3.1 Introduction to Integers

Learning objectives

Learning objectives

• Locate positive and negative numbers on the number line

• Order positive and negative numbers

• Find opposites

• Simplify expressions with absolute value

• Translate word phrases to expressions with 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, $-1\mathrm{^{\circ}F}$ (read as “negative one degree Fahrenheit”) is $1\,\mathrm{degree}$ below $0.$ A minus sign is shown before a number to indicate that it is negative. Figure 3.2 shows $-20\mathrm{^{\circ}F},$ which is $20\,\mathrm{degrees}$ 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 $\mathrm{0 feet}.$ 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 $1,302\,\mathrm{feet}$ below sea level, so the elevation of the Dead Sea can be represented as $-1,302\,\mathrm{feet}.$ See Figure 3.3.

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

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 $\mathrm{(1, 2, 3, …)}$ 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 $-1$ at the first mark to the left of $0,-2$ at the next mark, and so on. See Figure 3.6.

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.

Example 3.1

Solution

1. Draw the number line. Mark $0$ as the origin and label equal units to the left and the right so distances can be counted.
2. Plot $3$. Starting at $0$, move three units to the right and place the first point at $3$.
3. Plot $-3$. Starting at $0$, move three units to the left and place the second point at $-3$.
4. Plot $-2$. Starting at $0$, move two units to the left and place the third point at $-2$.
5. Check the locations. The positive point is right of zero and the negative points are left of zero at distances three, three, and two.

Takeaway: To plot a number on a number line, start at $0$ and move right for a positive value or left for a negative value, counting one unit per step.

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.

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.

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 $-3<-1.$

Example 3.2

Solution

1. Use the number-line rule. A number farther to the right is greater, so plot each pair before choosing the inequality sign.
2. Compare $14$ and $6$. The point for $14$ is to the right of the point for $6$, so $14>6$.
3. Compare $-1$ and $9$. The point for $-1$ is to the left of the point for $9$, so $-1<9$.
4. Compare $-1$ and $-4$. The point for $-1$ is to the right of the point for $-4$, so $-1>-4$.
5. Compare $2$ and $-20$. The point for $2$ is to the right of the point for $-20$, so $2>-20$.
6. Review all four orders. Each sign matches the relative position of the two plotted points on the number line.

Takeaway: On a number line, the number farther to the right is greater, so $14>6$, $-1<9$, $-1>-4$, and $2>-20$.

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 $-2$ 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).

Example 3.3

Solution

1. State the definition. The opposite of an integer is the same distance from $0$ on the other side of the number line.
2. Find the opposite of $7$. The point $7$ is seven units right of $0$, so its opposite is seven units left at $-7$.
3. Find the opposite of $-10$. The point $-10$ is ten units left of $0$, so its opposite is ten units right at $10$.
4. Check both distances. The pairs $7,-7$ and $-10,10$ are each equal distances from zero but lie on opposite sides.

Takeaway: The opposite of a number has the same distance from $0$ but lies on the other side, so the opposites of $7$ and $-10$ are $-7$ and $10$.

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 $\mathrm{"-",}$ in three different ways.

Example 3.4

Solution

1. Interpret the outside sign. The leading negative sign in $-(-6)$ asks for the opposite of the entire value inside the parentheses.
2. Identify the inner value. The number inside the parentheses is $-6$, which lies six units to the left of $0$ on a number line.
3. Take its opposite. Moving the same six units to the other side of $0$ gives the positive number $6$.
4. Check the simplification. Replacing $-6$ by its opposite produces $6$, so $-(-6)=6$.

Takeaway: Negating a negative number gives its positive opposite, so $-(-6)=6$.

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.

Example 3.5

Solution

1. Substitute the first value. Replace $x$ with $8$ in the expression $-x$, which gives $-(8)$.
2. Take the opposite. The opposite of $8$ is $-8$, so $-x=-8$ when $x=8$.
3. Substitute the second value. Replace $x$ with $-8$, which gives $-(-8)$.
4. Take the opposite again. The opposite of $-8$ is $8$, so $-x=8$ when $x=-8$.
5. State both results. The expression equals $-8$ for $x=8$ and $8$ for $x=-8$: substituting first and then taking the opposite reverses the sign of the given value.

Takeaway: To evaluate $-x$, substitute the value of $x$ first and then take its opposite, so the result carries the opposite sign from the given value.

Simplify Expressions with Absolute Value

We saw that numbers such as $5$ and $-5$ 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 $|5|,$ and the absolute value of $-5$ is written as $|-5|$ as shown in Figure 3.16.

