Rational Numbers

What You're Learning

In this section you will learn how to tell the difference between rational and irrational numbers, and how perfect squares and square roots connect to both categories.

Helpful Vocabulary

• Integer — any positive or negative whole number, including zero (…, $-2, -1, 0, 1, 2,$ …).

• Fraction — a number written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

• Decimal — another way to write a fraction (e.g., $0.75 = \frac{3}{4}$).

• Square root — a value that, when multiplied by itself, gives the original number ($\sqrt{n}$).

Rational Numbers

Definition

A number is rational if it can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

How to Recognize Them

Rational numbers have decimals that either:

• Terminate (end) — e.g., $0.5,\; 3.25,\; -7.0$

• Repeat a pattern forever — e.g., $0.\overline{3} = 0.333\ldots,\; 0.\overline{142857}$

Examples

• $6 = \frac{6}{1}$ ✓ rational

• $-\frac{2}{3}$ ✓ rational

• $0.\overline{7}$ ✓ rational (repeating decimal)

• $\sqrt{4} = 2 = \frac{2}{1}$ ✓ rational (perfect square)

Irrational Numbers

Definition

A number is irrational if it cannot be written as a fraction of two integers.

How to Recognize Them

The decimal form goes on forever without repeating.

Examples

• $\pi = 3.14159265\ldots$ ✗ never repeats → irrational

• $\sqrt{2} = 1.41421356\ldots$ ✗ never repeats → irrational

• $\sqrt{3},\; \sqrt{5},\; \sqrt{7}$ — all irrational (non-perfect-square radicands)

Perfect Squares

What Are They?

A perfect square is a number that is the result of an integer multiplied by itself.

List to Memorize

$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$

(i.e., $1^2, 2^2, 3^2, \ldots, 12^2$)

Why They Matter Here

• $\sqrt{\text{perfect square}}$ = integer → rational.

• $\sqrt{\text{non-perfect square}}$ = non-terminating, non-repeating decimal → irrational.

Negative Square Roots

• $\sqrt{25} = 5$ (the principal, or positive, square root).

• $-\sqrt{25} = -5$ (the negative square root).

• Both are rational because they equal integers.

• You cannot take the square root of a negative number using real numbers (e.g., $\sqrt{-4}$ is not real).

Common Mistakes

• Thinking all square roots are irrational — only roots of non-perfect squares are irrational.

• Thinking all decimals are irrational — terminating and repeating decimals are rational.

• Confusing "non-terminating" with "irrational" — $0.\overline{3}$ never ends but it repeats, so it's rational.

• Forgetting $0$ is rational — $0 = \frac{0}{1}$.

Quick Decision Checklist

1. Can the number be written as $\frac{a}{b}$ (integers, $b \neq 0$)?

   • Yes → rational.

2. Is it a square root?

   • Radicand is a perfect square → rational.

   • Radicand is not a perfect square → irrational.

3. Look at the decimal:

   • Terminates or repeats → rational.

   • Goes on forever without a pattern → irrational.