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, $\mathrm{-1}\text{°F}$ (read as “negative one degree Fahrenheit”) is $1\text{degree}$ below $0.$ A minus sign is shown before a number to indicate that it is negative. Figure 1 shows $\mathrm{-20}\text{°F},$ which is $20\text{degrees}$ below $0.$

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 $\text{0 feet}.$ 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 $\mathrm{1,302}\text{feet}$ below sea level, so the elevation of the Dead Sea can be represented as $\mathrm{-1,302}\text{feet}.$ See Figure 2.

Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of $500\text{feet}.$ Its position would then be $\mathrm{-500}\text{feet}$ as labeled in Figure 3.

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 4. The counting numbers $\text{(1, 2, 3, …)}$ 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.

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 $\mathrm{-1}$ at the first mark to the left of $0,\mathrm{-2}$ at the next mark, and so on. See Figure 5.

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.

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 9.

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 10.

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 $1<4.$

ⓑ

$\mathrm{-2}$ is to the left of $1$ on the number line, so $\mathrm{-2}<1.$

$1$ is to the right of $\mathrm{-2}$ on the number line, so $1>\mathrm{-2}.$

ⓒ

$\mathrm{-1}$ is to the right of $\mathrm{-3}$ on the number line, so $\mathrm{-1}>\mathrm{-3}.$

$\mathrm{-3}$ is to the left of $\mathrm{-1}$ on the number line, so $\mathrm{-3}<-1.$

Order each of the following pairs of numbers using $<$ or $\text{>:}$

1. ⓐ $14\_\_\_6$
2. ⓑ $\mathrm{-1}\_\_\_9$
3. ⓒ $\mathrm{-1}\_\_\_\mathrm{-4}$
4. ⓓ $2\_\_\_\mathrm{-20}$

Solution

Solution

Begin by plotting the numbers on a number line as shown in Figure 11.

Compares 14 and 6, illustrating the context and resulting mathematical inequality using a number line explanation.

ⓐ Compare 14 and 6.	$14\_\_\_6$
14 is to the right of 6 on the number line.	$14>6$

Comparison of -1 and 9 using number line reasoning and mathematical inequality.

ⓑ Compare −1 and 9.	$\mathrm{-1}\_\_\_9$
−1 is to the left of 9 on the number line.	$\mathrm{-1}<9$

Comparison of negative integers, demonstrating how their positions on a number line determine their inequality.

ⓒ Compare −1 and −4.	$\mathrm{-1}\_\_\_\mathrm{-4}$
−1 is to the right of −4 on the number line.	$\mathrm{-1}>\mathrm{-4}$

Compares 2 and -20, detailing the descriptive logic and the corresponding mathematical inequality, demonstrating how to compare positive and negative numbers.

ⓓ Compare 2 and −20.	$2\_\_\_\mathrm{-20}$
2 is to the right of −20 on the number line.	$2>\mathrm{-20}$

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 $\mathrm{-2}$ are the same distance from zero, they are called opposites. The opposite of $2$ is $\mathrm{-2},$ and the opposite of $\mathrm{-2}$ is $2$ as shown in Figure 12(a). Similarly, $3$ and $\mathrm{-3}$ are opposites as shown in Figure 12(b).

Integers

The set of counting numbers, their opposites, and $0$ is the set of integers.

We must be very careful with the signs when evaluating the opposite of a variable.

Evaluate $-x:$

1. ⓐ when $x=8$
2. ⓑ when $x=\mathrm{-8}.$

Solution

ⓐ To evaluate $-x$ when $x=8$, substitute $8$ for $x$.
	$-x$
Simplify.	$\mathrm{-8}$

ⓑ To evaluate $-x$ when $x=\mathrm{-8}$, substitute $\mathrm{-8}$ for $x$.
	$-x$
Simplify.	$8$

Simplify Expressions with Absolute Value

We saw that numbers such as $5$ and $\mathrm{-5}$ are opposites because they are the same distance from $0$ on the number line. They are both five units from $0.$ The distance between $0$ and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.

The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of $5$ is written as $\left|5\right|,$ and the absolute value of $\mathrm{-5}$ is written as $\left|\mathrm{-5}\right|$ as shown in Figure 15.

We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.

Evaluate:

1. ⓐ $\left|x\right|\text{when}x=\mathrm{-35}$
2. ⓑ $\left|\mathit{\text{−y}}\right|\text{when}y=\mathrm{-20}$
3. ⓒ $-\left|u\right|\text{when}u=12$
4. ⓓ $-\left|p\right|\text{when}p=\mathrm{-14}$

Solution

Solution

ⓐ To find $\left|x\right|$ when $x=\mathrm{-35}:$
	$\left|x\right|$
Take the absolute value.	$35$

ⓑ To find $|-y|$ when $y=\mathrm{-20}:$
	$|-y|$
Simplify.	$\left|20\right|$
Take the absolute value.	$20$

ⓒ To find $-\left|u\right|$ when $u=12:$
	$-\left|u\right|$
Take the absolute value.	$\mathrm{-12}$

ⓓ To find $-\left|p\right|$ when $p=\mathrm{-14}:$
	$-\left|p\right|$
Take the absolute value.	$\mathrm{-14}$

Notice that the result is negative only when there is a negative sign outside the absolute value symbol.

Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.

Translate Word Phrases into Expressions with Integers

Now we can translate word phrases into expressions with integers. Look for words that indicate a negative sign. For example, the word negative in “negative twenty” indicates $\mathrm{-20}.$ So does the word opposite in “the opposite of $20\text{.”}$

As we saw at the start of this section, negative numbers are needed to describe many real-world situations. We’ll look at some more applications of negative numbers in the next example.