Multiplying Integers
Every question with a fully worked solution.
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Theory
To multiply integers, multiply the sizes, then decide the sign: the same signs give a positive and different signs give a negative. This Year 7 maths page shows why \(3 \times (-4) = -12\) and \(-3 \times (-4) = 12\), how to multiply several integers, how to square a negative number and how to find a missing factor.
Multiplying is repeated addition. \(3 \times (-4)\) means three lots of \(-4\): \((-4) + (-4) + (-4) = -12\).
\(-3 \times (-4)\) is the opposite of three lots of \(-4\), so it is the opposite of \(-12\), which is \(12\).
The sign rule: two integers with the same sign multiply to a positive; two integers with different signs multiply to a negative.
With many factors, count the negative factors. An even number of negatives gives a positive; an odd number gives a negative.
A power is repeated multiplication. \((-6)^2 = (-6) \times (-6) = 36\), but \(-6^2 = -(6 \times 6) = -36\): without brackets only the \(6\) is squared.
Signs of a product:
| \(\times\) | positive | negative |
|---|---|---|
| positive | positive | negative |
| negative | negative | positive |
The pattern below shows why a negative times a negative is positive. Each answer goes up by \(2\):
| Product | Answer |
|---|---|
| \(2 \times (-2)\) | \(-4\) |
| \(1 \times (-2)\) | \(-2\) |
| \(0 \times (-2)\) | \(0\) |
| \(-1 \times (-2)\) | \(2\) |
| \(-2 \times (-2)\) | \(4\) |
How to multiply two integers
- Multiply the sizes, ignoring the signs.
- Same signs: the answer is positive. Different signs: the answer is negative.
How to multiply several integers
- Work out any powers first.
- Count the negative factors: even gives positive, odd gives negative.
- Multiply the sizes from left to right and attach the sign.
How to find a missing factor
- Size: divide the size of the answer by the size of the known factor.
- Sign: choose the sign that makes the sign rule work.
- Check by multiplying.
\(3 \times (-5)\) means three lots of \(-5\): \((-5) + (-5) + (-5)\).
| \((-5) + (-5)\) | \(=\) | \(-10\) |
| \(-10 + (-5)\) | \(=\) | \(-15\) |
\(3 \times (-5) = -15\)
(i) \(-7 \times 6\)
(ii) \(-9 \times (-4)\)
(iii) Which is larger, \(-8 \times 3\) or \(-8 \times (-3)\)?
| \(-7 \times 6\) | \(=\) | \(-42\) |
| \(-9 \times (-4)\) | \(=\) | \(36\) |
| \(-8 \times 3\) | \(=\) | \(-24\) |
| \(-8 \times (-3)\) | \(=\) | \(24\) |
(i) Different signs: \(-42\)
(ii) Same signs: \(36\)
(iii) \(-8 \times (-3)\), because \(24 > -24\).
(i) \(-3 \times 2 \times (-5)\)
(ii) \(-2 \times (-3) \times (-7)\)
| \(-3 \times 2 \times (-5)\) | \(=\) | \(30\) |
| \(-2 \times (-3) \times (-7)\) | \(=\) | \(-42\) |
(i) Two negatives (even): \(30\)
(ii) Three negatives (odd): \(-42\)
(i) \((-5)^2\)
(ii) \(-5^2\)
(iii) \((-2)^3\)
| \((-5)^2\) | \(=\) | \((-5) \times (-5)\) |
| \(\) | \(=\) | \(25\) |
| \(-5^2\) | \(=\) | \(-(5 \times 5)\) |
| \(\) | \(=\) | \(-25\) |
| \((-2)^3\) | \(=\) | \((-2) \times (-2) \times (-2)\) |
| \(\) | \(=\) | \(-8\) |
(i) \(25\)
(ii) \(-25\)
(iii) \(-8\)
| \((-2)^2\) | \(=\) | \(4\) |
| \(4 \times (-3) \times 5\) | \(=\) | \(-12 \times 5\) |
| \(\) | \(=\) | \(-60\) |
Power first, then one negative factor is left (odd), so the answer is \(-60\).
(i) \(\square \times (-6) = 54\)
(ii) \(-4 \times \square = -36\)
(iii) Two integers multiply to \(-15\) and add to \(2\). Find them.
(i) Size \(54 \div 6 = 9\). The answer is positive, so the signs match: \(-9\).
(ii) Size \(36 \div 4 = 9\). The answer is negative and \(-4\) is negative, so the missing number is positive.
(iii) One integer is negative. Pairs that multiply to \(-15\): \(1\) and \(-15\), \(-1\) and \(15\), \(3\) and \(-5\), \(-3\) and \(5\). Only \(5 + (-3) = 2\).
| \(-9 \times (-6)\) | \(=\) | \(54\) |
| \(-4 \times 9\) | \(=\) | \(-36\) |
(i) \(-9\)
(ii) \(9\)
(iii) \(5\) and \(-3\)
| \(7 \times (-2)\) | \(=\) | \(-14\) |
| \(3 + (-14)\) | \(=\) | \(-11\) |
A repeated drop is a product. The water is at \(-11\) cm, which is \(11\) cm below the marker.
Common pitfalls
Frequently asked questions
Why is a negative times a negative positive?
Follow the pattern \(1 \times (-2) = -2\), \(0 \times (-2) = 0\), \(-1 \times (-2) = 2\): each answer goes up by 2.
What is a positive times a negative?
A negative. \(5 \times (-3) = -15\).
How do you multiply three or more integers?
Count the negative factors: an even number gives a positive answer and an odd number gives a negative answer.
Is (-4) squared the same as -4 squared?
No. \((-4)^2 = 16\) but \(-4^2 = -16\), because without brackets only the 4 is squared.
How do you find a missing factor?
Divide the sizes, then choose the sign that fits. In \(\square \times (-6) = 54\), the missing factor is \(-9\).