1.5: Positive and Negative Numbers

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Dec 4, 2023
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- 1.4: Order of Operations
- 2: Fractions

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As we saw in the introduction to the chapter, negative numbers have a rich and storied history. One of the earliest applications of negative numbers had to do with credits and debits. For example, if $5 represents a credit or profit, then −$5 represents a debit or loss. Of course, the ancients had a different monetary system than ours, but you get the idea. Note that if a vendor experiences a profit of $5 on a sale, then a loss of −$5 on a second sale, the vendor breaks even. That is, the sum of $5 and −$5 is zero. In much the same way, every whole number has an opposite or negative counterpart.

The Opposite or Negative of a Whole Number

For every whole number a, there is a unique number −a, called the opposite or negative of a, such that a + (−a) = 0.

The opposite or negative of any whole number is easily located on the number line.

Number Line Locations

To locate the opposite (or negative) of any whole number, first locate the whole number on the number line. The opposite is the reflection of the whole number through the origin (zero).

Example 1

Locate the whole number 5 and its opposite (negative) on the number line.

Solution

Draw a number line, then plot the whole number 5 on the line as a shaded dot.

$Screen Shot 2019-08-12 at 5.58.13 PM.png$

To find its opposite, reflect the number 5 through the origin. This will be the location of the opposite (negative) of the whole number 5, which we indicate by the symbol −5.

$Screen Shot 2019-08-12 at 5.58.18 PM.png$

Note the symmetry. The whole number 5 is located five units to the right of zero. Its negative is located five units to the left of zero.

Exercise

Locate the number -7 and its opposite on the number line.

Important Pronunciation

The symbol −5 is pronounced in one of two ways: (1) “negative five,” or (2) “the opposite of five.”

In similar fashion, we can locate the opposite or negative of any whole number by reflecting the whole number through the origin (zero), which leads to the image shown in Figure 2.1.

Screen Shot 2019-08-12 at 6.01.48 PM.png — Figure 2.1: The opposite (negative) of any whole number is a reflection of that number through the origin (zero).

The Integers

The collection of numbers arranged on the number line in Figure 2.1 extend indefinitely to the right, and because the numbers on the left are reflections through the origin, the numbers also extend indefinitely to the left. This collection of numbers is called the set of integers.

The Integers

The infinite collection of numbers

{..., −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, ...}

is called the set of integers.

The ellipsis . . . at each end of this infinite collection means “etcetera,” as the integers continue indefinitely to the right and left. Thus, for example, both 23,456 and −117, 191 are elements of this set and are therefore integers.

Ordering the Integers

As we saw with the whole numbers, as you move to the right on the number line, the numbers get larger; as you move to the left, the numbers get smaller.

Order on the Number Line

Let a and b be integers located on the number line so that the point representing the integer a lies to the left of the point representing the integer b.

$Screen Shot 2019-08-12 at 6.03.49 PM.png$

Then the integer a is “less than” the integer b and we write

$a < b\nonumber$

Alternatively, we can also say that the integer b is “larger than” the integer a and write

$b > a.\nonumber$

Example 3

Replace each shaded box with < (less than) or > (greater than) so the resulting inequality is a true statement.

$Screen Shot 2019-08-12 at 6.07.02 PM.png$

Solution

For the first case, locate −3 and 5 on the number line as shaded dots.

$Screen Shot 2019-08-12 at 6.07.08 PM.png$

Note that −3 lies to the left of 5, so:

$Screen Shot 2019-08-12 at 6.07.11 PM.png$

That is, −3 is “less than” 5.

In the second case, locate −2 and −4 as shaded dots on the number line.

$Screen Shot 2019-08-12 at 6.09.21 PM.png$

Note that −2 lies to the right of −4, so:

$Screen Shot 2019-08-12 at 6.09.26 PM.png$

That is, −2 is “greater than” −4.

Exercise

Compare −12 and −11.

Answer: −12 < −11

Important Observation

In Example 3, note that the “pointy end” of the inequality symbol always points towards the smaller number.

We stated earlier that every integer has a unique number called its “opposite” or “negative.” Thus, the integer −5 is the opposite (negative) of the integer 5. Thus, we can say that the pair −5 and 5 are opposites. Each is the opposite of the other. Logically, this leads us to the conclusion that the opposite of −5 is 5. In symbols, we would write

$-(-5) = 5.\nonumber$

Opposites of Opposites

Let a be an integer. Then the “opposite of the opposite of a is a.” In symbols, we write

$-(-a) = a.\nonumber$

We can also state that the “negative of a negative a is a.

Example 4

Simplify −(−13) and −(−119).

