Can you divide zero by a negative integer?

Yes, you can divide zero by any nonzero integer, including negative integers. The result of dividing zero by any nonzero number is always zero. For example, 0-8 equals zero. This is because zero divided into any number of parts still results in zero. However, division by zero is undefined, so the divisor must not be zero.

Why is division by zero undefined?

Division by zero is undefined because it does not produce a meaningful or consistent result. When dividing a number by zero, there is no number that you can multiply by zero to get the original number (except zero itself). For example, -40 is undefined because no number multiplied by zero equals -4. This breaks the fundamental properties of division and multiplication, so division by zero is not allowed in mathematics.

2. Integers

Dividing Integers

2. Integers

Dividing Integers: Videos & Practice Problems

Video Lessons

Topic summary

Dividing Integers uses the same sign rules as multiplying integers because multiplication and division are closely related. When dividing integers with the same sign, the quotient is positive. When dividing integers with different signs, the quotient is negative. A helpful way to work is to divide the absolute values first, then decide the sign from the sign rules.

Key division terms include dividend, divisor, and quotient. You can check whether a division result is correct by using the related multiplication fact: $q \times d = n$ . Standard whole-number division ideas still apply to integers: division is done from left to right, dividing by $1$ keeps the number the same, dividing by $-1$ changes the sign, and $0$ divided by a nonzero integer is $0$.

A number can also be written in an algebraic expression as $\frac{x}{-16}$ . However, division by $0$ is undefined, so any expression with a zero divisor has no value.

Concept

Dividing Integers

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Problem

Divide.
(A) $\frac{- 24}{- 6}$

$4$

$negative 4$

$\frac{12}{3}$

$\frac{- 12}{- 3}$

Problem

Divide.
(B)

$17$

$21$

$negative 21$

$negative 17$

Problem

Divide.
(C) $\frac{- 16}{1}$

$16$

$negative 16$

$1$

$negative 1$

Problem

Translate each phrase, then simplify.
(A) The quotient of $negative 42$ and $6$

$\frac{1}{- 7}$

$\frac{1}{7}$

$7$

$negative 7$

Problem

Translate each phrase, then simplify.
(B) $300$ divided by $negative 3$

$1 over 100$

$100$

$negative 100$

$\frac{1}{- 100}$

Example

Dividing Integers Example 1

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Problem

Translate each phrase to an algebraic expression. Use $x$ to represent “a number.”
(A) Divide a number by $negative 16$

$\frac{x}{- 16}$

$\frac{- 16}{x}$

$negative 16 x$

The phrase can not be translated to algebraic expression

Problem

Translate each phrase to an algebraic expression. Use $x$ to represent “a number.”
(B) The product of $negative 5$ and a number.

$\frac{x}{- 5}$

$\frac{- 5}{x}$

$negative 5 x$

The phrase can not be translated to algebraic expression

Problem

If I owe \$480 on a loan that will be paid in equal installments over 6 months, what will be the amount charged each month?

-\$80

\$80

-\$8

\$8

Here's what students ask on this topic:

When dividing integers, the sign of the quotient depends on the signs of the dividend and divisor. If both integers have the same sign (both positive or both negative), the quotient is positive. If the integers have different signs (one positive and one negative), the quotient is negative. This rule is consistent with the sign rules for multiplying integers because multiplication and division are closely related operations. To simplify, you can divide the absolute values of the numbers first, then apply the sign rule to determine the final sign of the quotient.

To divide integers with different signs, first divide their absolute values as if they were positive numbers. Then, since the signs are different, the quotient will be negative. For example, dividing $- 18$ by $3$ involves dividing 18 by 3 to get 6, and because the signs differ, the final answer is $- 6$ . This follows the rule that dividing integers with different signs results in a negative quotient.

Yes, you can divide zero by any nonzero integer, including negative integers. The result of dividing zero by any nonzero number is always zero. For example, $\frac{0}{-}$ equals zero. This is because zero divided into any number of parts still results in zero. However, division by zero is undefined, so the divisor must not be zero.

Dividing any integer by $- 1$ changes the sign of that integer. For example, dividing $50$ by $- 1$ results in $- 50$ . This is because dividing by $- 1$ is equivalent to multiplying by $- 1$ , which flips the sign of the number. This property is useful when simplifying expressions or solving equations involving division by negative one.

Division by zero is undefined because it does not produce a meaningful or consistent result. When dividing a number by zero, there is no number that you can multiply by zero to get the original number (except zero itself). For example, $\frac{-}{4}$ is undefined because no number multiplied by zero equals $- 4$ . This breaks the fundamental properties of division and multiplication, so division by zero is not allowed in mathematics.