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Is the square root of 25 a Rational Number?


Is the Square root of 25 a rational number? Yes, square root of 25 is a rational number. For proving that, let's assume that √25 is a rational number. Then, we can write

√25 = a/b where, a, b are whole numbers and where, b is not zero.

√25 = a/b => +5 or -5

Because +5 or -5 are whole numbers and whole numbers are part of Rational Numbers. So, we can say that square root of 25 is a rational number.

Another way to proving that square root of 25 is a rational number - we square both sides in equation, it follows that:

25 = a2/b2,  


a2 = 25 * b2.

So, the square of a is an odd number because it is twenty five times the square of b . 

Here a is an odd number because a is 25 times some other number,


a = x,

Where, x is any number. Here, x is Integer or not doesn't matter because we deals with properties of a and b.  

If we substitute a = x into the original equation 25 = a2/b2, then we get:

     25 = (x)2/b2

     25.b= x2 = b x2

     => b = +x/5 or -x/5 

This means b is rational number because it is in a/b form where a and b are two integers, where b is not equal to 0. We started the whole process by saying that √25 is a Rational number which is in a/b form and now, it turns out that a is an odd number and b is a rational number form. So, we can say that √25 is a Rational number because any Combination where one number is a rational number is a rational number.