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1 Foundations 1 (Integers and Language of Algebra )

Topics Covered:

This module provides a review of those topics needed prior to the start of the semester.

Identify Counting Numbers and Whole Numbers[1]

Learning algebra is similar to learning a language. You start with a basic vocabulary and then add to it as you go along. You need to practice often until the vocabulary becomes easy to you. The more you use the vocabulary, the more familiar it becomes.

Algebra uses numbers and symbols to represent words and ideas. Let’s look at the numbers first. The most basic numbers used in algebra are those we use to count objects: 1, 2, 3, 4, 5, …  and so on. These are called counting numbers. The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly. Counting numbers are also called natural numbers.

Counting numbers start with 1 and continue.

1, 2, 3, 4, 5, …

 

Counting numbers and whole numbers can be visualized on a number line as shown below.


An image of a number line from 0 to 6 in increments of one. An arrow above the number line pointing to the right with the label “larger”. An arrow pointing to the left with the label “smaller”.The point labeled 0 is called the origin. The points are equally spaced to the right of 0 and labeled with the counting numbers. When a number is paired with a point, it is called the coordinate of the point.

The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the whole numbers.

Whole numbers are counting numbers and zero.

0, 1, 2, 3, 4, 5, …
We stopped at 5 when listing the first few counting numbers and whole numbers. We could have written more numbers if they were needed to make the patterns clear.

Try it!

Which of the following are  a. numbers? b. whole numbers?

0, \displaystyle \frac{1}{4}, 3, 5.2, 15, 105

Solution (click to reveal)

a. The counting numbers start at 1, so 0 is not a counting number. The numbers 3, 15, and 105 are all counting numbers.

b. Whole numbers are counting numbers and 0. The numbers 0, 3, 15, and 105 are whole numbers.The numbers \displaystyle \frac{1}{4}  and 5.2 are neither counting numbers nor whole numbers. We will discuss these numbers later.

Use Variables and Algebraic Symbols[2]

Greg and Alex have the same birthday, but they were born in different years. This year Greg is 20 years old and Alex is 23, so Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change or vary, so age is a variable. The 3 years between them always stay the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age g. Then we could use g + 3 to represent Alex’s age. See below.

Greg’s age Alex’s age
12 15
20 23
35 38
g g + 3

Letters are used to represent variables. Letters often used for variables are x, y, a, b, and c.

 A variable is a letter that represents a number or quantity whose value may change.

A constant is a number whose value always stays the same.

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. We will summarize the four basic arithmetic operations: addition, subtraction, multiplication, and division here, along with words we use for the operations and the result.

Operation Notation Say: The result is…
Addition a+b a plus b the sum of a and b
Subtraction a-b a minus b the difference of a and b
Multiplication a\cdot b, (a)(b), (a)b, a(b) a times b The product of a and b
Division \displaystyle a\div b, a/b, \frac{a}{b} a divided by b The quotient of a and b

In algebra, the cross symbol, \times, is not used to show multiplication because that symbol may cause confusion. Does 3xy mean (three times y) or 3 \cdot x \cdot y (three times x times y)? To make it clear, use • or parentheses for multiplication.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

The sum of 5 and 3 means add 5 plus 3, which we write as 5 + 3.

The difference of 9 and 2 means subtract 9 minus 2, which we write as 9 – 2.

The product of 4 and 8 means multiply 4 times 8, which we can write as 4 \boldsymbol{\cdot} 8.

The quotient of 20 and 5 means divide 20 by 5, which we can write as 20 \div 5.

When two quantities have the same value, we say they are equal and connect them with an equal sign.

a = b is read a is equal to b.

The symbol = is called the equal sign.

An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that b is greater than  a, it means that b is to the right of a on the number line. We use the symbols “<”  and “>” for inequalities.