Example 3.6

Solution

1. Apply the absolute-value definition. The value of $|x|$ is the nonnegative distance from $0$ to the number $x$.
2. Simplify $|3|$. The number $3$ is three units from $0$, and that distance is already nonnegative, so $|3|=3$.
3. Simplify $|-44|$. The number $-44$ is $44$ units from $0$, so its distance is $44$ and $|-44|=44$.
4. Simplify $|0|$. The number $0$ is already at $0$, so its distance from $0$ is $0$ and $|0|=0$.
5. Check the signs. Each answer is nonnegative because absolute value always measures distance rather than direction.

Takeaway: Absolute value gives the nonnegative distance from $0$, so $|3|=3$, $|-44|=44$, and $|0|=0$.

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

Example 3.7

Solution

1. Evaluate $|x|$. Substitute $x=-35$ to obtain $|-35|=35$, as shown by the figure displaying the absolute value of $-35$.
2. Evaluate $|-y|$. With $y=-20$, the expression becomes $-(-20)=20$, so $|-y|=|20|=20$.
3. Evaluate $-|u|$. With $u=12$, the absolute value is $|12|=12$, and the outside negative sign gives $-|12|=-12$.
4. Evaluate $-|p|$. With $p=-14$, the absolute value is $|-14|=14$, and applying the outside negative sign gives $-|-14|=-14$.
5. State the results. The four values are $35$, $20$, $-12$, and $-14$, respectively. The figures show that a negative sign inside the absolute value disappears, while a negative sign outside it determines the final sign.

Takeaway: A negative sign inside absolute value bars has no effect on the nonnegative result, but a negative sign outside the bars makes the final value negative.

Example 3.8

Solution

1. Simplify before comparing. Evaluate each absolute value first, and then compare the resulting signed integers.
2. Solve ⓐ. Since $|-5|=5$ and $-|-5|=-5$, the comparison becomes $5\mathrm{\_\_\_}-5$, and $5$ is greater than $-5$.
3. Solve ⓑ. Since $|-8|=8$, the comparison becomes $8\mathrm{\_\_\_}-8$, and $8$ is greater than $-8$.
4. Solve ⓒ. Since $|-9|=9$, the comparison becomes $-9\mathrm{\_\_\_}-9$, and the two values are equal.
5. Solve ⓓ. Since $|-7|=7$, the comparison becomes $-7\mathrm{\_\_\_}-7$, and the two values are equal.
6. Check all four signs. The simplified forms give $5>-5$, $8>-8$, $-9=-9$, and $-7=-7$ respectively.

Takeaway: Simplify absolute values before comparing, which yields $5>-5$, $8>-8$, $-9=-9$, and $-7=-7$.

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.

Example 3.9

Solution

1. Follow the order of operations. Simplify the expression inside each absolute value before applying the absolute value itself.
2. Simplify the first group. In $|9-3|$, subtract inside the bars to obtain $|6|$.
3. Take the first absolute value. Since $6$ is nonnegative and six units from $0$, $|6|=6$.
4. Simplify the second product. In $4|-2|$, find $|-2|=2$ and multiply by $4$ to obtain $8$.
5. Check both results. The simplified expressions are $|9-3|=6$ and $4|-2|=8$, as required.

Takeaway: Evaluate the expression inside each absolute value before taking its magnitude, giving $|9-3|=6$ and $4|-2|=8$.

Example 3.10

Solution

1. Identify the grouped values. The expression contains two absolute values, $|8+7|$ and $|5+6|$, which must be evaluated separately.
2. Simplify the first group. Adding inside the first bars gives $|8+7|=|15|=15$.
3. Simplify the second group. Adding inside the second bars gives $|5+6|=|11|=11$.
4. Subtract the magnitudes. The expression is now $15-11$, which equals $4$.
5. Check the nonnegative distances. Both absolute values produced positive distances before the final subtraction was performed.

Takeaway: Simplify inside both absolute values before subtracting them: $|8+7|-|5+6|=15-11=4$.

Example 3.11

Solution

1. Begin with the innermost parentheses. In $24-|19-3(6-2)|$, simplify $6-2=4$ before doing any other operation.
2. Multiply inside the absolute value. The inner product becomes $3(4)=12$, so the expression is $24-|19-12|$.
3. Subtract inside the bars. Compute $19-12=7$, giving $24-|7|$.
4. Evaluate the absolute value. Since $7$ is seven units from $0$, $|7|=7$ and the expression becomes $24-7$.
5. Perform the outer subtraction. Compute $24-7=17$, which is the value of the complete expression.
6. Check the order used. Each grouping was completed from the inside outward before its result entered the next operation.

Takeaway: Apply the order of operations from the innermost grouping outward, so $24-|19-3(6-2)|=24-|7|=17$.

