Is #0.25# a perfect square?

The number #0.25# can be written in the form #frac(25)(100)#.

If you notice, both the numerator #(25)# and the denominator #(100)# are perfect squares.

According to the Wikipedia article on square numbers, "the ratio of any two square integers is a square".

Therefore, #frac(25)(100)#, or #0.25#, is a perfect square.

Perfect square integers

If we are talking about integers, then we tend to be fairly clear what we mean by a perfect square, namely:

#0, 1, 4, 9, 16, 25, 36, 49,...#

That is - a perfect square is a number which is the square of an integer.

Perfect square rationals

When a number such as #0.25# is mentioned, we can immediately tell that we at least including rational numbers in our considerations. We find:

#0.25 = 1/4 = 1/2^2 = (1/2)^2 = 0.5^2#

So #0.25# is a rational number that is a square of a rational number.

So it does qualify as being called a perfect square.

In general we find that the only rational numbers which are squares of rational numbers can always be expressed in the form #p/q# where #p, q# are perfect square positive integers.

One step beyond...

Is #2# a perfect square number?

It is not the square of a rational number, so you would not normally count it as such, but consider the following:

Let #S# be the set of all numbers of the form #a+bsqrt(2)# where #a, b# are rational numbers.

You will find that #S# is closed under addition, subtraction, multiplication and division by non-zero elements. That is, if you perform any of these operations on elements of #S# then you will get an element of #S#.

#S# is said to form a field.

Then in #S#, the number #2# is a perfect square, being the square of #0+1sqrt(2)#.

...and another

In greater generality, any Complex number is - in a sense - a perfect square in that it is the square of a Complex number.

Summing up

Concepts like "perfect square" are sensitive to context. In the given example of #0.25# there is an implied context of rational numbers, for which it can be identified as a perfect square, but other cases may be less obvious.