Question

prove root 5 irrational ​

Answer 1

Answєr :-

Is Root 5 irrational number?

Yes √5 is irrational, here we go with the proof

Let us assume that √5 is rational

So, √5 is a rational number

All rational numbers are in p/q form

So, √5 = p/q , where p≠0 and p&q are integers

Squaring on both sides

(√5)²=(p/q)²

5=p²/q²

5q²=p²–(1)

5 divides p [p is any prime number and a is any positive integer and p divides

a² then p divides a]

So, p is a positive integer—(2)

Then, p=5r, r is a positive integer

Squaring on both sides

p²=(5r)²—(3)

By 1&3

(5r)²=5q²

25r²=5q²

5r²=q²

So, 5 divides q²

Then, 5 divides q [by above reason]

Then,q is a positive integer—(4)

By 2&4

p and q have common factor 5.

This is a contradiction since p and q are co-primes

So, our assumption is wrong

Therefore,

√5 is an irrational number.

Hᴏᴘᴇ Tʜɪs Hᴇʟᴘs Yᴏᴜ

Answer 2

Solution:

To prove √5 as an irrational number.

Assume √5 as rational number.

We know that,

Rational number are written in the form of a/b. Where, a and b are positive integers and b ≠ 0.

Therefore, √5 = a/b

Squaring on both sides.

=> (√5)² = (a/b)²

=> 5 = a²/b²

=> 5b² = a²

Now,

Let a = 5r be positive integer.

Squaring on both sides.

=> a² = (5r)²

=> (5r²) = 5b²

=> 25r² = 5b²

=> 5r² = b²

Here, 5 divides b².

So, b is a positive integer.

• a and b are common factor of 5.

Therefore, Our assumption is wrong that √5 is a rational number.

∴ √5 is an irrational number.