Reciprocals - Varsity Tutors

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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 divisionRemember 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

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Beginner Explanation

The reciprocal of a number $x$ is $\frac{1}{x}$.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

What is the reciprocal of $5$?

A$\frac{1}{5}$B$5$C$\frac{1}{10}$D$10$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

You have $\frac{3}{4}$ of a pizza and want to find its reciprocal.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$-\frac{8}{5}$B$\frac{5}{8}$C$\frac{8}{5}$D$-\frac{5}{8}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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