Question

Prove that √7 is irrational no.

Answer 1

$\underline{\Huge\mathfrak{Solution:}}$

Lets assume that √7 is rational number. ie √7=p/q.

suppose p/q have common factor then

we divide by the common factor to get √7 = a/b were a and b are co-prime number.

that is a and b have no common factor.

√7 =a/b co- prime number

√7= a/b

a=√7b

squaring

a²=7b² .......1

a² is divisible by 7

a=7c

substituting values in 1

(7c)²=7b²

49c²=7b²

7c²=b²

b²=7c²

b² is divisible by 7

that is a and b have atleast one common factor 7. This is contridite to the fact that a and b have no common factor.This is happen because of our wrong assumption.

√7 is irrational✔✔

Answer 2

Answer:

$let \sqrt{7 \: } be \: rational \: number \\ \\ \therefore \: \sqrt{7} = \frac{a}{b \:} \: ( \: where \: a \: and \: b \: are \: co - prime \: and \: b \: ≠0) \\ \\ \:\implies7 = \frac{ {a}^{2} }{ {b}^{2} } \\ \\ \implies7 {b}^{2} = {a}^{2} \\ \\ \therefore \: {a}^{2} \: is \: divisible \: by \: 7. \\ \\ hence \: a \: \: is \: also \: divisible \: by \: \: 7. \\ \\ again \: \: \: let \: a = 7c \\ \\ \implies {a}^{2} = {(7c)}^{2} \: \\ \\ \implies {7b}^{2} = {49c}^{2} ( \: putting \: {a}^{2} = {7b}^{2} ) \\ \\ \implies \: {b}^{2} = \frac{ {49c}^{2} }{7} \\ \\ \implies {b}^{2} = {7c}^{2} \\ \\ \therefore \: {b \: }^{2} is \: \: also \: divisible \: by \:7. \\ \\ hence \: \: b \: is \: also \: divisible \: by \: 7. \\ \\ but \: this \: contradicts \: the \: fact \: that \: a \: and \: b \: have \: no \: common \: factor \: other \: than \: 1. \\ as \: a \: and \: b \: have \: have \: 7 \: as \: a \: common \: factor. \\ hence \: \sqrt{7} \: \: is \: not \: rational. \\ \therefore \: \sqrt{7} \: is \: irrational.$