Example The Burning Candle
Consider the candle of length 18 cm which burns in 6 hrs. Then the
rate of burning is
18 cm
6 hr
= 3 cm/hr
.
Now if L is the length of that candle, L is changing at the
rate of -3 cm/hr.
Observe the candle when L = 15 cm. Ask the
question: What was the length of the candle 1 hour ago?
L
= 15 + (rate)·(timechange)
= 15 + (-3)·(-1)
= 18.
Thus (-3)·(-1) = +3.
Discussion of Products
We need to think about the meaning of multiplication. However we
interpret multiplication by positive integers, we should have a distributive
property , e. g.
(3 + 2)·a = 3 ·a + 2 ·a.
The trick is to interpret or "define" multiplication by negative
numbers in a way that the distributive property is maintained. If
we think of +1 ·a as adding a to whatever, we
could define -1 ·a as "undoing" the addition, i.e.
adding -a to whatever.
The crucial mathematical fact is that for a number a, the number -a,
negative or opposite of a is the same as the product
(-1)·a; i.e.,
-a
= -1·a.
The matter is rather delicate. If there is justice, we have the
distributive property, a(b + c) = ab + ac. This tells us that
a + -a
= 0
= 0·a
= (1 + -1)·a
= 1 ·a + -1·a
= a + -1·a.
It follows that
-a
= -1·a.
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