Page 49 - ArithBook5thEd ~ BCC

Chapter 2

Signed Numbers

We now start the process of extending the arithmetic of whole numbers to larger sets of numbers. In this
chapter, we include the negative numbers. We mostly concentrate on the integers,which consist of
whole numbers, 0, and the negative whole numbers. Later on, we’ll put fractions and other non-integers
into the mix.
The negative numbers form a kind of “mirror image” of the positive numbers, lying to the left of 0
on the number line. To distinguish them from the positive numbers, we give them a negative sign, −.
Thus, −3denotes the number “negative 3,”which lies 3units tothe left of 0.

| | | | | | | | | | |
−5 −4 −3 −2 −1 0 1 2 3 4 5
The positive numbers have the sign +, but we usually don’t write it, except for emphasis. Thus +5
means “positive 5,” but we usually just write 5.

2.1 Absolute value

The absolute value of a number is its distance from 0 on the number line. For example, −4lies at
a distance of 4 units from 0, so its absolute value is 4 (positive). Intuitively, distance is a nonnegative
quantity: the absolute value of a number is never negative, even if the number itself is negative. The
symbol for absolute value is a pair of vertical lines, ||. Thus we write

| −4 |=4.

Of course, the absolute value of 0 is 0
| 0 |=0,
√
since there is no distance at all between 0 and itself! Every other number, including numbers like 2
and π which are not integers, has a partner (its “opposite”) which lies at the same distance from, but
on the other side of 0, and therefore has the same absolute value.For example, both 5 (positive) and
−5have absolute value equal to 5

| −5 |=5 and | 5 |=5.

Similarly
√ √ √ √
| − 2 |= 2 and | 2 |= 2
| −π |= π and | π |= π.

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