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Reciprocals

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Reciprocals

Study Guide

Key Definition

The reciprocal or multiplicative inverse of a number $x$ is the number which, when multiplied by $x$, gives 1. For any non-zero number $n$, the reciprocal is $\frac{1}{n}$.

Important Notes

The product of a number and its reciprocal is always $1$
All real numbers have a reciprocal except $0$
The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$
Negative numbers also have reciprocals, e.g., the reciprocal of $-x$ is $-\frac{1}{x}$
Taking the reciprocal of a number twice returns the original number

Mathematical Notation

$\frac{1}{n}$ denotes the reciprocal of $n$

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

Remember to use proper notation when solving problems

Why It Works

Reciprocals work because multiplying a number by its reciprocal always equals $1$, demonstrating the concept of multiplicative inverses.

Remember

The reciprocal of $x$ is $\frac{1}{x}$, and multiplying them gives $1$.

Quick Reference

Property:$x \times \frac{1}{x} = 1$

Understanding Reciprocals

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The reciprocal of a number $x$ is $\frac{1}{x}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the reciprocal of $5$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have $\frac{3}{4}$ of a pizza and want to find its reciprocal.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If the reciprocal of $x$ is $\frac{1}{4}$, what is $x$?

Challenge Quiz

Single Choice Quiz

Advanced

What is the reciprocal of $-\frac{5}{8}$?

Recap

Watch & Learn

Review key concepts and takeaways