Solution

The opposite of the opposite of a number returns the original number. That is,

−(−13) = 13 and − (−119) = 11.

Exercise

Simplify: −(−50).

Answer: 50

Positive and Negative

We now define the terms positive integer and negative integer.

Positive Integer

If a is an integer that lies to the right of zero (the origin) on the number line, then a is a positive integer. This means that a is a positive integer if and only if a > 0.

Thus, 2, 5, and 117 are positive integers.

Negative Integer

If a is an integer that lies to the left of zero (the origin) on the number line, then a is a negative integer. This means that a is a negative integer if and only if a < 0.

Thus, −4, −8, and −1, 123 are negative integers.

Zero

The integer zero is neither positive nor negative.

Example 5

Classify each of the following numbers as negative, positive, or neither: 4, −6, and 0.

Solution

Locate 4, −6, and 0 on the number line.

$Screen Shot 2019-08-12 at 6.18.15 PM.png$

Thus:

4 lies to the right of zero. That is, 4 > 0, making 4 a positive integer.
−6 lies to the left of zero. That is, −6 < 0, making −6 a negative integer.
The number 0 is neutral. It is neither negative nor positive.

Exercise

Classify −11 as positive, negative, or neither.

Answer: Negative

Absolute Value

We define the absolute value of an integer.

Absolute Value

The absolute value of an integer is defined as its distance from the origin (zero).

It is important to note that distance is always a nonnegative quantity (not negative); i.e., distance is either positive or zero. As an example, we’ve shaded the integers −4 and 4 on a number line.

$Screen Shot 2019-08-12 at 6.21.51 PM.png$

The number line above shows two cases:

The integer −4 is 4 units from zero. Because absolute value measures the distance from zero, | − 4| = 4.
The integer 4 is also 4 units from zero. Again, absolute value measures the distance from zero, so |4| = 4.

Let’s look at another example.

Example 6

Determine the value of each expression: a) | − 7|, b) |3|, and c) |0|.

Solution

The absolute value of any integer is equal to the distance that number is from the origin (zero) on the number line. Thus:

a) The integer −7 is 7 units from the origin; hence, | − 7| = 7.

b) The integer 3 is 3 units from the origin; hence, |3| = 3.

c) The integer 0 is 0 units from the origin; hence, |0| =0.

Exercise

Simplify: | − 33|.

Answer: 33

Example 7

Determine the value of each expression: a) −(−8) and b) −| − 8|

Solution

These are distinctly different problems.

a) The opposite of −8 is 8. That is, −(−8) = 8.

b) However, in this case, we take the absolute value of −8 first, which is 8, then the opposite of that result to get −8. That is,

$\begin{aligned} - | -8 | = -(8) & \textcolor{red}{ \text{ First: } |-8| = 8.} \\ = -8 ~ & \textcolor{red}{ \text{ Second: The opposite of 8 is } -8.} \end{aligned}\nonumber$

Exercise

Simplify: −| − 50|.

Answer: 50

Exercises

In Exercises 13-24, for each of the following sets of integers, perform the following tasks:

a) Plot each of the integers on a numberline.

b) List the numbers in order, from smallest to largest.

15. 5, −6, 0, and 2

23. −2, −4, 3, and −6

In Exercises 25-36, in each of the following exercises, enter the inequality symbol < or the symbol > in the box in order that the resulting inequality is a true statement.

25. $-4 \square 0$

29. $-3 \square -1$

34. $0 \square -3$

36. $1 \square -4$

In Exercises 37-52, simplify each of the following expressions.

37. −(−4).

42. |3|

45. | − 5|

47. −| − 20|

In Exercises 53-64, for each of the following exercises, provide a number line sketch with your answer.

55. Find two integers on the number line that are 4 units away from the integer -1.

61. Find two integers on the number line that are 3 units away from the integer 0.

Answers

15.

i) Arrange the integers 5, −6, 0, and 2 on a number line.

$Screen Shot 2019-08-13 at 8.30.03 PM.png$

ii) −6, 0, 2, 5

23.

i) Arrange the integers −2, −4, 3, and −6 on a number line.

$Screen Shot 2019-08-13 at 8.35.31 PM.png$

ii) −6, −4, −2, 3

25. −4 <0

29. −3 < −1

34. 0 > -3

36. 1 > -4

37. 4

42. 3

45. 5

55. −7 and 1.

$Screen Shot 2019-08-13 at 8.45.27 PM.png$

61. −3 and 3.

$Screen Shot 2019-08-13 at 8.40.40 PM.png$

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The Integers

Ordering the Integers

Positive and Negative

Absolute Value

Exercises

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