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 $-20.$ So does the word opposite in “the opposite of $20\mathrm{."}$

Example 3.12

Solution

1. Translate ⓐ The opposite of positive fourteen is $-14$.

2. Translate ⓑ The opposite of $-11$ is $-(-11)=11$, because two negatives make a positive.

3. Translate ⓒ "Negative sixteen" is written directly as $-16$.

4. Translate ⓓ "Two minus negative seven" is $2-(-7)$, which keeps both the subtraction and the negative sign.

Takeaway: "Opposite of" turns $14$ into $-14$ and $-11$ into $11$, while "negative" and "minus" keep their signs, giving $-16$ and $2-(-7)$.

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.

Example 3.13

Solution

1. Locate the comparison phrases. Words such as below zero, gain, below sea level, and overdrawn determine whether each quantity is positive or negative.
2. Translate ⓐ. Since the temperature is $12$ degrees below zero, represent it as $-12$ degrees Fahrenheit.
3. Translate ⓑ. A gain of $3$ yards increases the distance, so represent it as the positive quantity $3$ yards.
4. Translate ⓒ. The elevation is $1,302$ feet below sea level, so represent it as $-1,302$ feet.
5. Translate ⓓ. An overdrawn account is $40$ dollars below a zero balance, so represent it as $-40$ dollars.
6. Check the signs. Each expression agrees with the direction described by its key phrase and includes the stated unit.

Takeaway: Words such as “below zero,” “below sea level,” and “overdrawn” signal negative quantities, so the four expressions are $-12$ degrees Fahrenheit, $3$ yards, $-1,302$ feet, and $-40$ dollars.

Section 3.1 Exercises

Section 3.1 Exercises

Locate Positive and Negative Numbers on the Number Line

For the following exercises, draw a number line and locate and label the given points on that number line.

Order Positive and Negative Numbers on the Number Line

In the following exercises, order each of the following pairs of numbers, using $<$ or $\mathrm{>.}$

Find Opposites

In the following exercises, find the opposite of each number.

In the following exercises, simplify.

In the following exercises, evaluate.

Simplify Expressions with Absolute Value

In the following exercises, simplify each absolute value expression.

In the following exercises, evaluate each absolute value expression.

In the following exercises, fill in $<,>,\mathrm{or}=$ to compare each expression.

In the following exercises, simplify each expression.

Translate Word Phrases into Expressions with Integers

Translate each phrase into an expression with integers. Do not simplify.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

