Square 2 and you get 4. So what if we ask the reverse question — "what number, squared, gives you 4?" The answer is 2. A number that, squared, produces a given number is called that number's square root. The square root of 4 is 2 (and −2 counts too, since squaring it also gives 4). The idea itself is simple, but here's what makes it interesting: not every number's square root comes out as a clean integer like 2 does.
Take the square root of 3. Since 1²=1 and 2²=4, the square root of 3 has to sit somewhere between 1 and 2. Work it out and you get 1.7320508..., with digits after the decimal point that go on forever with no repeating pattern. A number like this — one that can't be written exactly as a fraction, and whose decimal expansion never settles into a repeating pattern — is called an irrational number. Numbers that CAN be written as a fraction (integers, terminating decimals, repeating decimals) are all rational numbers. Rational and irrational numbers together are called real numbers — the name literally means "every number that actually exists."
For a number's square root to be rational, that number has to be a perfect square (the square of an integer: 1, 4, 9, 16, 25...). The square roots of every other natural number that isn't a perfect square turn out to be irrational. And here's the interesting part: perfect squares are actually quite rare among all the integers. Among the numbers 1 through 100, there are only 10 perfect squares (1, 4, 9, ..., 100) — meaning the other 90 all have irrational square roots.
One useful skill when working with square roots is simplifying the radical. √12 might look complicated at first glance, but since 12 = 4×3 and 4 is a perfect square, you can rewrite it as √12 = √4×√3 = 2√3. Pulling every perfect-square factor out from under the radical as far as possible gives you a much simpler, more workable form. This simplifying step turns out to be essential later too — when adding or subtracting square roots, you can only combine them (like combining like terms) if the number under the radical matches exactly.
Comparing the size of square roots is easier than it sounds. For any two positive numbers a and b, if a < b then √a < √b always holds. In other words, the bigger the number under the radical, the bigger the square root. So without ever calculating a decimal, you can tell instantly which square root is larger just by comparing the numbers under the radical signs.
Irrational numbers might feel unfamiliar at first, but they're actually hiding all around us. The diagonal of a square, and the ratio of a circle's circumference to its diameter (π), are both irrational. Rational numbers alone can't completely fill the number line — it takes irrational numbers to fill in the gaps before the number line is truly packed solid with real numbers. That's the heart of why the concept of "real numbers" is needed in the first place.
On our activity page, you can change a number with a slider and check whether its square root is rational or irrational, whether the radical can be simplified further, and exactly where it sits on the number line. Try the quiz too, and practice comparing the size of square roots.