Inequality:

Visual of line graph
 Translation of graph
The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.
a < b is read a is less than b

a is to the left of b on the number line
The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right. a > b is read a is greater than b

b is to the left of a on the number line

The expressions a < b and a > b can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,

Algebraic Notation Say
a < b is equivalent to b > a For example, 7 < 11 is equivalent to 11 > 7.
a > b is equivalent to b < a For example, 17 > 4 is equivalent to 4 < 17.

When we write an inequality symbol with a line under it, such as  a ≤ b, it means a < b or a = b.  We read this a is less than or equal to b. Also, if we put a slash through an equal sign, ≠, it means not equal.

We summarize the symbols of equality and inequality in the table below.

Algebraic Notation Say
a = b a is equal to b
a ≠ b a is not equal to b
a < b a is less than b
a > b a is greater than b
a ≤ b a is less than or equal to b
a ≥ b a is greater than or equal to b

Symbols < and >: The symbols < and > each have a smaller side and a larger side.

The smaller side of the symbol faces the smaller number and the larger faces the larger number.

smaller side < larger side

larger side > smaller side

Try it!

Translate from algebra to words:

a. 20 ≤ 35     b. 11 ≠ 15 − 3      c. 9 > 10 ÷ 2       d. x + 2 < 10

Solution A (click to reveal)

a. 20 ≤ 35

20 is less than or equal to 35

 

 

Solution B (click to reveal)
b.
11 ≠ 15 − 3
11 is not equal to 15 minus 3

 

Solution C (click to reveal)
c.
9 > 10 ÷ 2
9 is greater than 10 divided by 2

 

Solution D (click to reveal)
d.
x + 2 < 10
x plus 2 is less than 10
 Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The following table lists three of the most commonly used grouping symbols in algebra.
Common Grouping Symbols
parentheses (    )
brackets [   ]
braces {   }

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section..

8(14 − 8)

21 − 3[2 + 4(9 − 8)]

24 ÷ {13 − 2[1(6 − 5) + 4]}

Use Negatives and Opposites[3]

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.

A number line extends from negative 4 to 4. A bracket is under the values “negative 4” to “0” and is labeled “Negative numbers”. Another bracket is under the values 0 to 4 and labeled “positive numbers”. There is an arrow in between both brackets pointing upward to zero.
The number line shows the location of positive and negative numbers.

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.

A number line ranges from negative 4 to 4. An arrow above the number line extends from negative 1 towards 4 and is labeled “larger”. An arrow below the number line extends from 1 towards negative 4 and is labeled “smaller”.
The numbers on a number line increase in value going from left to right and decrease in value going from right to left.

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.

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 which showed the whole numbers to include negative numbers, too. The numbers marked by points in the image below are called the integers. The integers are the numbers …−3, −2, −1, 0, 1, 2, 3 …

A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four
All the marked numbers are called integers.

Try it!

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

a. 14\rule{1cm}{1pt}6

b. −1\rule{1cm}{1pt}9

c. −1\rule{1cm}{1pt}−4

d. 2\rule{1cm}{1pt} −20.

Solution (click to reveal)

It may be helpful to refer to the number line shown.

A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.

Steps
 Algebraic
a.  14 is to the right of 6 on the number line. 14\rule{1cm}{1pt}6 = 14 > 6
b.  −1 is to the left of 9 on the number line. −1\rule{1cm}{1pt}9 = −1 < 9
c.  −1 is to the right of −4 on the number line. −1\rule{1cm}{1pt}−4 = −1 > −4
d. 2 is to the right of −20 on the number line. 2\rule{1cm}{1pt}−20 = 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.

Opposite

The Oppositeof a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.
The opposite of 3 is −3.

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.

Algebraic

Translation
10 − 4 Between two numbers, it indicates the operation of subtraction.

We read 10 − 4 as “10 minus 4.”
−8 In front of a number, it indicates a negative number.

We read −8 as “negative eight.”
−x In front of a variable, it indicates the opposite.

