Question

2
Prove that
17 is an irrational number.​

Answer 1

Check your Question because 17 is a rational number

Rational numbers: are numbers that can be written in the form of p/q where p and q are coprime integers and q≠0,

Here , 17 is given which can be written as 17/1 which is in the rational form, hence 17 is rational.

Answer 2

Q. Prove that √17 is an irrational number.

Let us assume that √17 can be written in form of p/q where (p, q) ∊ Z and q ≠ 0.

Also, a/b = √17 where (a, b) ∊ Z, b ≠ 0 and a, b are co-prime.

∴ a = b√17

By squaring both sides,

⇒ a² = 17b² ...(1)

⇒ b² = a²/17 or, a² is divisible by 17.

Thus, a is also divisible by 17.

Now, let a = 17c

⇒ a² = 17 × 17c² ...(2)

Now, putting the value of a² from (2) to (1),

∴ 17 × 17c² = 17b²

⇒ b² = 17c²

⇒ b²/17 = c² or, b² is divisible by 17 too.

Thus, b is also divisible by 17.

But, erstwhile, we assumed a and b to be co-prime, i.e., have HCF = 1. But now they are not so.

∴ We reached at a contradiction. Or, √17 is an irrational number.