Question

Is root 23 a rational number or irrational number?

Answer 1

√23 is an irrational number

Given :

The number √23

To find :

√23 is a rational number or an irrational number

Solution :

Step 1 of 2 :

Write down the given number

Here the given number is √23

Step 2 of 2 :

Check √23 is a rational number or an irrational number

Let us assume that √23 is a rational number.

then, as we know a rational number should be in the form of p/q

where p and q are co- prime number.

⇒ √23 = p/q { where p and q are co- prime}

⇒ √23q = p

Now, by squaring both the side

we get,

(√23q)² = p²

⇒ 23q² = p² - - - - - - (1)

Now if 23 is the factor of p²

Then , 23 is also a factor of p - - - - - - (2)

⇒ Let p = 23m { where m is any integer }

squaring both sides

p² = (23m)²

p² = 529m²

Putting the value of p² in Equation 1 we get

23q² = p²

⇒ 23q² = 529m²

⇒ q² = 23m²

Now if 23 is factor of q²

Then , 23 is also factor of q

Since

23 is factor of p & q both

So, our assumption that p & q are co- prime is wrong

Hence √23 is an irrational number

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