1. Order each of the following pairs of numbers using $<$ or $\mathrm{>.}$

1. ⓐ $\,15\mathrm{\_\_\_}7\,$

2. ⓑ $\,-2\mathrm{\_\_\_}5\,$

3. ⓒ $\,-3\mathrm{\_\_\_}-7\,$

4. ⓓ $\,5\mathrm{\_\_\_}-17$

2. Order each of the following pairs of numbers using $<$ or $\mathrm{>.}$

1. ⓐ $\,8\mathrm{\_\_\_}13\,$

2. ⓑ $\,3\mathrm{\_\_\_}-4\,$

3. ⓒ $\,-5\mathrm{\_\_\_}-2\,$

4. ⓓ $\,9\mathrm{\_\_\_}-21$

3. Find the opposite of each number:

1. ⓐ $\,4\,$

2. ⓑ $\,-3$

1. Find the opposite of each number:

1. ⓐ $\,8\,$

2. ⓑ $\,-5$

2. Evaluate $-n:\,$

1. ⓐ $\,\mathrm{when}\,n=4\,$

2. ⓑ $\,\mathrm{when}\,n=-4$

3. Evaluate: $-m:\,$

1. ⓐ $\,\mathrm{when}\,m=11\,$

2. ⓑ $\,\mathrm{when}\,m=-11$

1. Simplify:

1. ⓐ $\,|12|\,$

2. ⓑ $\,-|-28|$

2. Simplify:

1. ⓐ $\,|9|\,$

2. ⓑ $-|37|$

3. Evaluate:

1. ⓐ $\,|x|\,\mathrm{when}\,x=-17\,$

2. ⓑ $\,|\mathrm{-y}|\,\mathrm{when}\,y=-39\,$

3. ⓒ $\,-|m|\,\mathrm{when}\,m=22\,$

4. ⓓ $\,-|p|\,\mathrm{when}\,p=-11$

1. 1. ⓐ $\,|y|\,\mathrm{when}\,y=-23\,$

2. ⓑ $\,|-y|\,\mathrm{when}\,y=-21$

3. ⓒ $\,-|n|\,\mathrm{when}\,n=37\,$

4. ⓓ $\,-|q|\,\mathrm{when}\,q=-49$

2. Fill in $<,>,\mathrm{or}=\,\mathrm{for each of the following:}$

1. ⓐ $\,|-9|\,\mathrm{\_\_\_}-|-9|\,$

2. ⓑ $\,2\mathrm{\_\_\_}-|-2|\,$

3. ⓒ $\,-8\mathrm{\_\_\_}|-8|\,$

4. ⓓ $\,-|-5|\mathrm{\_\_\_}-5$

3. Fill in $<,>,\mathrm{or}=$ for each of the following:

1. ⓐ $\,7\mathrm{\_\_\_}-|-7|\,$

2. ⓑ $\,-|-11|\mathrm{\_\_\_}-11\,$

3. ⓒ $\,|-4|\mathrm{\_\_\_}-|-4|\,$

4. ⓓ $\,-1\mathrm{\_\_\_}|-1|$

1. Simplify:

1. ⓐ $\,|12-9|\,$

2. ⓑ $\,3|-6|$

2. Simplify:

1. ⓐ $\,|27-16|\,$

2. ⓑ $\,9|-7|$

3. Translate each phrase into an expression with integers:

1. ⓐ the opposite of positive nine

2. ⓑ the opposite of $-15$

3. ⓒ negative twenty

4. ⓓ eleven minus negative four

1. Translate each phrase into an expression with integers:

1. ⓐ the opposite of negative nineteen

2. ⓑ the opposite of twenty-two

3. ⓒ negative nine

4. ⓓ negative eight minus negative five

2. Translate into an expression with integers:

The football team had a gain of $5\,\mathrm{yards.}$

3. Translate into an expression with integers:

The scuba diver was $30\,\mathrm{feet}$ below the surface of the water.

1. 1. ⓐ $\,9\mathrm{\_\_}4\,$

2. ⓑ $\,-3\mathrm{\_\_}6\,$

3. ⓒ $\,-8\mathrm{\_\_}-2\,$

4. ⓓ $\,1\mathrm{\_\_}-10$

2. 1. ⓐ $\,-5\mathrm{\_\_}1;\,$

2. ⓑ $\,-4\mathrm{\_\_}-9;\,$

3. ⓒ $\,6\mathrm{\_\_}10;\,$

4. ⓓ $\,3\mathrm{\_\_}-8$

3. 1. ⓐ $\,2\,$

2. ⓑ $\,-6$

1. 1. ⓐ $\,-8\,$

2. ⓑ $\,1$

2. $-m\,\mathrm{when}\,$

1. ⓐ $\,m=3\,$

2. ⓑ $\,m=-3$

3. $-c\,\mathrm{when}\,$

1. ⓐ $\,c=12\,$

2. ⓑ $\,c=-12$

1. 1. ⓐ $\,|7|\,$

2. ⓑ $\,|-25|\,$

3. ⓒ$\,|0|$

2. 1. ⓐ $\,|-32|\,$

2. ⓑ $\,|-18|\,$

3. ⓒ $\,|16|$

3. 1. ⓐ $\,|x|\,\mathrm{when}\,x=-28$

2. ⓑ $\,|-u|\,\mathrm{when}\,u=-15$

1. 1. ⓐ $\,-|p|\,\mathrm{when}\,p=19$

2. ⓑ $\,-|q|\,\mathrm{when}\,q=-33$

2. 1. ⓐ $\,-6\mathrm{\_\_}|-6|\,$

2. ⓑ $\,-|-3|\mathrm{\_\_}-3$

3. 1. ⓐ $\,|-3|\mathrm{\_\_}-|-3|\,$

2. ⓑ $\,4\mathrm{\_\_}-|-4|$

1. 1. ⓐ the opposite of $8$

2. ⓑ the opposite of $-6$

3. ⓒ negative three

4. ⓓ $4$ minus negative $3$

2. 1. ⓐ the opposite of $20$

2. ⓑ the opposite of $-5$

3. ⓒ negative twelve

4. ⓓ $18$ minus negative $7$

3. a temperature of $6\,\mathrm{degrees}$ below zero

1. an elevation of $40\,\mathrm{feet}$ below sea level

2. a football play loss of $12\,\mathrm{yards}$

3. a stock gain of $\text{\textdollar 3}$

1. a golf score one above par

2. Elevation The highest elevation in the United States is Mount McKinley, Alaska, at $20,320\,\mathrm{feet}$ above sea level. The lowest elevation is Death Valley, California, at $282\,\mathrm{feet}$ below sea level. Use integers to write the elevation of:

1. ⓐ Mount McKinley

2. ⓑ Death Valley

3. State budgets In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of $\text{\textdollar 540 million.}$ That same month, Texas estimated it would have a budget deficit of $\text{\textdollar 27 billion.}$ Use integers to write the budget:

1. ⓐ surplus

2. ⓑ deficit

1. Give an example of a negative number from your life experience.