Question

Prove that √7 is irrational

Answer 1

Step-by-step explanation:

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

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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

√7= a/ba=√7b

√7= a/ba=√7bsquaring

√7= a/ba=√7bsquaringa²=7b² .......1

a² is divisible by 7

a=7c

substituting values in 1

(7c)²=7b²

(7c)²=7b²49c²=7b²

(7c)²=7b²49c²=7b²7c²=b²

(7c)²=7b²49c²=7b²7c²=b²b²=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

$\huge\green{Thank\:you}$