# Is #0.25# a perfect square?

##### 2 Answers

Yes,

#### Explanation:

The number

If you notice, both the numerator

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

Therefore,

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

So

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

**One step beyond...**

Is

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

Let

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

*field*.

Then in *is* a perfect square, being the square of

**...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