Lesson 2-3: Real Numbers

Property

Together, rational and irrational numbers make up the real numbers. A rational number can be written as the ratio of two integers, $\frac{a}{b}$ where $b \neq 0$, and its decimal form either stops or repeats. An irrational number cannot be written as a ratio of two integers, and its decimal form never stops and never repeats. When a positive integer is not a perfect square, its square root is an irrational number.

Examples

• The numbers $5$, $-\frac{3}{8}$, and $0.333...$ are rational because they can be written as fractions ($\frac{5}{1}$, $-\frac{3}{8}$, $\frac{1}{3}$) and their decimals terminate or repeat. $\sqrt{81}$ is also rational because $9^2 = 81$, so $\sqrt{81} = 9$.
• The number $\sqrt{50}$ is irrational because $50$ is not a perfect square, so its decimal form goes on forever without repeating.
• $\sqrt{2}$, $\sqrt{7}$, and $\sqrt{15}$ are all irrational numbers because the numbers under the radical are not perfect squares.

Explanation

Property

Real Numbers are composed of Rational Numbers (can be written as a fraction $\frac{a}{b}$) and Irrational Numbers (cannot). Rational numbers contain Integers, which include Whole Numbers, which in turn include Natural Numbers.

Examples

• The number $9$ is a Natural, Whole, Integer, Rational, and Real number.
• The number $-4.5$ is a Rational and Real number because it can be written as $-\frac{9}{2}$.
• The number $5\pi$ is an Irrational and Real number because its decimal is endless and non-repeating.

Explanation

Think of it like a big family tree! At the top, you have the Real Numbers. This family has two main branches: the Rationals and the Irrationals. The Rational branch has its own smaller families nested inside. The Integers are part of the Rationals, the Wholes are part of the Integers, and the Naturals are the babies of the whole group.