Welcome back to The Real Number System and Irrationality! This is Lesson 3 of 5, so we are well past the halfway mark. So far, we have defined irrational numbers by their non-terminating, non-repeating decimals and then studied two celebrity examples — π and e. Those constants are fascinating, but they might leave us wondering: are irrational numbers a rare curiosity, or are they hiding all around us?

In this lesson, we answer that question by turning to an entire family of irrational numbers: square roots. You will learn a single, reliable test — based on whether or not a number is a perfect square — that lets you classify any square root of a positive integer as rational or irrational in seconds. We will also practice writing clear justifications for our answers, a skill the upcoming exercises will put to the test.

π and e are special because they appear in countless formulas, but their fame can create a misleading impression — that irrational numbers are exotic outliers. In truth, irrational numbers are far more plentiful than rational ones, and you do not have to search the world of circles or compound interest to find them. One of the richest and most accessible sources is something we meet early in math: square roots.

Think about a simple, everyday situation. A carpenter measuring the diagonal of a square tile that is $1$ foot on each side will find, by the Pythagorean theorem, that the diagonal measures $\sqrt{2}$ feet. That length is irrational — it cannot be expressed exactly as a fraction of inches or feet. As we will see next, nearly every positive integer produces an irrational square root, making square roots a practically unlimited supply of irrational numbers.

Before we can classify square roots, we need to be comfortable recognizing perfect squares. A perfect square is a positive integer that equals some whole number multiplied by itself. For example, $9$ is a perfect square because $3 \times 3 = 9$, and $25$ is a perfect square because $5 \times 5 = 25$.

What happens when the number under the radical sign is not a perfect square? Consider $\sqrt{2}$. There is no integer whose square equals $2$ — we know $1^2 = 1$ and , so falls somewhere between and . Its decimal expansion begins:

The test in practice boils down to one question: Is there a whole number whose square equals the given integer? Let's walk through several examples using the flowchart below.

Example 1 — $\sqrt{49}$: Is there an integer $k$ with $$? Yes — . So , which is .

Being able to classify a square root is one thing; explaining why is just as important. A strong justification is short and anchored to the definition. Here is a model response for a non-perfect square:

"$\sqrt{72}$ is irrational because $72$ is not a perfect square. The closest perfect squares are $64 = 8^2$ and , so no integer multiplied by itself equals . Therefore cannot be expressed as a ratio of integers, and its decimal expansion neither terminates nor repeats."

In this lesson, we discovered that square roots are one of the most abundant sources of irrational numbers. The rule is elegantly simple: if the positive integer under the radical is a perfect square, its square root is a whole number and therefore rational; if it is not a perfect square, its square root is irrational. We also practiced writing concise justifications that point directly to the perfect-square test.

Up next are four practice exercises where you will identify perfect squares, match integers to the rational or irrational status of their square roots, classify a mixed set of square-root expressions, and write your own justification for a specific case. Let's put this test to work!