 3 {tex}\times{/tex} 3 = 33
81 = 3 {tex}\times{/tex} 3 {tex}\times{/tex} 3 {tex}\times{/tex} 3 = 34
So, the HCF (9, 27, 81) = 32 = 9 and LCM (9, 27, 81) = 34 = 81
{tex}\Rightarrow \text { HCF } \left( \frac { 8 } { 9 } , \frac { 10 } { 27 } , \frac { 16 } { 81 } \right) = \frac { 2 } { 81 }{/tex} and {tex}\operatorname { LCM } \left( \frac { 8 } { 9 } , \frac { 10 } { 27 } , \frac { 16 } { 81 } \right) = \frac { 80 } { 9 }{/tex}
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Sia ? 6 years, 4 months ago
Suppose {tex}\sqrt2{/tex} is a rational number. That is , {tex}\sqrt2{/tex} = {tex}\frac{p}{q}{/tex} for some p{tex}\in{/tex}Z and q {tex}\in{/tex}Z. We can assume the fraction is in lowest fraction, That is p and q shares no common factors.
Then {tex}\sqrt2q=p{/tex}
Squaring both side we get,
{tex}2q^2=p^2{/tex}
So {tex}p^2{/tex} is a multiple of 2,
let's assume {tex}p=2m{/tex}
Then, {tex}2q^2=\left(2m\right)^2{/tex}
{tex}2q^2=4m^2{/tex}
Or {tex}q^2=2m^2{/tex}
So {tex}q^2{/tex} is a multiple of 2,
{tex}\therefore{/tex} q is multiple of 2
Thus p and q shares a common factor.This is contradiction.
{tex}\Rightarrow {/tex}{tex}\sqrt { 2 }{/tex} is an irrational number.
Posted by Nilesh Kumar 7 years, 7 months ago
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Nivedita Gupta 7 years, 7 months ago
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