Add and Subtract Integers

Use Negatives and Opposites of Integers

Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than $0$. The negative numbers are to the left of zero on the number line.

The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.

Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.

Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value.

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, and $a > b$ (read “$a$ is greater than $b$”) when $a$ is to the right of $b$ on the number line.

Now we need to extend the number line to include negative numbers, too. The numbers marked by points on this extended line are called the integers. The integers are the numbers $\dots -3, -2, -1, 0, 1, 2, 3 \dots$

Example. Order each of the following pairs of numbers, using $<$ or $>$: (a) $14 \_\_\_ 6$ (b) $-1 \_\_\_ 9$ (c) $-1 \_\_\_ -4$ (d) $2 \_\_\_ -20$.

It may be helpful to refer to a number line.

(a) $14$ is to the right of $6$ on the number line, so $14 > 6$.

(b) $-1$ is to the left of $9$ on the number line, so $-1 < 9$.

(c) $-1$ is to the right of $-4$ on the number line, so $-1 > -4$.

(d) $2$ is to the right of $-20$ on the number line, so $2 > -20$.

You may have noticed that, 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 $2$.

Sometimes in algebra the same symbol has different meanings. Just like some words in English, the specific meaning becomes clear by looking at how it is used. You have seen the symbol “$-$” used in three different ways:

Opposite notation. $-a$ means the opposite of the number $a$. The notation $-a$ is read as “the opposite of $a$.”

Example. Find: (a) the opposite of $7$ (b) the opposite of $-10$ (c) $-(-6)$.

(a) $-7$ is the same distance from $0$ as $7$, but on the opposite side of $0$. The opposite of $7$ is $-7$.

(b) $10$ is the same distance from $0$ as $-10$, but on the opposite side of $0$. The opposite of $-10$ is $10$.

(c) $-(-6)$ is the opposite of $-6$, which is $6$.

When evaluating the opposite of a variable, we must be careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether $-x$ is positive or negative.

Example. Evaluate (a) $-x$, when $x = 8$ (b) $-x$, when $x = -8$.

(a) To evaluate when $x = 8$ means to substitute $8$ for $x$, then write the opposite: $-x = -(8) = -8$.

(b) To evaluate when $x = -8$ means to substitute $-8$ for $x$, then write the opposite: $-x = -(-8) = 8$.

Simplify Expressions with Absolute Value

We saw that numbers such as $2$ and $-2$ are opposites because they are the same distance from $0$ on the number line. They are both two units from $0$. The distance between $0$ and any number on the number line is called the absolute value of that number.

For example, $-5$ is $5$ units away from $0$, so $|-5| = 5$. And $5$ is $5$ units away from $0$, so $|5| = 5$.

The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from $0$ to $0$ on the number line is zero units.

Example. Simplify: (a) $|3|$ (b) $|-44|$ (c) $|0|$.

The absolute value of a number is the distance between the number and zero. Distance is never negative, so absolute value is never negative.

(a) $|3| = 3$

(b) $|-44| = 44$

(c) $|0| = 0$

In the next example, we’ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers.

Example. Fill in $<, >,$ or $=$ for each of the following pairs of numbers: (a) $|-5| \_\_\_ -|-5|$ (b) $8 \_\_\_ -|-8|$ (c) $-9 \_\_\_ -|-9|$ (d) $-(-16) \_\_\_ -|-16|$.

(a) Simplify each side: $|-5| = 5$ and $-|-5| = -5$. Since $5 > -5$, $|-5| > -|-5|$.

(b) Simplify: $8$ stays $8$, and $-|-8| = -8$. Since $8 > -8$, $8 > -|-8|$.

(c) Simplify: $-9$ stays $-9$, and $-|-9| = -9$. Since $-9 = -9$, $-9 = -|-9|$.

(d) Simplify: $-(-16) = 16$, and $-|-16| = -16$. Since $16 > -16$, $-(-16) > -|-16|$.

We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.

In the next example, we simplify the expressions inside absolute value bars first, just as we do with parentheses.

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

Next we evaluate absolute value expressions with a variable.

Example. Evaluate: (a) $|x|$ when $x = -35$ (b) $-|y|$ when $y = -20$ (c) $-|u|$ when $u = 12$ (d) $-|p|$ when $p = -14$.

(a) Substitute $-35$ for $x$, then take the absolute value: $|x| = |-35| = 35$.

(b) Substitute $-20$ for $y$. Simplify inside the bars first, then negate: $-|y| = -|-20| = -(20) = -20$.

(c) Substitute $12$ for $u$, take the absolute value, then negate: $-|u| = -|12| = -12$.

(d) Substitute $-14$ for $p$, simplify inside the bars, then negate: $-|p| = -|-14| = -14$.

Add Integers

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

We can model addition and subtraction of integers with positive and negative counters. One color (blue) represents positive counters; the other color (red) represents negative counters. If we have one positive counter and one negative counter, the value of the pair is zero — they form a neutral pair.

We use the counters to model the four addition facts using the numbers $5$, $-5$, $3$, and $-3$: $5+3$, $-5+(-3)$, $-5+3$, and $5+(-3)$.

To add $5 + 3$, we start with $5$ positive counters, then add $3$ more positive counters, giving $8$ positives. Likewise, to add $-5 + (-3)$, we start with $5$ negative counters, then add $3$ more negative counters, giving $8$ negatives.

In each case, when the signs were the same, the counters were all the same color, and so we added them.

So what happens when the signs are different? Let’s add $-5 + 3$. We start with $5$ negative counters, then add $3$ positive counters. Three of the negatives pair up with the three positives to form neutral pairs, which we remove. That leaves $2$ negatives, so $-5 + 3 = -2$. Notice there were more negatives than positives, so the result was negative.

