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Glacius
FoundationsConcept reference

Dividing signed numbers

The quotient of an integer by a nonzero integer is the number that multiplies the divisor to recover the dividend.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. Division asks for a missing factor
  3. Check your understanding
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01Division asks for a missing factor#

Division asks for a missing factor. To find −12÷3-12\div3, ask: what number times 33 gives −12-12? Three copies of −4-4 do, so the quotient is −4-4.

The quotient −4 is checked by multiplying it by the divisor 3 to recover −12.The quotient −4 is checked by multiplying it by the divisor 3 to recover −12.
Figure 1The quotient −4 is checked by multiplying it by the divisor 3 to recover −12.
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With two negative inputs, the missing factor is positive: (−3)×4=−12(-3)\times4=-12, so −12÷(−3)=4-12\div(-3)=4. For nonzero inputs, matching signs give a positive quotient; different signs give a negative one.

−12÷(−3)=4-12\div(-3)=4

Zero divided by a nonzero number is zero. Division by zero is undefined: no number times zero can recover 66. This lesson uses divisions with integer answers.

0÷(−3)=00\div(-3)=0
Check your reasoning

Calculate −28÷(−7)-28\div(-7).

Show answer and explanation
4

The quotient is 4 because (−7)×4=−28(-7)\times4=-28.

02Check your understanding#

Check your reasoning

Which expression is undefined?

  1. A0÷50\div5
  2. B5÷05\div0
  3. C−5÷5-5\div5
Show answer and explanation
5÷05\div0

The divisor cannot be zero. Zero divided by 5 is allowed and equals zero.

03Check your understanding#

Check your reasoning

A counter changes by −35-35 across 77 identical steps. What is the signed change per step?

Show answer and explanation
-5

Divide total change by the number of steps: −35÷7=−5-35\div7=-5.

Key takeaway

Check a quotient by multiplying it by the divisor.

  • Divide signed integers when the quotient is an integer and the divisor is nonzero

Further questions

Why is zero divided by zero undefined?
Every number multiplied by zero gives zero, so no single quotient is selected. Division requires one well-defined result.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Dividing signed numbers.” Math behind ML. /learn/f-divide-signed