8.1: Addition and Subtraction

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Whole numbers and fractions are valuable tools for solving a wide range of problems in everyday life. However, many real-world situations call for the use of negative numbers. For instance, negative numbers are essential for representing temperatures below zero, elevations below sea level, financial losses in the stock market, and overdrawn bank accounts. In this chapter, we explore the set of integers—numbers that include all whole numbers along with their negative counterparts, which are the opposites of the nonzero whole numbers.

Figure \(\PageIndex{1}\): Temperatures below zero are described by negative numbers.

Figure \(\PageIndex{2}\): The surface of the Mediterranean Sea has an elevation of $0\text{ft}.$ The diagram shows that nearby mountains have higher (positive) elevations whereas the Dead Sea has a lower (negative) elevation.

Figure \(\PageIndex{3}\): Depths below sea level are described by negative numbers. A submarine $500\text{ft}$ below sea level is at $\mathrm{-500}\text{ft}.$

Wonder, Play, Grow

Before we start learning the specifics of integers, let's consider the different situations and scenarios where negative numbers arise. Write a word problem for each of the following expressions.

1. -20 + 8
2. -20 - 8
3. -20 - (-8)

Be prepared to share with the class.

The opposite or negative of any whole number is easily located on the number line.

Number Line Locations

To locate the opposite (or negative) of any whole number, first locate the whole number on the number line. The opposite is the reflection of the whole number through the origin (zero).

To find its opposite, reflect the number 5 through the origin. This will be the location of the opposite (negative) of the whole number 5, which we indicate by the symbol −5.

The symbol −5 is pronounced in one of two ways: (1) “negative five,” or (2) “the opposite of five.”

In similar fashion, we can locate the opposite or negative of any whole number by reflecting the whole number through the origin (zero), which leads to the image shown in Figure 8.1.4.

Figure 8.1.4: The opposite (negative) of any whole number is a reflection of that number through the origin (zero).

The Integers

The collection of numbers arranged on the number line in Figure 8.4.1 extend indefinitely to the right, and because the numbers on the left are reflections through the origin, the numbers also extend indefinitely to the left. This collection of numbers is called the set of integers.

The Integers

The infinite collection of numbers

{..., −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, ...}

is called the set of integers.

The ellipsis . . . at each end of this infinite collection means “etcetera,” as the integers continue indefinitely to the right and left. Thus, for example, both 23,456 and −117, 191 are elements of this set and are therefore integers.

Ordering the Integers

As we saw with the whole numbers, as you move to the right on the number line, the numbers get larger; as you move to the left, the numbers get smaller. Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers, Figure 8.1.5.

Figure \(\PageIndex{5}\)

Order on the Number Line

Let a and b be integers located on the number line so that the point representing the integer a lies to the left of the point representing the integer b.

Then the integer a is “less than” the integer b and we write

\[ a < b\nonumber \]

Alternatively, we can also say that the integer b is “larger than” the integer a and write

\[ b > a.\nonumber \]

Example \(\PageIndex{1}\)

Replace each shaded box with < (less than) or > (greater than) so the resulting inequality is a true statement.

Answer

Note that −3 lies to the left of 5, so: -3 < 5.

Note that −2 lies to the right of −4, so: -2 > -4

We stated earlier that every integer has a unique number called its “opposite” or “negative.” Thus, the integer −5 is the opposite (negative) of the integer 5. Thus, we can say that the pair −5 and 5 are opposites. Each is the opposite of the other. Logically, this leads us to the conclusion that the opposite of −5 is 5. In symbols, we would write

\[ -(-5) = 5.\nonumber \]

Opposites of Opposites

Let a be an integer. Then the “opposite of the opposite of a is a.” In symbols, we write

\[ -(-a) = a.\nonumber \]

We can also state that the “negative of a negative a is a.

For example: −(−13) = 13

Model Addition of Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with 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 difficult. This difficulty relates to the way the brain learns.

The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of $2-1.$ Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.

Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.

Set Model

Addition means to put together or form the union of two disjoint sets. We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.

