Notes

introduction to signed numbers
To use a real number, we need to understand their sign. Many situations use negative and positive numbers. The phrase signed numbers represents both positive and negative numbers. A positive number is any number greater than 0. A negative number is any number less than 0.

Note, zero is neither positive nor negative.
The set of whole numbers includes the numbers 0, 1, 2, 3, 4, 5, 6, and so on. The opposites of these numbers are 0, minus1, minus2, minus3, minus4, minus5, minus6, and so on. The set of signed numbers is the set consisting of whole numbers and their opposites. Signed numbers are either positive or negative. Although you can write positive 13 as plus13, a positive number is typically written without the plus (+) sign. A number without a sign is assumed to be positive. In contrast, negative numbers will always have the negative sign in front.
A number line represents real numbers in order, with every point representing a real number and every real number corresponding to a point on the line. On a horizontal number line, it is usual to indicate positive numbers in increasing order moving to the right from a point labeled 0, and negative numbers in decreasing order, moving to the left from 0.

number line, negative 10 to 10

A signed number gives both direction and magnitude (distance). You can think of a positive number as moving forward and a negative number as moving backward—that is, to the right and to the left of 0, respectively. For example, if you wanted to represent a person walking backward 10 steps, you could use the signed number minus10. If another person is going forward 20 steps, you could simply represent it as 20.

Consider the signed numbers 5 and minus5. On a number line, 5 is located five units away from 0 to the right. Similarly, minus5 is located five units away from 0 but to the left. They are both the same number of units from 0. Note that “five units away” is a measure of distance. A signed number gives you both the direction and distance that a number is from 0. A positive number is to the right of 0, and a negative number is to the left of 0. The distance is the number itself. The distance of a number x from 0 is the absolute value of the number, and it is denoted by |x|. The absolute value of 5 and minus5 are the same, as shown on the number line below:

Therefore, vertical bar5 vertical bar equals5 a n d vertical bar minus5 vertical bar equals5

Similarly, minus13 is 13 units from 0 to the left, so open vertical bracket minus13 close vertical bracket equals13

The number 20 is 20 units from 0 to the right, soopen vertical bracket 20 close vertical bracket equals20

The absolute value of a number is really measuring distance, even though it seems to make negative numbers magically positive. Notice that the absolute value of a number is always 0 or greater.

Find open vertical bracket minus37 close vertical bracket.

solution

The open vertical bracket minus37 close vertical bracket means the distance between minus37 and 0 on the number line. Therefore,open vertical bracket minus37 close vertical bracket equals37.

example

Find open vertical bracket 99 close vertical bracket.

solution

The open vertical bracket 99 close vertical bracket means the distance between 99 and 0 on the number line. Therefore, open vertical bracket 99 close vertical bracket equals99

example

Find minus open vertical bracket 99 close vertical bracket.

solution

Notice that the negative sign is on the outside, so although open vertical bracket 99 close vertical bracket equals99, you have to keep the negative sign on the outside, as follows:

minus open vertical bracket 99 close vertical bracket equals minus99

example
John is going to run 5 miles to his friend's house. His starting position is represented by 0. After running 4 miles, he realizes that he dropped his keys and has to go back the other direction, or minus3 miles. He then continues on this path to his friend's house. How many miles did John run in total?

solution

To find the total amount of miles that John ran, the direction that he ran doesn’t matter – the distance matters.

When John picks up his keys, he will have to run towards his friend’s house those 3 miles again. Then, he will have to run one more mile to get to his friend’s house. (Remember that his friend’s house is 5 miles away.)

Add the absolute value of each segment of his run, as follows:

open vertical bracket 4 close vertical bracket plus open vertical bracket minus3 close vertical bracket plus open vertical bracket 3 close vertical bracket plus open vertical bracket 1 close vertical bracket equals4 plus3 plus3 plus1 equals11 miles

Therefore, John ran 11 miles in total.

example
The following table gives the weight change for participants on a diet:

Name Weight Change
Sue minus15 pounds
John 2 pounds
Bill minus30 pounds
Stephanie minus5 pounds
Who lost the most weight?
Whose weight changed the least?
Negative values represent weight loss. It is clear to see that Bill lost the most weight since he lost 30 pounds.

To find out whose weight changed the least, the absolute value of each weight change must be found. The least absolute value will give the weight that changed the least.

Name Weight Change Absolute Value
Sue minus15 pounds open vertical bracket minus15 close vertical bracket equals15
John 2 pounds open vertical bracket 2 close vertical bracket equals2
Bill minus30 pounds open vertical bracket minus30 close vertical bracket equals30
Stephanie minus5 pounds open vertical bracket minus5 close vertical bracket equals5
John's weight changed by only 2 pounds. Therefore, his weight changed the least.

example

When operating a car, the gears for drive and reverse are opposites—that is, drive moves the car forward, whereas reverse moves the car backward.

Suppose that the car starts at 0 feet along the number line. The driver shifts the car into drive and moves forward 600 feet (see Figure 1). From the original position, the car is now at +600, or 600, feet along the number line.

number line negative 800 to positive 800 with car image moving 0 to positive 600

In drive gear, a car moves forward 600 feet to +600 on the number line.

Suppose that a second car starts from the same point at 0 feet along the number line. The driver shifts the car into reverse and moves backward 300 feet (see Figure 2). From the original position, the car is now at minus300 feet along the number line.

number line negative 800 to positive 800 with car image moving 0 to negative 300

Figure 2: In reverse gear, a second car moves backward 300 feet to -300 on the number line.