Now add the last combination, $5 + (-3)$. Start with $5$ positive counters, then add $3$ negative counters. Three neutral pairs form and are removed, leaving $2$ positives, so $5 + (-3) = 2$.

5 + (-3)more positives — sum is positive5 + (-3) = 2

Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers. When you need to add numbers such as $37 + (-53)$, you really don’t want to count out 37 blue counters and 53 red counters. Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be negative. How many more red counters would there be? Because $53 - 37 = 16$, there are 16 more red counters. Therefore, the sum of $37 + (-53)$ is $-16$.

Let’s try another one. We’ll add $-74 + (-27)$. Imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is $-101$.

Example. Simplify: (a) $19 + (-47)$ (b) $-14 + (-36)$.

(a) Since the signs are different, we subtract $19$ from $47$. The answer will be negative because there are more negatives than positives: $19 + (-47) = -28$.

(b) Since the signs are the same, we add. The answer is negative because there are only negatives: $-14 + (-36) = -50$.

The techniques used up to now extend to more complicated problems, like the ones we’ve seen before. Remember to follow the order of operations!

Example. Simplify: $-5 + 3(-2 + 7)$.

Subtract Integers

We continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers. Perhaps when you were younger, you read “$5 - 3$” as “5 take away 3.” When you use counters, you can think of subtraction the same way!

To subtract $5-3$, restate the problem as “5 take away 3”: start with $5$ positive counters, take away $3$ positive counters, and $2$ positives are left. The difference of $5$ and $3$ is $2$.

To subtract $-5-(-3)$, restate it as “$-5$ take away $-3$”: start with $5$ negative counters, take away $3$ negative counters, and $2$ negatives are left. The difference of $-5$ and $-3$ is $-2$.

-5 - (-3) = -22 negatives left

Notice that these two examples are much alike: in the first, we subtract $3$ positives from $5$ positives and end up with $2$ positives. In the second, we subtract $3$ negatives from $5$ negatives and end up with $2$ negatives. Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters, as well as some neutral pairs. Adding a neutral pair does not change the value — it’s like changing quarters to nickels, the value is the same, but it looks different.

To subtract $-5-3$, restate it as “$-5$ take away $3$.” We start with $5$ negatives. We need to take away $3$ positives, but we don’t have any positives to take away. Since a neutral pair has value zero, we can add neutral pairs without changing the value. We add $3$ neutral pairs to the $5$ negatives, giving us $3$ positives to take away. Removing them leaves $8$ negatives: $-5 - 3 = -8$.

For the fourth case, $5-(-3)$, restate it as “$5$ take away $-3$.” We start with $5$ positives. We need to take away $3$ negatives, but there are no negatives to take away, so we add $3$ neutral pairs, giving us $3$ negatives to take away. Removing them leaves $8$ positives: $5-(-3) = 8$.

5 - (-3) = 8removing the 3 negatives leaves8 positives

Example. Subtract: (a) $-3-1$ (b) $3-(-1)$.

(a) We need $1$ more negative than we have, so add a neutral pair, then take $1$ positive away from that added pair: $-3-1 = -4$.

(b) We need $1$ more positive than we have, so add a neutral pair, then take $1$ negative away from that added pair: $3-(-1) = 4$.

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In the last example, $-3-1$ is the same as $-3+(-1)$, and $3-(-1)$ is the same as $3+1$. You will often see this idea, the subtraction property, written as follows:

Look at $6-4$ and $6+(-4)$: both give $2$. Of course, when you have a subtraction problem with only positive numbers, like $6-4$, you just do the subtraction — you already knew how to subtract $6-4$ long ago. But knowing that $6-4$ gives the same answer as $6+(-4)$ helps when you are subtracting negative numbers.

Example. Simplify: (a) $13-8$ and $13+(-8)$ (b) $-17-9$ and $-17+(-9)$.

(a) $13 - 8 = 5$ and $13 + (-8) = 5$ — the same.

(b) $-17-9 = -26$ and $-17+(-9)=-26$ — the same.

Now look at what happens when we subtract a negative: $8-(-5)$ gives the same result as $8+5$, namely $13$. Subtracting a negative number is like adding a positive! You will often see this written as $a - (-b) = a + b$.

Example. Simplify: (a) $9-(-15)$ and $9+15$ (b) $-7-(-4)$ and $-7+4$.

(a) $9-(-15) = 24$ and $9+15 = 24$ — the same.

(b) $-7-(-4) = -3$ and $-7+4=-3$ — the same.

Let’s look again at the results of subtracting the different combinations of $5$, $-5$, $3$, and $-3$.

What happens when there are more than two integers? We just use the order of operations as usual.

Example. Simplify: $7-(-4-3)-9$.

Key terms

negative numbers — numbers less than $0$; they lie to the left of zero on the number line. integers — the whole numbers and their opposites: $\dots -3, -2, -1, 0, 1, 2, 3 \dots$. opposite — the number that is the same distance from zero on the number line, but on the opposite side. absolute value — the distance between a number and $0$ on the number line, written $|n|$; always greater than or equal to zero. neutral pair — one positive counter and one negative counter together, with combined value zero. subtraction property — $a - b = a + (-b)$; subtracting a number is the same as adding its opposite.

This section is adapted from Elementary Algebra 2e, Section 1.3: Add and Subtract Integers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the number-line and positive/negative-counters figures as accessible inline graphics, and the grouping-symbols list as a table; omitted the Manipulative Mathematics callouts, media links, Self Check checklist, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.