Question

Is `0.25` a perfect square?

Answer 1

Yes, `0.25` is a perfect square.

Explanation:

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.

Answer 2

Yes, but it's worth a few remarks...

Explanation:

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.