How far is each car from the starting point?

solution

The first car is at +600, or 600, so it is 600 feet from the starting point. The second car is at minus300, so it is 300 feet from the starting point. To say that the car is minus300 feet from the starting point does not make sense. When the context is distance, a negative number does not make sense because distance is always positive.

Absolute value bars are used to indicate the absolute value of a number. Referring to the car going forward (see Figure 1), it is open vertical bracket 600 close vertical bracket equals600 feet from the starting point. In terms of the second car going in reverse, it is open vertical bracket minus300 close vertical bracket equals300 feet from the starting point.

Video resource Watch the following video to learn more about absolute values of specific numbers:

Math is all around you. You can see math from the time that you look at a clock when you wake up in the morning to when you calculate the volume of toothpaste left before you go to bed. You can use math throughout your day and not even notice. People use addition and subtraction of signed numbers when balancing their checkbooks, checking the temperature, or even determining their score in golf! In this lesson, you will learn to add and subtract signed numbers.

When adding two integers, it is important to keep track of the signs of the numbers.

The sum of two positive numbers is always positive.
Examples of adding positive integers
integers sum
2+9= 11
17+5= 22
29+16= 45
The sum of two negative numbers is always negative.
Examples of adding negative integers
integers sum
-2+-9= -11
-5+-8= -13
-13+-7= -20
Steps for adding a positive and a negative number are as follows:

1.Find the absolute value of each integer.
2.Subtract the smaller absolute value from the larger absolute value.
3.The result from step 2 takes the same sign of the number with the greater absolute value.

Example 1:

Add minus10 plus8.


Solution:

One of the numbers is negative and the other is positive, so look at the absolute value of each number.

open vertical bracket minus10 close vertical bracket equals10

open vertical bracket 8 close vertical bracket equals8

minus10 has the larger absolute value so the final answer will be negative. Find the difference between the larger absolute value and smaller absolute value. Then, make the final sign negative.

10 minus8 equals2

The final answer is minus2.

minus10 plus8 equals minus2
The final answer is minus2.

minus10 plus8 equals minus2

Example 2:

Add open bracket 25 close bracket plus open bracket minus7 close bracket.
The difference of two numbers is the result of subtracting one number from the other. When the word difference is used in a mathematical situation, it implies the operation of subtraction.

Subtraction undoes addition. Addition and subtraction are sometimes called opposite operations.

To subtract signed numbers, simply add the opposite and then follow the rules of addition.

examples

Example 1

Subtract 10 minus8.

Multiplication of signed numbers

The product of numbers (or expressions) is the result of multiplying them. For example, the product of 7 and 4 is 28, or 7 times4 equals28.

When whole numbers are multiplied together, they are called factors of the product. For example, 7 and 4 are factors of 28 because 7 times4 equals28. The number 2 is a factor of 4 because 2 times2 equals4. The number 2 is also a factor of 28 since you could write 7 times2 times2 equals28.

The most common symbols used to represent multiplication are times, ·, and *. A quantity that is written next to a set of parentheses also denotes multiplication. For example, the following expressions represent 7 multiplied by 4:

7 times 4
7 · 4
7 * 4
7(4)
(7)(4)
For variables, simply write the variables next to each other to denote multiplication. For example, the product of x and y is written as x y

There are two main methods of multiplying numbers by hand: the expanded form and the carrying method. In the next example, two numbers are multiplied using both methods.

example
Multiply 468 and 4.

solution

Expanded Form

To multiply a multidigit number by a single-digit number, write the multidigit number in expanded form. The number 468 can be written as 400 plus60 plus8.

Next, multiply 4 by every number in the expanded form, as follows:

4 times400 equals1 comma600

4 times60 equals240

4 times8 equals32
Finally, add the products together to get the final answer, as follows:

1 comma600 plus240 plus32 equals1 comma872

Therefore, 468 times4 equals1 comma872.

Figure 1 shows the same multiplication written vertically. When writing the product vertically, the ones, tens, hundreds, and thousands place of the numbers are aligned vertically. This is very important: The place values must always be aligned vertically.


Multiplying the Expanded Form of a Multidigit Number

Figure 1: Multiplying the Expanded Form - Vertically
Finally, add the products together to get the final answer, as follows:

1 comma600 plus240 plus32 equals1 comma872

Therefore, 468 times4 equals1 comma872.

Figure 1 shows the same multiplication written vertically. When writing the product vertically, the ones, tens, hundreds, and thousands place of the numbers are aligned vertically. This is very important: The place values must always be aligned vertically.


Multiplying the Expanded Form of a Multidigit Number

Figure 1: Multiplying the Expanded Form - VerticallyFinally, add the products together to get the final answer, as follows:

1 comma600 plus240 plus32 equals1 comma872

Therefore, 468 times4 equals1 comma872.

Figure 1 shows the same multiplication written vertically. When writing the product vertically, the ones, tens, hundreds, and thousands place of the numbers are aligned vertically. This is very important: The place values must always be aligned vertically.


Multiplying the Expanded Form of a Multidigit Number

Figure 1: Multiplying the Expanded Form - VerticallyFinally, add the products together to get the final answer, as follows:

1 comma600 plus240 plus32 equals1 comma872

Therefore, 468 times4 equals1 comma872.

Figure 1 shows the same multiplication written vertically. When writing the product vertically, the ones, tens, hundreds, and thousands place of the numbers are aligned vertically. This is very important: The place values must always be aligned vertically.


Multiplying the Expanded Form of a Multidigit Number

Figure 1: Multiplying the Expanded Form - Vertically