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

Book overview

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Chapter 1: Properties of Arithmetic

  1. Lesson 1

    Lesson 1: Why Start with Arithmetic?

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  2. Lesson 2

    Lesson 2: Addition

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  3. Lesson 3

    Lesson 3: Multiplication

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  4. Lesson 4

    Lesson 4: Negation

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  5. Lesson 5

    Lesson 5: Subtraction

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  6. Lesson 6Current

    Lesson 6: Reciprocals

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  7. Lesson 7

    Lesson 7: Division

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Lesson overview

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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

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 1: Properties of Arithmetic

  1. Lesson 1

    Lesson 1: Why Start with Arithmetic?

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  2. Lesson 2

    Lesson 2: Addition

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  3. Lesson 3

    Lesson 3: Multiplication

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  4. Lesson 4

    Lesson 4: Negation

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  5. Lesson 5

    Lesson 5: Subtraction

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  6. Lesson 6Current

    Lesson 6: Reciprocals

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  7. Lesson 7

    Lesson 7: Division

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