MISSION · Number sense
Multiplying and Dividing Negatives
Why two negatives make a positive
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LEVEL 2 · LEARN
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The idea in one sentence
Multiplying by a negative reverses direction, so multiplying by two negatives reverses twice and lands you facing forwards again.
The rules
\((+) \times (+) = (+) \qquad (+) \times (-) = (-)\) \((-) \times (+) = (-) \qquad (-) \times (-) = (+)\)
Division behaves identically. Same signs give a positive, different signs give a negative — one sentence covering both operations.
For longer strings, just count the negatives. An even number of them gives a positive answer, an odd number gives a negative one. So \((-2)(-3)(-1)\) has three negatives and comes out negative: \(-6\).
Why negative times positive is negative
\(3 \times (-4)\) is three lots of \(-4\), which is \(-12\). That one is just repeated addition and nobody objects to it.
Why negative times negative is positive
This is the one that needs an argument, because “repeated addition” makes no sense a negative number of times.
The pattern argument. Watch a column count down:
\(3 \times (-4) = -12\) \(2 \times (-4) = -8\) \(1 \times (-4) = -4\) \(0 \times (-4) = 0\) \((-1) \times (-4) = \;?\)
Each step up the column adds 4. Keeping the pattern going, the next answer must be \(+4\). Any other value breaks the arithmetic that the rows above already obey.
The direction argument. Multiplying by \(-1\) flips a number to the other side of zero. Flip twice and you’re back where you started, so \(-(-4) = 4\). That’s the same fact as “taking away a debt leaves you better off” — removing a negative is a positive change.
The distributive argument, for anyone still unconvinced. We know \(4 + (-4) = 0\), so multiplying that whole thing by \(-1\) must still give 0:
\[(-1)(4) + (-1)(-4) = 0\]The first term is \(-4\), so the second must be \(+4\) for the total to be zero. Nothing else works.
The rules aren’t a choice
That last argument is the important one. Mathematicians did not decide that two negatives make a positive because it seemed tidy. Any other rule would break the distributive property — and with it ordinary algebra, the ability to expand brackets, and most of arithmetic. The rule is forced.
Common mix-ups
- Applying addition rules to multiplication. \(-3 + -4 = -7\), but \(-3 \times -4 = +12\). Different operations, different rules.
- Mishandling \((-3)^2\) against \(-3^2\). The first is 9; the second is \(-9\), because the power binds tighter than the minus sign.
- Losing a negative in a long chain. Count them.
- Assuming a negative answer must be wrong. It’s often the right one.
Check yourself
Work out \((-6) \times (-2) \div (-3)\), and explain \((-5)^2\).
\((-6)(-2) = 12\), then \(12 \div (-3) = -4\) — three negatives, so a negative answer. \((-5)^2 = (-5)(-5) = 25\), since the brackets mean the whole of \(-5\) is squared.
LEVEL 3 · REFLECT
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