Introduction to Integers

Read this text, which explains the concept of negative numbers using number lines. Pay close attention to the section "Opposite Notation", as this notation will come up frequently in the next few sections of this course. Complete the practice questions and check your answers.

Introduction to Integers

Locate Positive and Negative Numbers on the Number Line

Do you live in a place that has very cold winters? Have you ever experienced a temperature below zero? If so, you are already familiar with negative numbers. A negative number is a number that is less than 0.
Very cold temperatures are measured in degrees below zero and can be described by negative numbers. For example, −1°F (read as "negative one degree Fahrenheit") is 1 degree below 0. A minus sign is shown before a number to indicate that it is negative. Figure 3.2 shows −20°F, which is 20 degrees below 0.

This figure is a thermometer scaled in degrees Fahrenheit. The thermometer has a reading of 20 degrees.

Figure 3.2 Temperatures below zero are described by negative numbers.

Temperatures are not the only negative numbers. A bank overdraft is another example of a negative number. If a person writes a check for more than he has in his account, his balance will be negative.

Elevations can also be represented by negative numbers. The elevation at sea level is 0 feet.

Elevations above sea level are positive and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about 1,302 feet below sea level, so the elevation of the Dead Sea can be represented as −1,302 feet. See Figure 3.3.

This figure is a drawing of a side view of the coast of Israel, showing different elevations. The Mediterranean Sea is labele

Figure 3.3 The surface of the Mediterranean Sea has an elevation of 0 ft. The diagram shows that nearby mountains have higher (positive) elevations whereas the Dead Sea has a lower (negative) elevation.

Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of 500 feet.
Its position would then be −500 feet as labeled in Figure 3.4.

This figure is a drawing of a submarine underwater. In the water is also a vertical number line, scaled in feet. The number l

Figure 3.4 Depths below sea level are described by negative numbers. A submarine 500ft below sea level is at −500ft.

Both positive and negative numbers can be represented on a number line. Recall that the number line created in Add Whole Numbers started at 0 and showed the counting numbers increasing to the right as shown in Figure 3.5. The counting numbers (1, 2, 3, …) on the number line are all positive. We could write a plus sign, +, before a positive number such as +2 or +3, but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.

Figure 3.5

Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with −1 at the first mark to the left of 0,−2 at the next mark, and so on. See Figure 3.6.

This figure is a number line with 0 in the middle. Then, the scaling has positive numbers 1 to 4 to the right of 0 and negati

Figure 3.6 On a number line, positive numbers are to the right of zero. Negative numbers are to the left of zero. What about zero? Zero is neither positive nor negative.

The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number.

Manipulative Mathematics

Doing the Manipulative Mathematics activity "Number Line-part 2" will help you develop a better understanding of integers.

Order Positive and Negative Numbers

We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See Figure 3.10.

This figure is a number line. Above the number line there is an arrow pointing to the right labeled increasing. Below the num

Figure 3.10

Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers. Remember that we use the notation $a < b$ (read $a$ is less than $b$ when a is to the left of b on the number line. We write $a > b$ (read $a$ is greater than $b$ ) when $a$ is to the right of $b$ on the number line. This is shown for the numbers 3 and 5 in Figure 3.11.

This figure is a number line with points 3 and 5 labeled with dots. Below the number line is the statements 3 is less than 5

Figure 3.11 The number 3 is to the left of 5 on the number line. So 3 is less than 5, and 5 is greater than 3.

The numbers lines to follow show a few more examples.

ⓐ

4 is to the right of 1 on the number line, so $4 > 1$ .

1 is to the left of 4 on the number line, so $1 < 4$ .

ⓑ

−2 is to the left of 1 on the number line, so $−2 < 1$ .

1 is to the right of −2 on the number line, so $1 > −2$ .

ⓒ

−1 is to the right of −3 on the number line, so $−1 > −3$ .

−3 is to the left of −1 on the number line, so $−3 < −1$ .

Find Opposites

On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers 2 and −2 are the same distance from zero, they are called opposites. The opposite of 2 is −2, and the opposite of −2 is 2 as shown in Figure 3.13(a). Similarly, 3 and −3 are opposites as shown in Figure 3.13(b).

This figure shows two number lines. The first has points negative 2 and positive 2 labeled. Below the first line the statemen

Figure 3.13

Opposite

The opposite of a number is the number that is the same distance from zero on the number line, but on the opposite side of zero.

Opposite Notation

Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol "−", in three different ways.


10−4	Between two numbers, the symbol indicates the operation of subtraction. We read 10−4 as 10 minus 4.
−8	In front of a number, the symbol indicates a negative number. We read −8 as negative eight.
$−x$	In front of a variable or a number, it indicates the opposite. We read $−x$ as the opposite of $x$ .
−(−2)	Here we have two signs. The sign in the parentheses indicates that the number is negative 2. The sign outside the parentheses indicates the opposite. We read −(−2) as the opposite of −2.