If we have one positive and one negative counter, the value of the pair is zero. The counters form a neutral pair. The value of this neutral pair is zero as summarized in Figure 8.1.6.

Figure 8.1.6 A blue counter represents $+1.$ A red counter represents $\mathrm{-1}.$ Together they add to zero.

We will model using

$$5+3\mathrm{-5}+\left(\mathrm{-3}\right)\mathrm{-5}+3$$

Set Model Addition

Model: 5 + 3 .

Interpret the expression.	$5+3$ means the sum of $5$ and $3$.
Model the first number. Start with 5 positives.
Model the second number. Add 3 positives.
Count the total number of counters.
The sum of 5 and 3 is 8.	$5+3=8$

Model: −5 + ( −3 ) .

Interpret the expression.	$\mathrm{-5}+\left(\mathrm{-3}\right)$ means the sum of $\mathrm{-5}$ and $\mathrm{-3}$.
Model the first number. Start with 5 negatives.
Model the second number. Add 3 negatives.
Count the total number of counters.
The sum of −5 and −3 is −8.	$\mathrm{-5}+\mathrm{-3}=\mathrm{-8}$

Model: −5 + 3 .

Interpret the expression.	$\mathrm{-5}+3$ means the sum of $\mathrm{-5}$ and $3$.
Model the first number. Start with 5 negatives.
Model the second number. Add 3 positives.
Remove any neutral pairs.
Count the result.
The sum of −5 and 3 is −2.	$\mathrm{-5}+3=\mathrm{-2}$

Notice that there were more negatives than positives, so the result is negative.

Measurement Model

Addition means to put directed arrows end to end starting at zero. Note that positive integers are represented by arrows pointing to the right and negative integers by arrows pointing to the left (Figure 8.1.7-9).

Figure \(\PageIndex{7}\): 2 + 1

Figure \(\PageIndex{8}\): -3 + (-1)

Figure \(\PageIndex{9}\): 3 + (-4)

Example \(\PageIndex{2}\)

Model each addition.

a) −3 + 6

b) 4 + (−5)

c) -2 + (−3)

Answer

a) −3 + 6 = 3

b) 4 + ( −5 ) = −1

c) −2 + ( −3 ) = −5

 

This conceptual work leads us to the following definition of integer addition.

Definition: Addition of Integers

Let a and b be integers.

1. Adding zero: a + 0 = 0 + a = a.

2. Adding two positives: If a and b are positive, they are added as whole numbers.

3. Adding two negatives: If a and b are positive (hence −a and −b are negative), then (−a) + (−b) = −(a + b), where a + b is the whole-number sum of a and b.

4. Adding a positive and a negative:

a. If a and b are positive and a ≥ b, then a + (−b) = a − b, where a − b is thewhole-number difference of a and b.

b. If a and b are positive and a < b, then a + (−b) = −(b − a), where b − a is the whole-number difference of a and b.

Example \(\PageIndex{3}\)

Calculate the following using the definition of integer addition.

a) 3 + 0 b) 3 + 4 c) (−3) + (−4)

d) 7 + (−3) e) 3 + (−7) f) 5 + (−5)

Answer

a. Adding zero: 3 + 0 = 3

b. Adding two positives: 3 + 4 = 7

c. Adding two negatives: (−3) + (−4) = −(3 + 4) = −7

d. Adding a positive and a negative: 7 + (−3) = 7 − 3 = 4

e. Adding a positive and a negative: 3 + (−7) = −(7 − 3) = −4

f. Adding a number and its opposite: 5 + (−5) = 0

Additive Inverse Property for Integer Addition

For each integer a there is a unique integer, written −a, such that a + (−a) = 0. The integer −a is called the additive inverse of a. In words, this property states that any number plus its additive inverse is zero.

Algebraic Reasoning: When solving an equation such as x + 4 = 3, the additive inverse of 4, namely −4, is added to both sides of the equation as follows: x + 4 + (−4) = 3 + (−4). This shows that x = −1.

