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

Lesson 7: Division

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra, students learn the formal definition of division as multiplication by a reciprocal, expressed as a ÷ b = a · (1/b). The lesson covers key properties including why division by zero is undefined, how division distributes over addition and subtraction, and the rules for dividing negative numbers. Students also explore concepts such as canceling common factors and why division is neither commutative nor associative.

Section 1

Division as the Inverse of Multiplication

Property

Division is the inverse operation of multiplication. We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend.

Examples

To solve $56 \div 7$ , we can think 'what number times 7 equals 56?'. The answer is 8. Check: $8 \cdot 7 = 56$ .
To solve $\frac{48}{8}$ , we can think 'what number times 8 equals 48?'. The answer is 6. Check: $6 \cdot 8 = 48$ .
To solve $54 \div 9$ , we can think 'what number times 9 equals 54?'. The answer is 6. Check: $6 \cdot 9 = 54$ .

Explanation

Division and multiplication are opposites that undo each other. You can use your multiplication facts to solve division problems and to check if your answer is correct. It's a great way to be sure you have the right quotient!

Section 2

Division with zero

Property

Division of Zero: For any real number $a$ , $a \neq 0$

\frac{0}{a} = 0

Zero divided by any real number, except itself, is zero.

Division by Zero: For any real number $a$ ,

\frac{a}{0} \text{ is undefined.}

Division by zero is undefined.

Examples

The expression $\frac{0}{15}$ simplifies to $0$ , as zero divided by any non-zero number is zero.
The expression $\frac{-8}{0}$ is undefined, because division by zero is not a defined operation in mathematics.
For any non-zero value of $k$ , the expression $0 \div k$ will always equal $0$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Division as the Inverse of Multiplication

Property

Division is the inverse operation of multiplication. We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend.

Examples

To solve $56 \div 7$ , we can think 'what number times 7 equals 56?'. The answer is 8. Check: $8 \cdot 7 = 56$ .
To solve $\frac{48}{8}$ , we can think 'what number times 8 equals 48?'. The answer is 6. Check: $6 \cdot 8 = 48$ .
To solve $54 \div 9$ , we can think 'what number times 9 equals 54?'. The answer is 6. Check: $6 \cdot 9 = 54$ .

Explanation

Section 2

Division with zero

Property

Division of Zero: For any real number $a$ , $a \neq 0$

\frac{0}{a} = 0

Zero divided by any real number, except itself, is zero.

Division by Zero: For any real number $a$ ,

\frac{a}{0} \text{ is undefined.}

Division by zero is undefined.

Examples

The expression $\frac{0}{15}$ simplifies to $0$ , as zero divided by any non-zero number is zero.
The expression $\frac{-8}{0}$ is undefined, because division by zero is not a defined operation in mathematics.
For any non-zero value of $k$ , the expression $0 \div k$ will always equal $0$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Division as the Inverse of Multiplication

Property

Division is the inverse operation of multiplication. We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend.

Examples

To solve $56 \div 7$ , we can think 'what number times 7 equals 56?'. The answer is 8. Check: $8 \cdot 7 = 56$ .
To solve $\frac{48}{8}$ , we can think 'what number times 8 equals 48?'. The answer is 6. Check: $6 \cdot 8 = 48$ .
To solve $54 \div 9$ , we can think 'what number times 9 equals 54?'. The answer is 6. Check: $6 \cdot 9 = 54$ .

Explanation

Section 2

Division with zero

Property

Division of Zero: For any real number $a$ , $a \neq 0$

\frac{0}{a} = 0

Zero divided by any real number, except itself, is zero.

Division by Zero: For any real number $a$ ,

\frac{a}{0} \text{ is undefined.}

Division by zero is undefined.

Examples

The expression $\frac{0}{15}$ simplifies to $0$ , as zero divided by any non-zero number is zero.
The expression $\frac{-8}{0}$ is undefined, because division by zero is not a defined operation in mathematics.
For any non-zero value of $k$ , the expression $0 \div k$ will always equal $0$ .