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Rationals (Advanced)

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Rationals (Advanced)

Study Guide

Key Definition

Rational numbers are numbers that can be expressed as a ratio $\frac{a}{b}$ where $a$ and <a href="integers"> $b$ are integers</a> and $b \neq 0$.

Important Notes

All integers are rational numbers because they can be expressed as $\frac{z}{1}$.
Decimals that terminate are rational numbers because they can be expressed as fractions.
Repeating decimals are also rational numbers because they can be expressed as fractions.
The set of rational numbers is closed under addition, subtraction, multiplication, and division (except division by zero).
If a number cannot be expressed as a ratio of two integers, it is called irrational.

Mathematical Notation

$\frac{a}{b}$ represents a fraction where $a$ is the numerator and $b$ is the denominator

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

The operations on rational numbers work because any operation on a ratio of two integers results in another ratio of integers, maintaining the property of rationality: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$.

Remember

Always simplify fractions to their lowest terms using $\text{GCD}$ (Greatest Common Divisor).

Quick Reference

Closure Property:Rational numbers are closed under $+$, $-$, $\times$, and $\div$ (except by zero).

Understanding Rationals (Advanced)

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Rational numbers can be expressed as simple fractions like $\frac{1}{2}$.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

Which of the following is a rational number? $\sqrt{2}$, $\frac{3}{4}$, $\pi$, $\sqrt{3}$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You bought a pizza and ate $\frac{1}{3}$ of it. What fraction of the pizza is left?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Prove that the sum of two rational numbers is rational with $\frac{a}{b} + \frac{c}{d}$.

Challenge Quiz

Single Choice Quiz

Advanced

Which option is the fraction form of the repeating decimal $0.\overline{666}$?

Recap

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