rational and irrational numbers, math for prep 2

Rational and Irrational Numbers

A Complete Guide to Understanding Rational and Irrational Numbers with Practical Examples

Published: July 8, 2025

🔢 Rational Numbers

Definition of Rational Numbers

A rational number is any number that can be written in the form $\frac{a}{b}$ where $a$ and $b$ are integers, and $b \neq 0$

Rational Number = $\frac{a}{b}$ where $a$, $b \in \mathbb{Z}$ and $b \neq 0$

The set of rational numbers is denoted by the symbol $\mathbb{Q}$

Examples of Rational Numbers

$3.\overline{6} = \frac{11}{3}$

$5 = \frac{5}{1}$

$-12 = \frac{-12}{1}$

$0.5 = \frac{1}{2}$

$0.25 = \frac{1}{4}$

🔣 Irrational Numbers

Definition of Irrational Numbers

An irrational number is a number that cannot be written in the form $\frac{a}{b}$ where $a$ and $b$ are integers, and $b \neq 0$

Irrational Number $\neq \frac{a}{b}$ where $a$, $b \in \mathbb{Z}$ and $b \neq 0$

The set of irrational numbers is denoted by the symbol $\mathbb{Q}'$

Note: The calculator displays an approximate value for these numbers, not the exact value. Therefore, such numbers cannot be written as rational numbers.

Examples of Irrational Numbers

Square Root: Any number that is not a perfect square, like $\sqrt{2}$ ≈ 1.414213562...

Where $\sqrt{2}$ ≈ 1.414213562...

Cube Root: Any number that is not a perfect cube, like $\sqrt[3]{5}$ ≈ 1.709975947...

Where $\sqrt[3]{5}$ ≈ 1.709975947...

📝 Comprehensive Example

Question: Which of the following numbers is rational and which is irrational, with reasoning:

$\sqrt[3]{-27}$ , $\sqrt{13}$ , $\sqrt{25}$ , $-12$ , $3.\overline{6}$ , $0.125$ , $-\sqrt[3]{10}$ , $2\frac{2}{5}$

Think: Can each of these numbers be written as rational and irrational?

Solution:

Each of the following numbers is rational:

$3.\overline{6}$ , $0.125$ , $2\frac{2}{5}$ , $\sqrt[3]{-27}$ , $\sqrt{25}$ , $-12$

Because each can be written in the form $\frac{a}{b}$ where $a$ and $b$ are integers, $b \neq 0$, as follows:

$\sqrt{25} = 5 = \frac{5}{1}$

$-12 = \frac{-12}{1}$

$2\frac{2}{5} = \frac{12}{5}$

$\sqrt[3]{-27} = -3 = \frac{-3}{1}$

$3.\overline{6} = \frac{11}{3}$

$0.125 = \frac{125}{1000} = \frac{1}{8}$

Remember: The number $3.\overline{6}$ is a **repeating decimal** and is a rational number.
$3.\overline{6} = 3.6666... = \frac{11}{3}$

The remaining numbers are irrational:

$\sqrt{13}$ ≈ 3.605551275...

Not a perfect square

$-\sqrt[3]{10}$ ≈ -2.154434690...

Not a perfect cube

💡 Important Tips to Remember

Important Note: The calculator displays approximate values for irrational numbers, not exact values. Therefore, such numbers cannot be written as rational numbers.

Square Root Examples

Any number that is not a perfect square such as:

$\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7}, \sqrt{8}, \sqrt{10}, \sqrt{11}, \sqrt{12}...$

Cube Root Examples

Any number that is not a perfect cube such as:

$\sqrt[3]{2}, \sqrt[3]{3}, \sqrt[3]{4}, \sqrt[3]{5}, \sqrt[3]{6}, \sqrt[3]{7}, \sqrt[3]{9}, \sqrt[3]{10}...$

🎯 Lesson Summary

Key Points

• Rational numbers can be written as fractions $\frac{a}{b}$
• Irrational numbers cannot be written as fractions
• Terminating and repeating decimals are rational numbers
• Non-perfect square roots and cube roots are irrational numbers
• $\pi$ and $e$ are among the most famous irrational numbers
• The set of all real numbers consists of both rational and irrational numbers

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Created by Mr. Ayman Haasn