Is 0.25 a perfect square?

Sep 18, 2017

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 $\left(25\right)$ and the denominator $\left(100\right)$ 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.

Sep 18, 2017

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:

$0 , 1 , 4 , 9 , 16 , 25 , 36 , 49 , \ldots$

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:

$0.25 = \frac{1}{4} = \frac{1}{2} ^ 2 = {\left(\frac{1}{2}\right)}^{2} = {0.5}^{2}$

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 $\frac{p}{q}$ 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 $a + b \sqrt{2}$ 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 $0 + 1 \sqrt{2}$.

...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.