We read −x as “the opposite of x.“
−(−2) Here there are two “−” signs. The one in the parentheses tells us the number is negative 2. The one outside the parentheses tells us to take the opposite of −2.

We read −(−2) as “the opposite of negative two.”
Opposite Notation
−a means the opposite of the number a.
The notation −a is read as “the opposite of a.”

Try it!

Find:  a. the opposite of 7   b. the opposite of −10   c. −(−6).

Solution (click to reveal)

 

Translation
 Algebraic
a. −7 is the same distance from 0 as 7, but on the opposite side of 0. This figure has three parts: a , b, and c. Each part has text on the left and a corresponding number line on the right. The text for part “a” reads “negative seven is the same distance from zero as seven, but on the opposite side of zero.” To the right is a number line ranging from negative seven to seven. There are two brackets above the number line. The first bracket is from negative 7 to zero. Above this bracket reads “10”. The second bracket is from 0 to seven. Above this bracket reads “10”. Under the number line it reads “The opposite of seven is negative seven”. Points are plotted on negative seven and seven. The text for part “b” reads “ten is the same distance from zero as negative 10, but on the opposite side of zero.” To the right is is a number line ranging from negative ten to ten. There are two brackets above the number line. The first bracket is from negative 10 to zero. Above this bracket reads “10”. The second bracket is from 0 to ten. Above this bracket reads “10”. Under the number line it reads “The opposite of negative ten is ten”. Points are plotted on negative 10 and 10. The text for part “c” is an algebraic term: negative sign, open parenthesis, negative 6, close parenthesis. To the right is a number line ranging from negative six to six. There are two brackets above the number line. The first bracket is from negative 6 to zero. Above this bracket reads “6”. The second bracket is from 0 to 6. Above this bracket reads “six”. Under the number line it reads “The opposite of negative 6 is six”. Points are plotted on negative 6 and 6.

The opposite of 7 is −7.
b. 10 is the same distance from 0 as −10, but on the opposite side of 0. This figure has three parts: a , b, and c. Each part has text on the left and a corresponding number line on the right. The text for part “a” reads “negative seven is the same distance from zero as seven, but on the opposite side of zero.” To the right is a number line ranging from negative seven to seven. There are two brackets above the number line. The first bracket is from negative 7 to zero. Above this bracket reads “10”. The second bracket is from 0 to seven. Above this bracket reads “10”. Under the number line it reads “The opposite of seven is negative seven”. Points are plotted on negative seven and seven. The text for part “b” reads “ten is the same distance from zero as negative 10, but on the opposite side of zero.” To the right is is a number line ranging from negative ten to ten. There are two brackets above the number line. The first bracket is from negative 10 to zero. Above this bracket reads “10”. The second bracket is from 0 to ten. Above this bracket reads “10”. Under the number line it reads “The opposite of negative ten is ten”. Points are plotted on negative 10 and 10. The text for part “c” is an algebraic term: negative sign, open parenthesis, negative 6, close parenthesis. To the right is a number line ranging from negative six to six. There are two brackets above the number line. The first bracket is from negative 6 to zero. Above this bracket reads “6”. The second bracket is from 0 to 6. Above this bracket reads “six”. Under the number line it reads “The opposite of negative 6 is six”. Points are plotted on negative 6 and 6.

