Lesson 1: Rational and Irrational Numbers

New Concept

This lesson expands your number toolkit by distinguishing between rational numbers (writable as $\frac{p}{q}$) and irrationals (non-repeating/non-terminating decimals). Mastering this helps you classify every real number you encounter.

What’s next

Now, you'll put this into practice! Get ready for a series of interactive examples and challenge problems designed to make you an expert at classifying 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.

Property

To determine if a number is rational or irrational, examine its decimal form.

• If the decimal form of a number stops or repeats, the number is rational.
• If the decimal form of a number does not stop and does not repeat, the number is irrational.

Examples

• The number $0.777...$ or $0.\overline{7}$ is rational because its decimal form repeats.
• The number $\sqrt{64}$ is rational because it simplifies to the integer $8$.
• The number $4.1234567...$ is irrational because the ellipsis indicates it continues without stopping or repeating.

Explanation

A simple test for classifying a number is to look at its decimal form. If the decimal terminates (ends) or has a repeating block of digits, it's rational. If it goes on forever with no pattern, it's irrational.

Property

Real numbers are numbers that are either rational or irrational. The set of real numbers is formed by combining the set of rational numbers and the set of irrational numbers. It includes counting numbers, whole numbers, integers, fractions, and decimals.

Examples

• The number $100$ is a whole number, an integer, a rational number, and a real number.
• The number $\frac{17}{4}$ is a rational number and a real number.
• The number $-\sqrt{15}$ is an irrational number and a real number.

Explanation

Real numbers represent all the points on the number line. This is a vast category that includes every number you've worked with so far, from neat integers to messy irrational numbers. Everything rational or irrational is a real number.