3.19 Multiplication with negative numbers
Now that you have rules for addition and subtraction of negative numbers, think about multiplication and division.
Example 27
Describe each of the following in terms of the number line and the value of Thomas’s piggy bank:
(a) the multiplication of 3 times 5 (that is, 3 × 5);
(b) the multiplication of 3 times −5 (that is, 3 × −5).
Answer
(a) Three times 5 means adding three lots of 5, or do the move ‘5 units to the right’ three times.
Thomas’s piggy bank has three £5 notes in it. The value of the piggy bank in pounds is 3 × 5 = 15.
(b) 3 times −5 means adding three lots of −5, or do 3 times ‘move 5 units to the left’.
Thomas’s piggy bank has 3 IOUs each for £5 in it. The value of the piggy bank in pounds is 3 × −5 = −15.
So multiplying a negative number by a positive number gives a negative answer. But what does it mean to multiply by a negative number – how can you add something a negative number of times?
Think of multiplication by a positive number as repeated addition, and of multiplication by a negative number as repeated subtraction. If Thomas’ mother says she’ll take away three of his £5 IOUs, it is equivalent to giving him three £5 notes. So −3 × −5 = 3 × 5 = 15. He is £15 better off.
Example 28
Evaluate each of the following in terms of the number line and the value of Thomas’s piggy bank.
(a) 5 × −3
(b) −5 × −3
Answer
(a) Think of 5 times −3 as meaning ‘5 lots of −3’. So 5 × −3 means: do 5 times ‘move 3 units to the left’.
So 5 × −3 = −15.
In terms of the piggy bank Thomas has five £3 IOUs in it. The value in pounds is 5 × −3 = −15, which is equivalent to one £15 IOU.
(b) Thinking in terms of repeated subtraction, do 5 times ‘subtract −3’. Subtracting −3 means adding 3, or moving 3 units to the right. So do 5 times ‘move 3 units to the right’.
Subtracting −3 is the same as adding 3.
So −5 × −3 = 5 × 3 = 15.
Thomas’s mother takes away five £3 IOUs. This is equivalent to giving him £15.
The calculation in pounds is −5 × −3 = 5 × 3 = 15.