The opposite of −10 is 10.
c. −(−6) This figure has three parts: a , b, and c. Each part has text on the left and a corresponding number line on the right. The text for part “a” reads “negative seven is the same distance from zero as seven, but on the opposite side of zero.” To the right is a number line ranging from negative seven to seven. There are two brackets above the number line. The first bracket is from negative 7 to zero. Above this bracket reads “10”. The second bracket is from 0 to seven. Above this bracket reads “10”. Under the number line it reads “The opposite of seven is negative seven”. Points are plotted on negative seven and seven. The text for part “b” reads “ten is the same distance from zero as negative 10, but on the opposite side of zero.” To the right is is a number line ranging from negative ten to ten. There are two brackets above the number line. The first bracket is from negative 10 to zero. Above this bracket reads “10”. The second bracket is from 0 to ten. Above this bracket reads “10”. Under the number line it reads “The opposite of negative ten is ten”. Points are plotted on negative 10 and 10. The text for part “c” is an algebraic term: negative sign, open parenthesis, negative 6, close parenthesis. To the right is a number line ranging from negative six to six. There are two brackets above the number line. The first bracket is from negative 6 to zero. Above this bracket reads “6”. The second bracket is from 0 to 6. Above this bracket reads “six”. Under the number line it reads “The opposite of negative 6 is six”. Points are plotted on negative 6 and 6.

The opposite of −(−6) is −6.

Our work with opposites gives us a way to define the integers. The whole numbers and their opposites are called the integers. The integers are the numbers …−3, −2, −1, 0, 1, 2, 3…

Integers

 Whole numbers and their opposites are called integers.

Integers are the numbers

…−3, −2, −1, 0, 1, 2, 3…

When evaluating the opposite of a variable, we must be very careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether −x is positive or negative. We can see this in the example below.

Try it!

Evaluate:

 a. −x, when x = 8

b.  −x, when x = −8.

Solution A (click to reveal)

a. To evaluate when x = 8 means to substitute 8 for x.

Steps  Algebraic
Example −x
Substitute 8 for x.      −(8)
Write the opposite of 8.          −8                        

 

Solution B (click to reveal)

b. To evaluate when x = −8 means to substitute −8 for x.

Steps  Algebraic
Example −x
Substitute –8 for x.      −(-8)
Write the opposite of -8.         8

Add and Subtract Integers[4]

So far, we have only used counting numbers and whole numbers.

Counting numbers 1, 2, 3…
Whole numbers 0, 1, 2, 3…

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 will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one color (blue) represent positive. The other color (red) will represent the negatives.

Figure show two circles labeled positive blue and negative red.

If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

Figure shows a blue circle and a red circle encircled in a larger shape. This is labeled 1 plus minus 1 equals 0.

We will use the counters to show how to add:

a. 5 + 3,

b. −5 + (−3),

c. −5 + 3,

d. 5 + (−3)

The first example, 5 + 3, adds 5 positives and 3 positives—both positives.

The second example, −5 + (−3), adds 5 negatives and 3 negatives—both negatives.

When the signs are the same, the counters are all the same color, and so we add them. In each case we get 8—either 8 positives or 8 negatives.

Figure on the left is labeled 5 plus 3. It shows 8 blue circles. 5 plus 3 equals 8. Figure on the right is labeled minus 5 plus open parentheses minus 3 close parentheses. It shows 8 blue circles labeled 8 negatives. Minus 5 plus open parentheses minus 3 close parentheses equals minus 8.

So what happens when the signs are different? Let’s add −5 + 3 and 5 + (−3).

When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.

Figure on the left is labeled minus 5 plus 3. It has 5 red circles and 3 blue circles. Three pairs of red and blue circles are formed. More negatives means the sum is negative. The figure on the right is labeled 5 plus minus 3. It has 5 blue and 3 red circles. Three pairs of red and blue circles are formed. More positives means the sum is positive.

Try it!

Add: a. −1 + (−4)       b. −1 + 5          c. 1 + (−5).

Solution

a.

Steps  Algebraic
Original example −1 +(−4)
Show counters Figure shows five red circles. It is labeled minus 1 minus 4. 1 negative plus 4 negatives is 5 negatives. The answer is minus 5
1 negative plus 4 negatives is 5 negatives -5

b.

Steps  Algebraic
Original example −1 + 5
Show counters Figure is labeled minus 1 plus 5. It has 5 blue circles and 1 red circle. 1 pair of red and blue circles is formed. There are more positives, so the sum is positive. The answer is 4.
There are more positives, so the sum is positive. 4

c.

