Operations with Signed Numbers
So far this semester, we have dealt only with positive numbers and zero. We now extend our
number system to include negative numbers as well by making the following definition:
Definition: For any integer x, the integer “negative x”, denoted by
integer with the property that
. In other words,
, is the unique
is the additive inverse of
x.
1. Give examples of situations in which negative numbers arise naturally.
One way to represent positive and negative integers is to use filled in circles to represent
positive numbers and open circles to represent negative numbers (This is sometimes called the
chip model for representing signed integers.) For example we would represent 5 by ●●●●● and
by ○○○○○. We think of each white dot as canceling a black dot, so that each integer can be
represented in many ways. For example two different ways to represent 0 are: ●○ and
●●●○○○. Use the chip model to represent
in two different ways.
Another way to represent signed numbers is by using a number line. A positive number n is a
jump to the right of length n units (from any starting point on the number line). If n is positive,
then
(negative n) is represented by a jump to the left of length n units. As before, the
starting point does not matter. Use a number line to represent
in two different ways.
2. Use one of the models above (or a different model for signed numbers, if you prefer) to
illustrate following statements:
(a) The sum of two positive numbers is positive.
(b) The sum of two negative numbers is negative.
3. Recall that subtracting
can be thought of as taking b units away from a (the “take-
away” view of subtraction) or it can be thought of as the number you must add to b to get
a (the “missing addend” view of subtraction). Use a model of signed numbers and one or
more ways of viewing subtraction to illustrate the property that
.
(Note
that once this fact has been established, it is no longer necessary to distinguish between
the minus sign and the negative sign.)
4. Explain, why subtracting a negative number is the same as adding a positive number.
Illustrate for the following differences. In each case, make up a problem whose solution
is the given difference.
(a)
(b)
5. Recall that multiplication by an integer can be thought of as repeated addition. For
example
. Use this view of multiplication and a model of signed
numbers to explain why the product of a positive integer and a negative integer is
negative. Illustrate with
. How would you interpret the product
? What
property of multiplication are you using?
6. Can you explain why the product of two negative numbers is positive? In particular,
how would you convince a skeptic that
? We don’t have a good model
for the product of two negative numbers, but here are a couple ideas you might try:
(i)
Look for a pattern: Multiply
pattern you see, what should
?
,
,
be? What about
,
. Based on the
?
?
(ii)
Use the properties of integers: What is
? What expression do you
get when you expand this product using the distributive law? Use the result of
problem 8 and the definition of the additive inverse to show that
7. What can you say about
when
(i)
a and b are both positive
(ii)
a is positive and b is negative
(iii)
a is negative and b is positive
(iv)
a and b are both negative
Justify your responses.