## Section2.1Signed Numbers

### SubsectionTypes of Numbers

Numbers greater than zero are called positive numbers, and numbers less than zero are negative numbers. We use a number line to illustrate relationships among numbers.

To the right of zero we mark the numbers 1, 2, 3, 4, ... at evenly spaced intervals. These numbers are called the natural or counting numbers. The whole numbers include the natural numbers and zero: 0, 1, 2, 3, ... .

The natural numbers, zero, and the negatives of the natural numbers are called the integers:

\begin{equation*} \cdots -3,~ {-2},~ {-1},~ 0,~ 1,~ 2,~ 3 \cdots \end{equation*}

Fractions, such as $\dfrac{2}{3}, -\dfrac{5}{4}$ and $3.6\text{,}$ lie between the integers on the number line.

###### 1.

What are integers?

As we move from left to right on a number line, the numbers increase. In the figure below, the graph of $-6$ lies to the left of the graph of $-2\text{.}$ Therefore, $-6$ is less than $-2\text{,}$ or, equivalently, $-2$ is greater than $-6\text{.}$

We use special symbols to indicate order:

\begin{equation*} \begin{aligned} \blert{\lt} \amp ~~~~ \text{means}~~~~ \blert{\text{is less than}}\\ \blert{\gt} \amp ~~~~ \text{means}~~~~ \blert{\text{is greater than}} \end{aligned} \end{equation*}

For example,

\begin{align*} -6 \lt -2 \amp ~~~~ \text{means}~~~~ {-6}~~ \text{is less than}~~ {-2}\\ -2 \gt -6 \amp ~~~~ \text{means}~~~~ {-2}~~ \text{is greater than}~~ {-6} \end{align*}

The small ends of the symbols $\lt$ and $\gt$ always point to the smaller number.

###### Example2.1.
1. Which is the lower altitude, $-81$ feet or $-94$ feet?
2. Express the relationship using one of the order symbols.
Solution
1. Negative altitudes correspond to feet below sea level, and 94 feet is farther below sea level than 81 feet. Therefore, $-91$ feet is the lower altitude.
2. $-94 \lt -81\text{,}$ or $-81 \gt -94$

### SubsectionAdding Two Numbers with the Same Sign

We'll use number lines to review operations on signed numbers.

###### Case 1: The sum of two positive numbers.

Illustrate the sum $5+3$ on a number line.

1. Graph the first number, $+5\text{,}$ as shown below.
2. Move $3$ units in the positive direction, or to the right.
3. This brings us to the sum, which is $+8\text{.}$ Thus, $5+3=8\text{.}$
###### Case 2: The sum of two negative numbers.

Illustrate the sum $(-5)+(-3)$ on a number line.

1. Graph the first number, $-5\text{,}$ as shown below.
2. Move $3$ units in the negative direction, or to the left.
3. This brings us to the sum, $-8\text{,}$ as expected. Thus, $(-5)+(-3)=-8\text{.}$
###### Look Closer.

In the examples above, we see that:

\begin{equation*} \begin{aligned} \amp \blert{\text{The sum of two positive numbers is positive.}}\\ \amp \blert{\text{The sum of two negative numbers is negative.}} \end{aligned} \end{equation*}

Thus, the sign of the sum is the same as the sign of the two terms. To find the value of the sum, we add the absolute values, or unsigned parts, of the two terms.

###### Example2.2.

Add $(-4)+(-7)$

Solution

The sum is negative. We add the absolute values of $-4$ and $-7$ (the numbers 4 and 7 without their signs) to get 11, then make the sum negative: $(-4)+(-7)=-11$

### SubsectionAdding Two Numbers with Opposite Signs

Suppose you have debts of $9, and assets of$6. (Your "assets" are your possessions.) What is your net worth? We can model this situation by the sum

\begin{equation*} -9+(+6) \end{equation*}

where debts are represented by negative numbers and assets by positive numbers. If you use your $6 to pay off part of your debt, you will still owe$3, so your net worth is $-3\text{.}$ It makes sense that if your debts are greater than your assets, your net worth is negative.

Illustrate the sum $(-9)+(+6)$ on a number line.

1. Graph the first number, $-9\text{,}$ as shown below.
2. Move $6$ units in the positive direction, or to the right.
3. This brings us to the sum, $-3\text{.}$ Thus, $(-9)+(+6)=-3\text{.}$
###### Look Closer.

In the example above, we see that the sum has the same sign as the number with the larger absolute value. (Can you explain this in terms of debts and assets?) When the two numbers have opposite signs, we find the value of the sum by subtracting the absolute values of the two numbers.

###### Example2.3.
1. Add $(-5)+(+9)$
2. Add $(+5)+(-9)$
Solution
1. The two numbers have opposite signs, so we subtract 5 from 9 to get 4, then make the answer positive, because $+9$ has a larger absolute value than $-5\text{.}$ Thus, $(-5)+(+9)=4.$
2. We subtract 5 from 9 to get 4, then make the answer negative, because $-9$ has a larger absolute value than $+5\text{.}$ Thus,$(+5)+(-9)=-4.$

###### 2.

What is the absolute value of a number?

###### 3.

When we add two numbers with opposite signs, we their absolute values.

We now have two rules for adding signed numbers.

1. To add two numbers with the same sign, add their absolute values. The sum has the same sign as the numbers.
2. To add two numbers with opposite signs, subtract their absolute values. The sum has the same sign as the number with the larger absolute value.

### SubsectionSubtracting a Positive Number

Let's compare the two problems:

\begin{equation*} 9+(-5)~~~~\text{and}~~~~9-(+5) \end{equation*}

Illustrate the sum $9+(-5)$ on a number line.

1. Plot $9$ on the number line.
2. Move $5$ units in the negative direction, or to the left.
3. The sum is $4\text{.}$

Illustrate the sum $9-(+5)$ on a number line.

1. Plot $9$ on the number line.
2. Move $5$ units to the left, to indicate subtraction.
3. The result is $4\text{.}$ The picture is identical to the graph above.
###### Look Closer.

We see that subtracting a positive number has the same result as adding the negative number with the same absolute value. Thus, to subtract a positive number, we add its opposite.

### SubsectionSubtracting a Negative Number

When we add a negative number we move to the left on the number line. There are only two directions we can move on a number line, so when we subtract a negative number we must move to the right.

Compare the graphs for adding $-3$ and subtracting $-3$

• Addition: $~~~~5+(-3)~~~~$ Move 3 units to the left, to get 2.
• Subtraction: $~~~~5-(-3)~~~~$ Move 3 units to the right, to get 8.
###### Look Closer.

Notice that

\begin{equation*} 5-(-3)=8 ~~~~~ \text{and} ~~~~~5+(+3)=8 \end{equation*}

Subtracting a negative number has the same result as adding the positive number with the same absolute value. Thus, to subtract a negative number, we add its opposite.

###### 4.

True or False: $-x$ always represents a negative number. Give an example to support your answer.

###### 5.

Subtracting a negative number is the same as with the same absolute value.

From the examples above, we see that to subtract any number, positive or negative, we add its opposite. We can do this in steps as follows.

###### Rules for Subtracting Integers.

To subtract $b$ from $a\text{:}$

1. Change the sign of $b\text{.}$
2. Change the subtraction to addition.