Steps  Algebraic
Original example 1 + (−5)
Show counters The figure is labeled 1 plus minus 5. There is 1 blue circle and 5 red circles. 1 pair of blue and red circles is formed. There are more negatives, so the sum is negative. The answer is minus 4
There are more negatives, so the sum is negative. -4

We will continue to use counters to model the subtraction. 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!

We will use the counters to show how to subtract:

Ex. 1 Ex. 2 Ex. 3 Ex. 4
5 – 3 −5 – (−3) −5 – 3 5 – (−3)

The first example, 5 − 3 we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, −5 − (−3), 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.

Figure on the left is labeled 5 minus 3 equals 2. There are 5 blue circles. Three of these are encircled and an arrow indicates that they are taken away. The figure on the right is labeled minus 5 minus open parentheses minus 3 close parentheses equals minus 2. There are 5 red circles. Three of these are encircled and an arrow indicates that they are taken away.

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. If we don’t have the number of counters needed to take away, we add neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.

Let’s look at −5 − 3 and 5 − (−3).

Steps Ex. 1 Ex. 2
Examples −5 − 3 5 − (−3)
Model the first number. Calculations of two expressions are shown. The expression on the left is minus 5 minus 3. We first model the first number. So, we have 5 red counters Calculations of two expressions are shown. The expression on the right is 5 minus open parentheses minus 3 close parentheses. We first model the first number. So, we have 5 white counters.
We now add the needed neutral pairs. We have 5 red counters and now add three neutral pairs . So we have 8 red and 3 white counters. We now add three neutral pairs for both expressions. So we have 8 white and 3 red counters.
We remove the number of counters modeled by the second number. We now remove the number of counters modeled by the second number. Three white counters are removed, shown circled in red. We now remove the number of counters modeled by the second number. Three red counters are removed, shown circled in red.
Count what is left. Image shows 8 red circles. So, minus 5 minus 3 is minus 8 We now have 8 white counters. Hence, 5 minus negative 3,is 8.
Translate it −5 − 3 = -8 5 − (−3) = 8
Simplify -8 8

Try it!

Subtract: a. 3 − 1         b. −3 − (−1)         c. −3 − 1          d. 3 − (−1).

Solution

 

a.

Steps  Algebraic
 The expression is 3 minus 1. Three white counters are shown. An arrow indicates that 1 is taken away from these, and 2 remain. The answer is 2.

3 – 1
Take 1 positive from 3 positives and get 2 positives. 2

b.

Steps  Algebraic
The expression is minus 3 minus minus 1. Three red counters are shown. An arrow indicates that 1 is taken away from these, and 2 remain. The answer is minus 2. −3 − (−1)
Take 1 positive from 3 negatives and get 2 negatives. -2

c.

Steps  Algebraic
Counters The expression is minus 3 minus 1. 4 red counters are shown. −3 − 1
Take 1 positive from the one added neutral pair. The expression is minus 3 minus 1. 1 blue counter is shown. An arrow indicates that 1 blue counter is taken away from the 4 red counters above. The answer is minus 4.    -4

d.

Steps  Algebraic
Counters The expression is 3 minus minus 1. 4 blue counters are shown. 3 − (−1)
Take 1 negative from the one added neutral pair. The expression is 3 minus minus 1.1 red counters are shown. An arrow indicates that 1 red counter is taken away from the. 4 blue counters above. The answer is 4.     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:

Subtraction Property
a − b = a + (−b)
Subtracting a number is the same as adding its opposite.

Try it!

Simplify: a. 13 − 8 and 13 + (−8)   b. −17 − 9 and −17 + (−9)   c. 9 − (−15) and 9 + 15   d. −7 − (−4) and −7 + 4.

Solution

a.

Steps Ex. 1 Ex. 2
Example 13 − 8 13 + (−8)
Subtract.     5   5

b.

