Question

Prove √5 is an irrational ​

Answer 1

Step-by-step explanation:

suppose root 5 is rational number equal to another rational number a/b where A and B are Co prime and b is is not equal to zero.

√5 = a/b

multiply them by 5

5 = a/b. (5s of a and b candles each other)

b = a/5

so a is divisible by 5

now let a = 5c

√5 = 5c/b

square both sides

5 = 25c^2/b^2

b^2 = 25c^2/5

so b is also divisible by 5

but we assume that A and B are coprime so our statement is proved wrong and root 5 has proved irrational

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Answer 2

$\huge \red{\bf Answer}$

We need to prove that √5 is irrational

$\pink{\rm Proof:}$

Let us assume that √5 is a rational number.

Sp it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√5=p/q

On squaring both the sides we get,

⇒5=p²/q²

⇒5q²=p² —————–(i)

p²/5= q²

So 5 divides p

ANSWER

p is a multiple of 5

⇒p=5m

⇒p²=25m² ————-(ii)

From equations (i) and (ii), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5

⇒q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

$\green{ \bf \sqrt{5} \: is \: a \: irrational \: number}$