Model Subtraction of Integers

Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.

Subtraction of integers can be viewed in several ways.

Pattern Approach: What do you notice? Come up with your own examples after you see what is going on.

Subtracting a negative number is the same as adding the positive version of that number.

Adding the Opposite: Find 2 - 5.

Let a and b be any integers. Then a − b = a + (−b).

2 - 5 = 2 + (-5) = -3

Take Away Approach:

Remember, the blue counters represent positive numbers and the red counters represent negative numbers. Perhaps when you were younger, you read $5-3$ as five take away three. When we use counters, we can think of subtraction the same way.

Model: 5 − 3 . 5 - 3 = 2.

Model: −5 −(-3) . −5 − ( −3 ) = −2.

Model: - 5 − 3 .

But there are no positives to take away. Add neutral pairs until you have 3 positives.

−5 − 3 = −8

TRY 5 - (-3)

5 − ( −3 ) = 8

Let's see what Dr. Howie Hua has to say about the Take-Away Approach.

Example \(\PageIndex{4}\)

Model each subtraction using a different approach.

a) −5 − 4

b) 6 − (−6)

c) −8 − (−3)

Answer

, , , −5 − 4 = −9

b) 6 + (6) = 12

,,, 6 − ( −6 ) = 12

c) -8 + ( 3) = -5

Interactive Element

Think About and Discuss

Recall that another approach to subtraction, the missing-addend approach, was used in whole-number subtraction. For example, 6 − 2 = n if and only if 6 = 2 + n. In this way, subtraction can be done by referring to addition. This method can also be extended to integer subtraction.

Missing-Addend Approach: Let a, b, and c be any integers. Then a − b = c if and only if a = b + c.

Think about and discuss with your peers how the Missing-Addend Approach can be used to find 4 - (-2). Then come up with another example that solidifies your understanding.

Answer

4 − (−2) = c if and only if 4 = (−2) + c. But 4 = −2 + 6. Therefore, c=6.

Absolute Value (Optional Section for this Course - But Still Important for Future Teachers)

We define the absolute value of an integer.

Absolute Value

The absolute value of an integer is defined as its distance from the origin (zero).

It is important to note that distance is always a nonnegative quantity (not negative); i.e., distance is either positive or zero. As an example, we’ve shaded the integers −4 and 4 on a number line.

The number line above shows two cases:

• The integer −4 is 4 units from zero. Because absolute value measures the distance from zero, | − 4| = 4.
• The integer 4 is also 4 units from zero. Again, absolute value measures the distance from zero, so |4| = 4.

Let’s look at another example.

Example \(\PageIndex{5}\)

Determine the value of each expression: a) | − 7|, b) |3|, and c) |0|.

Answer

The absolute value of any integer is equal to the distance that number is from the origin (zero) on the number line. Thus:

a) The integer −7 is 7 units from the origin; hence, | − 7| = 7.

b) The integer 3 is 3 units from the origin; hence, |3| = 3.

c) The integer 0 is 0 units from the origin; hence, |0| =0.

Example \(\PageIndex{6}\)

Determine the value of each expression: a) −(−8) and b) −| − 8|.

Answer

These are distinctly different problems.

a) The opposite of −8 is 8. That is, −(−8) = 8.

b) However, in this case, we take the absolute value of −8 first, which is 8, then the opposite of that result to get −8. That is,

\[ \begin{aligned} - | -8 | = -(8) & \textcolor{red}{ \text{ First: } |-8| = 8.} \\ = -8 ~ & \textcolor{red}{ \text{ Second: The opposite of 8 is } -8.} \end{aligned}\nonumber \]

An alternate definition of absolute value is: \[
|a| =
\begin{cases}
a & \text{if } a \text{ is positive or zero}. \\
-a & \text{if } a \text{ is negative}.
\end{cases}
\]

Example \(\PageIndex{7}\)

Using this definition, calculate the following values.

a) |− 3|

b) |7|

c) |0|

Answer

a) 3 b) 7 c) 0

Interactive Element