Master

Irrational Numbers

Master irrational numbers with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Irrational Numbers

Study Guide

Key Definition

An irrational number is any real number that cannot be expressed in the form $\frac{a}{b}$ when $a$ and $b$ are integers and $b \neq 0$. In decimal form, they never terminate or repeat.

Important Notes

Irrational numbers cannot be written as a fraction of two integers.
The decimal form of an irrational number is non-terminating and non-repeating.
The square root of any non-perfect square is an irrational number, like $\sqrt{2}$.
The number $\pi$ is a well-known irrational number.
Irrational numbers are real numbers.

Mathematical Notation

$\frac{a}{b}$ represents a fraction

$\sqrt{x}$ represents the square root

$\pi$ represents the constant pi

$\neq$ means not equal to

Remember to use proper notation when solving problems

Why It Works

Irrational numbers are essential because they complete the real number system, allowing us to solve equations like $x^2 = 2$.

Remember

Irrational numbers include numbers like $\sqrt{2}$ and $\pi$, which are necessary for many mathematical calculations.

Quick Reference

Square Root of 2:$\sqrt{2} \approx 1.414$

Pi:$\pi \approx 3.14159$

Understanding Irrational Numbers

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

An irrational number cannot be expressed as a fraction of integers. Its decimal expansion never terminates or repeats. For example, $\sqrt{2} ≈ 1.4142135...$ continues without pattern, so it’s irrational.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following is an irrational number?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have a square with side length 1. What is the length of the diagonal?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Prove that $\pi$ is an irrational number.

Challenge Quiz

Single Choice Quiz

Advanced

Which expression is irrational?

Recap

Watch & Learn

Review key concepts and takeaways