Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 1: Properties of Arithmetic

Lesson 6: Reciprocals

Property.

Section 1

Multiplicative Inverse (Reciprocal) Definition

Property

Multiplicative Inverse (Reciprocal): For any non-zero real number aa, there is a reciprocal, 1a\frac{1}{a}, such that their product is one.

a⋅1a=1(a≠0)a \cdot \frac{1}{a} = 1 \quad (a \neq 0)

This property is essential for solving equations and simplifying expressions involving multiplication and division.

Section 2

Reciprocal of a Reciprocal

Property

The reciprocal of a reciprocal returns the original number:

11x=x\frac{1}{\frac{1}{x}} = x

Examples

Section 3

Reciprocal of a Product

Property

The reciprocal of a product equals the product of the reciprocals:

1xy=1x×1y\frac{1}{xy} = \frac{1}{x} \times \frac{1}{y}

Examples

Section 4

Reciprocals of Negative Numbers

Property

The reciprocal of a negative number equals the negative of the reciprocal:

1−x=−1x\frac{1}{-x} = -\frac{1}{x}

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Multiplicative Inverse (Reciprocal) Definition

Property

Multiplicative Inverse (Reciprocal): For any non-zero real number aa, there is a reciprocal, 1a\frac{1}{a}, such that their product is one.

a⋅1a=1(a≠0)a \cdot \frac{1}{a} = 1 \quad (a \neq 0)

This property is essential for solving equations and simplifying expressions involving multiplication and division.

Section 2

Reciprocal of a Reciprocal

Property

The reciprocal of a reciprocal returns the original number:

11x=x\frac{1}{\frac{1}{x}} = x

Examples

Section 3

Reciprocal of a Product

Property

The reciprocal of a product equals the product of the reciprocals:

1xy=1x×1y\frac{1}{xy} = \frac{1}{x} \times \frac{1}{y}

Examples

Section 4

Reciprocals of Negative Numbers

Property

The reciprocal of a negative number equals the negative of the reciprocal:

1−x=−1x\frac{1}{-x} = -\frac{1}{x}

Examples