Introduction to Integers

3.1 Introduction to Integers

The topics covered in this section are:

  1. Locate positive and negative numbers on the number line
  2. Order positive and negative numbers
  3. Find opposites
  4. Simplify expressions with absolute value
  5. Translate word phrases to expressions with integers

3.1.1 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. The figure below 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.

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 the figure below.

This figure is a drawing of a side view of the coast of Israel, showing different elevations. The Mediterranean Sea is labeled 0 feet elevation and the Dead Sea is labeled negative 1302 feet elevation. The country of Jordan is also labeled in the figure.
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 the figure below.

This figure is a drawing of a submarine underwater. In the water is also a vertical number line, scaled in feet. The number line has 0 feet at the surface and negative 500 feet below the water where the submarine is located.
Depths below sea level are described by negative numbers. A submarine $500$ ft below sea level is at $-500$ ft.

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 the figure below. 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.

This figure is a number line scaled from 0 to 6.

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 the figure below.

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 negative numbers, negative 1 to negative 4 to the left of 0.
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.

Example 1

Plot the numbers on a number line:

  1. $3$
  2. $-3$
  3. $-2$
Solution

Draw a number line. Mark $0$ in the center and label several units to the left and right.

Part 1. To plot $3$, start at $0$ and count three units to the right. Place a point show in the figure below.

This figure is a number line scaled from negative 4 to 4, with the point 3 labeled with a dot.

Part 2. To plot $-3$, start at $0$ and count three units to the left. Place a point as shown in the figure below.

This figure is a number line scaled from negative 4 to 4, with the point negative 3 labeled with a dot.

Part 3. To plot $-2$, start at $0$ and count two units to the left. Place a point as shown in the figure below.

This figure is a number line scaled from negative 4 to 4, with the point negative 2 labeled with a dot.

3.1.2 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 the figure below.

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

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 show for the numbers $3$ and $5$ in the figure below.

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 and 5 is greater than 3.
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.


This figure is a number line with points 1 and 4 labeled with dots.

$4$ 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 $1<4$.


This figure is a number line with points negative 2 and 1 labeled with dots.

$-2$ 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 $1>-2$.


This figure is a number line with points negative 3 and negative 1 labeled with dots.

$-1$ 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 2

Order each of the following pairs of numbers using $<$ or $>$:

  1. $14$____$6$
  2. $-1$____$9$
  3. $-1$____$-4$
  4. $2$____$-20$
Solution

Begin by plotting the numbers on a number line as show in the figure below.

This figure is a number line with points negative 20, negative 4, negative 1, 2, 6, 9, and 14 labeled with dots.
Part 1. Compare $14$ and $6$.$14$____$6$
$14$ is to the right of $6$ on the number line.$14>6$
Part 2. Compare $-1$ and $9$.$-1$____$9$
$-1$ is to the left of $9$ on the number line.$-1<9$
Part 3. Compare $-1$ and $-4$.$-1$____$-4$
$-1$ is to the right of $-4$ on the number line.$-1>-4$
Part 4. Compare $2$ and $-20$.$2$____$-20$
$2$ is to the right of $-20$ on the number line.$2>-20$

3.1.3 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(a) below. Similarly, $3$ and $-3$ are opposites as show in Figure(b) below.

This figure shows two number lines. The first has points negative 2 and positive 2 labeled. Below the first line the statement is the numbers negative 2 and 2 are opposites. The second number line has the points negative 3 and 3 labeled. Below the number line is the statement negative 3 and 3 are opposites.

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.

Example 3

Find the opposite of each number:

  1. $7$
  2. $-10$
Solution

Part 1. The number $-7$ is the same distance from $0$ as $7$, but on the opposite side of $0$. So $-7$ is the opposite of $7$ as show in the figure below.

This figure is a number line. The points negative 7 and 7 are labeled. Above the line it is shown the distance from 0 to negative 7 and the distance from 0 to 7 are both 7.

Part 2. The number $10$ is the same distance from $0$ as $-10$, but on the opposite side of $0$. So $10$ is the opposite of $-10$ as shown in the figure below.

This figure is a number line. The points negative 10 and 10 are labeled. Above the line it is shown the distance from 0 to negative 10 and the distance from 0 to 10 are both 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 “$-$”, 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 $-2$.

Opposite Notation

$-a$ means the opposite of the number $a$

The notation $-a$ is read the opposite of $a$.

Example 4

Simplify: $-(-6)$

Solution

$-(-6)$
The opposite of $-6$ is $6$.$6$

Integers

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

Integers

Integers are counting numbers, their opposites, and zero.