Steps Ex. 1 Ex. 2
Example −17 − 9 −17 + (−9)
Subtract.    −26   −26

c.

Steps Ex. 1 Ex. 2
Example 9 − (−15) 9 + 15
Subtract.     24 24

d.

Steps Ex. 1 Ex. 2
Example −7 − (−4) −7 + 4
Subtract.      −3 −3
 What happens when there are more than three integers? We just use the order of operations as usual.

Try it!

Simplify: 7 − (−4 − 3) − 9

Solution

Steps Algebraic
Example 7 − (−4 − 3) − 9
Simplify inside the parentheses first. 7 − (−7) − 9
Subtract left to right. 14 − 9
Subtract 5

Multiply and Divide Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

We remember that a · b means add a, b times. Here, we are using the model just to help us discover the pattern.


Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under this is the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”The next two examples are more interesting. What does it mean to multiply 5 by -3? It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace.

This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.

In summary:

Ex. 1 Ex. 2
5 · 3 = 15 −5(3) = −15
5(−3) = −15 (−5)(−3) = 15

Notice that for multiplication of two signed numbers, when the:

Rules
signs are the same, the product is positive.
signs are different, the product is negative.

What about division? Division is the inverse operation of multiplication. So, 15 ÷ 3 = 5 because 5 · 3 = 15. In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.

Ex. 1 Ex. 2
5 · 3 = 15      so  15 ÷ 3 = 5     −5 (3) = −15 so  −15 ÷ 3 = −5
(−5)(−3) = 15 so  15 ÷ (−3) = −5     5(−3) = −15 so   −15 ÷ (−3) = 5

Division follows the same rules as multiplication with regard to signs.

Multiplication of Signed Numbers

For multiplication of two signed numbers:

Same signs Product Example
Two positives
 Positive
 7 · 4 = 28
Two negatives Positive −8(−6) = 48
Different signs Product Example
Positive · negative
 Negative
 7(−9) = −63
Negative · positive Negative −5 · 10 = −50

Try it!

Multiply or divide: a. −100 ÷ (−4)       b. 7 · 6         c. 4(−8)       d. –27 ÷ 3.

Solution

a.

Steps Algebraic
Example −100 ÷ (−4)
Divide, with signs that are the same the quotient is positive. 25

b.

Steps Algebraic
Example 7 · 6  
Multiply, with same signs 42     

c.

Steps Algebraic
Example 4(−8)  
Multiply, with different signs. −32    

d.

Steps Algebraic
Example −27 ÷ 3  
Divide, with different signs, the quotient is negative. −9      

When we multiply a number by 1, the result is the same number. What happens when we multiply a number by −1? Let’s multiply a positive number and then a negative number by −1 to see what we get.

Steps Ex. 1 Ex. 2
Example −1 · 4 −1(−3)
Multiply. -4 3
Solution −4 is the opposite of 4. 3 is the opposite of −3.

Each time we multiply a number by −1, we get its opposite!

Multiplication by −1

−1a = −a

Multiplying a number by −1 gives its opposite.

Try it!

Multiply: a. -1 \cdot 7         b. -1(-11)

Solution
Steps Algebraic
a. Multiply, noting that the signs are different so the product is negative. −1 · 7    
-7  
-7 is the opposite of 7.
b. Multiply, noting that the signs are the same so the product is positive.


 −1(−11)       
11
11 is the opposite of −11.

Identify Multiples of a Number[5]

The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2.

Multiples of 2: 2 times 1 is 2, 2 times 2 is 4, 2 times 3 is 6, 2 times 4 is 8, 2 times 5 is 10, 2 times 6 is 12 and so on.

Similarly, a multiple of 3 would be the product of a counting number and 3.

Multiples of 3: 3 times 1 is 3, 3 times 2 is 6, 3 times 3 is 9, 3 times 4 is 12, 3 times 5 is 15, 3 times 6 is 18 and so on.

