Lesson 1: Operations on Real Numbers

Property

A rational number is a number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. All fractions, both positive and negative, are rational numbers.
Since any integer, terminating decimal, or repeating decimal can be written as a ratio of two integers, they are all rational numbers.

Examples

• To write the integer $-25$ as a ratio of two integers, express it as a fraction with a denominator of 1: $\frac{-25}{1}$.
• The decimal $9.37$ can be written as a mixed number $9\frac{37}{100}$, which converts to the improper fraction $\frac{937}{100}$.
• The mixed number $-4\frac{2}{3}$ is equivalent to the improper fraction $-\frac{14}{3}$.

Explanation

Think of 'rational' as 'ratio-nal.' Any number that can be expressed as a simple fraction or ratio between two integers is a rational number. This includes whole numbers, integers, and decimals that either end or repeat predictably.

Property

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.
Famous examples include $\pi$ and the square roots of numbers that are not perfect squares.

Examples

• The number $\pi$ is a famous irrational number, beginning with $3.14159...$ and continuing infinitely without repetition.
• The square root of 3, $\sqrt{3}$, is irrational because 3 is not a perfect square. Its decimal form is $1.7320508...$.
• A decimal that continues without a pattern, such as $67.121231234...$, is an irrational number.

Explanation

Irrational numbers cannot be written as a simple fraction. Their decimal representations are infinite and non-repeating, meaning they go on forever without any predictable pattern. Think of them as the 'wild' numbers on the number line.