$…-3,-2,-1,0,1,2,3…$

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

Example 5

Evaluate $-x$:

  1. when $x=8$.
  2. when $x=-8$.
Solution

Part 1. To evaluate $-x$ when $x=8$, substitute $8$ for $x$.$-x$
Substitute $8$ for $x$.$-(8)$
Simplify.$-8$
Part 2. To evaluate $-x$ when $x=-8$, substitute $-8$ for $x$.$-x$
Substitute $-8$ for $x$.$-(-8)$
Simplify.$8$

3.1.4 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 the figure below.

This figure is a number line. The points negative 5 and 5 are labeled. Above the number line the distance from negative 5 to 0 is labeled as 5 units. Also above the number line the distance from 0 to 5 is labeled as 5 units.

Absolute Value

The absolute value of a number is its distance from $0$ on the number line.

The absolute value of a number $n$ is written as $|n|$.

$|n| \geq 0$ for all numbers

Example 6

Simplify:

  1. $|3|$
  2. $|-44|$
  3. $|0|$
Solution

Part 1.$|3|$
$3$ is $3$ units from zero.$3$
Part 2.$|-44|$
$-44$ is $44$ units from zero.$44$
Part 3.$|0|$
$0$ is already at zero.$0$

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

Example 7

Evaluate:

  1. $|x|$ when $x=-35$
  2. $|-y|$ when $y=-20$
  3. $-|u|$ when $u=12$
  4. $-|p|$ when $p=-14$
Solution

Part 1. To find $|x|$ when $x=-35$:$|x|$
Substitute $-35$ for $x$.$|-35|$
Take the absolute value.$35$
Part 2. To find $|-y|$ when $y=-20$:$|-y|$
Substitute $-20$ for $y$.$|-(-20)|$
Simplify.$|20|$
Take the absolute value.$20$
Part 3. To find $-|u|$ when $u=12$:$-|u|$
Substitute $12$ for $u$.$-|12|$
Take the absolute value.$-12$
Part 4. To find $-|p|$ when $p=-14$:$-|p|$
Substitute $-14$ for $p$.$-|-14|$
Take the absolute value.$-14$

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.

Example 8

Simplify:

  1. $|9-3|$
  2. $4|-2|$
Solution

For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.

Part 1.$|9-3|$
Simplify inside the absolute value sign.$|6|$
Take the absolute value.$6$
Part 2.$4|-2|$
Take the absolute value.$4 \cdot 2$
Multiply.$8$

Example 9

Simplify: $|8+7|-|5+6|$.

Solution

For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.

$|8+7|-|5+6|$
Simplify inside each absolute value sign.$|15|-|11|$
Simplify.$4$

Example 10

Simplify: $24-|19-3(6-2)|$.

Solution

We use the order of operations. Remember to simplify grouping symbols first, so parentheses inside absolute value symbols would be first.

$24-|19-3(6-2)|$
Simplify in the parentheses first.$24-|19-3(4)|$
Multiply $3(4)$.$24-|19-12|$
Subtract inside the absolute value sign.$24-|7|$
Take the absolute value.$24-7$
Subtract.$17$

3.1.5 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 $20$.”

Example 11

Translate each prhase into an expression with integers:

  1. the opposite of positive fourteen
  2. the opposite of $-11$
  3. negative sixteen
  4. two minus negative seven
Solution
  1. the opposite of positive fourteen
    • $-14$
  2. the opposite of $-11$
    • $-(-11)=11$
  3. negative sixteen
    • $-16$
  4. two minus negative seven
    • $2-(-7)$

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 12

Translate into an expression with integers:

  1. The temperature is $12$ degrees Fahrenheit below zero.
  2. The football team had gain of $3$ yards.
  3. The elevation of the Dead Sea is $1,302$ feet below sea level.
  4. A checking account is overdrawn by $ \$40$.
Solution

Look for key phrases in each sentence. Then look for words that indicate negative signs. Don’t forget to include units of measurement described in the sentence.

Part 1.The temperature is $12$ degrees Fahrenheit below zero.
Below zero tells us that $12$ is a negative number.$-12$ºF
Part 2.The football team had a gain of $3$ yards.
gain tells us that $3$ is a positive number.$3$ yards
Part 3.The elevation of the Dead Sea is $1,302$ feet below sea level.
Below sea level tells us that $1,302$ is a negative number.$-1,302$ feet
Part 4.A checking account is overdrawn by $ \$ 40$.
Overdrawn tells us that $40$ is a negative number.$- \$ 40$
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