We could find the multiples of any number by continuing this process.

Counting Number 1 2 3 4 5 6 7 8 9 10 11 12
Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24
Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36
Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48
Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60
Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72
Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84
Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96
Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108

Multiple of a Number

A number is a multiple of n if it is the product of a counting number and n.

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, 15 ÷ 3 is 5, so 15 is 5 · 3.

Use Common Divisibility Tests[6]

Another way to say that 375 is a multiple of 5 is to say that 375 is divisible by 5. In fact, 375 ÷ 5 is  75, so 375 is 5 ⋅ 75. However, 10,519 is not a multiple of 3. When we divide 10,519 by 3 we do not get a counting number, so 10,519 is not divisible by 3.

Divisibility

If a number m is a multiple of n, then we say that m is divisible by n.

Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. The table below summarizes divisibility tests for some of the counting numbers between one and ten.

Divisibility Tests
A number is divisible by
                     2 if the last digit is 0, 2, 4, 6, or 8
                     3 if the sum of the digits is divisible by 3
                     5 if the last digit is 5 or 0
                     6 if divisible by both 2 and 3
                    10 if the last digit is 0

Try it!

  1. Determine whether 1,290 is divisible by 2, 3, 5, and 10.
Solution

The table applies the divisibility tests to 1,290. In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number.

Divisible by…? Test Divisible? Check
2 Is last digit 0, 2, 4, 6, or 8? Yes. yes 1290 ÷ 2 = 645
3 Is sum of digits divisible by 3?
1 + 2 + 9 + 0 = 12   Yes.		 yes 1290 ÷ 3 = 430
5 Is last digit 5 or 0? Yes. yes 1290 ÷ 5 = 258
10 Is last digit 0? Yes. yes 1290 ÷ 10 = 129

Thus, 1,290 is divisible by 2, 3, 5, and 10.

2. Determine whether 5,625 is divisible by 2, 3, 5, and 10.

Solution

The table applies the divisibility tests to 5,625 and tests the results by finding the quotients.

Divisible by…? Test Divisible? Check
2 Is last digit 0, 2, 4, 6, or 8? No. no 5625 ÷ 2 = 2812.5
3 Is sum of digits divisible by 3?

5 + 6 + 2 + 5 = 18      Yes.

 yes 5625 ÷ 3 = 1875
5 Is last digit 5 or 0?  Yes. yes 5625 ÷ 5 = 1125
10 Is last digit 0?  No. no 5625 ÷ 10 = 562.5

Thus, 5,625 is divisible by 3 and 5, but not 2, or 10.

Find all the Factors of the Given Number[7]

There are often several ways to talk about the same idea. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. We know that 72 is the product of 8 and 9, so we can say 72 is a multiple of 8 and 72 is a multiple of 9. We can also say 72 is divisible by 8 and by 9. Another way to talk about this is to say that 8 and 9 are factors of 72. When we write 72 = 8 ⋅ 9 we can say that we have factored 72.

The image shows the equation 8 times 9 equals 72. The 8 and 9 are labeled as factors and the 72 is labeled product.

Factors

If a ⋅ b = m, then a and b are factors of m, and m is the product of  a and b.

In algebra, it can be useful to determine all of the factors of a number. This is called factoring a number, and it can help us solve many kinds of problems.

For example, suppose a choreographer is planning a dance for a ballet recital. There are 24 dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.

In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of 24. The table summarizes the different ways that the choreographer can arrange the dancers.

Number of Groups Dancers per Group Total Dancers
                     1                       24           1 ⋅ 24 = 24
                     2                       12           2 ⋅ 12 = 24
                     3                        8           3⋅ 8 = 24
                     4                        6           4 ⋅ 6 = 24
                     6                        4           6 ⋅ 4 = 24
                     8                        3           8 ⋅ 3 = 24
                     12                        2           12 ⋅ 2 = 24
                     24                        1           24 ⋅ 1 = 24

What patterns do you see in the table? Did you notice that the number of groups times the number of dancers per group is always 24? This makes sense, since there are always 24 dancers.

