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Glacius
FoundationsConcept reference

Multiplying signed numbers

A product of two nonzero signed integers is positive when their signs match and negative when they differ.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. Multiplying 3 by -2 adds three copies of -2
  3. Check your understanding
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01Multiplying 3 by -2 adds three copies of -2#

Multiplying 33 by −2-2 adds three copies of −2-2. The total is −6-6. Reversing the factor order gives the same product.

(−2)+(−2)+(−2)=−6\begin{gathered}(-2)+(-2)+(-2)\\=-6\end{gathered}

Decrease the first factor in multiples of −3-3: 22, 11, 00, then −1-1. The products increase by 33 each time. Continuing that pattern past zero gives a positive product for two negative factors.

As the first factor falls by one, the product increases by three: −6, −3, 0, 3.As the first factor falls by one, the product increases by three: −6, −3, 0, 3.
Figure 1As the first factor falls by one, the product increases by three: −6, −3, 0, 3.
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For two nonzero factors, matching signs give a positive product; different signs give a negative product. Multiply the sizes, then choose the sign. If either factor is zero, the product is zero.

(−4)×(−3)=12(-4)\times(-3)=12
Check your reasoning

Calculate (−5)×(−2)(-5)\times(-2).

Show answer and explanation
10

Two negative factors give a positive product; 5×2=105\times2=10.

02Check your understanding#

Check your reasoning

What is 0×(−8)0\times(-8)?

  1. A−8-8
  2. B88
  3. C00
Show answer and explanation
00

Zero copies contribute zero. A zero factor makes the product zero.

03Check your understanding#

Check your reasoning

A process repeats 66 identical steps. Each step changes a counter by −2-2. What is the total counter change?

Show answer and explanation
-12

Six changes of −2 give 6×(−2)=−126\times(-2)=-12.

Key takeaway

Matching signs give a positive product; different signs give a negative product.

  • Multiply two signed integers

Further questions

Can a negative times a negative be negative?
For two ordinary negative real numbers, their product is positive. Zero is neither positive nor negative and has its own zero-product case.

Sources & further reading

  1. [1]

Reference this concept

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

Glacius. “Multiplying signed numbers.” Math behind ML. /learn/f-multiply-signed