You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers—but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of 24, which are:

1, 2, 3, 4, 6, 8, 12, 24

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with 1. If the quotient is also a counting number, then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.

How to find all the factors of a counting number.

  1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
    • If the quotient is a counting number, the divisor and quotient are a pair of factors.
    • If the quotient is not a counting number, the divisor is not a factor.
  2. List all the factor pairs.
  3. Write all the factors in order from smallest to largest.

Try it!

Find all the factors of 72.

Solution

Divide  72 by each of the counting numbers starting with 1. If the quotient is a whole number, the divisor and quotient are a pair of factors.

The figure shows a table with ten rows and four columns. The first row is a header row and labels the rows “Dividend”, “Divisor”, “Quotient”, and “Factors”. Under the “Dividend” column all rows show the number 72. In the second row the “Divisor” column is 1, the “Quotient” column is 72 and the “Factors” column is 1 and 72. In the third row the “Divisor” column is 2, the “Quotient” column is 36 and the “Factors” column is 2 and 36. In the fourth row the “Divisor” column is 3, the “Quotient” column is 24 and the “Factors” column is 3 and 24. In the fifth row the “Divisor” column is 4, the “Quotient” column is 18 and the “Factors” column is 4 and 18. In the sixth row the “Divisor” column is 5, the “Quotient” column is 14.4 and the “Factors” column is blank. In the seventh row the “Divisor” column is 6, the “Quotient” column is 12 and the “Factors” column is 6 and 12. In the eighth row the “Divisor” column is 7, the “Quotient” column is about 10.29 and the “Factors” column is blank. In the ninth row the “Divisor” column is 8, the “Quotient” column is 9 and the “Factors” column is 8 and 9. In the tenth row the “Divisor” column is 9, the “Quotient” column is 8 and the “Factors” column is 9 and 8.

The next line would have a divisor of 9 and a quotient of 8. The quotient would be smaller than the divisor, so we stop. If we continued, we would end up only listing the same factors again in reverse order. Listing all the factors from smallest to greatest, we have

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72

Key Concepts

Operation Notation Say: The result is…
Addition a + b a plus b the sum of a and b
Subtraction a − b a minus b the difference of a and b
Multiplication a·b, (a)(b), (a)b, a(b) a times b The product of a and b
Division a ÷ b, a/b, \displaystyle \frac{a}{b}, b⟌a a divided by b The quotient of a and b
  • Equality Symbol:
    • a = b is read a is equal to b
    • The symbol = is called the equal sign.
  • Inequality
The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.
  • a < b is read a is less than b
  • a is to the left of b on the number line
The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.
  • a > b is read a is greater than b
  • a is to the right of b on the number line
    Algebraic Notation Say
    a = b a is equal to b
    a ≠ b a is not equal to b
    a < b a is less than b
    a > b a is greater than b
    a ≤ b a is less than or equal to b
    a ≥ b a is greater than or equal to b
    • Subtraction Property: Subtracting a number is the same as adding its opposite.
                                 Divisibility Tests
    A number is divisible by
    2 if the last digit is 0, 2, 4, 6, or 8
    3 if the sum of the digits is divisible by 3
    5 if the last digit is 5 or 0
    6 if divisible by both 2 and 3
    10 if the last digit is 0
    • Factors If a ⋅ b = m, then a and b are factors of m, and m is the product of a and b.
    • Find all the factors of a counting number.
      1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
        1. If the quotient is a counting number, the divisor and quotient are a pair of factors.
        2. If the quotient is not a counting number, the divisor is not a factor.
      2. List all the factor pairs.
      3. Write all the factors in order from smallest to largest.
    • Determine if a number is prime.
      1. Test each of the primes, in order, to see if it is a factor of the number.
      2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
      3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
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