In mathematics symbols are used to obtain a clearer and shorter presentation. The first of these symbols is the ellipses (\(\ldots\)). When we use this symbol in mathematics, it means “continuing in this manner.” When a pattern is evident, we can use the ellipses (\(\ldots\)) to indicate that the pattern continues. We use this to define the integers.
The integer \(0\) is not considered to be positive or negative.
In the video in Figure 1.1 we give an introduction to the integers and statements.
Figure1.1.The Integers (Part I: Definition and Statements) by Matt Farmer and Stephen Steward
Figure 1.2 shows the integers, natural numbers, and negative integers on the number line.
The number line with an ellipsis to the left and to the right and with the integers from -4 to 10 labeled
(a)The integers on the number line extend to the left and right.
The number line with integers from \(1\) to \(10\) labeled followed by an ellipsis
(b)The natural numbers (or positive integers) on the number line extend only to the right starting at \(1\text{.}\)
The number line with an ellipsis to the left and integers from \(-4\) to \(-1\) labeled
(c)The negative integers on the number line extend only to the left starting at \(-1\text{.}\)
Figure1.2.Number lines
SubsectionComparing Integers
The symbols \(=\text{,}\)\(\ne\text{,}\)\(\lt\text{,}\)\(\le\text{,}\)\(>\text{,}\) and \(\ge\) are used to compare integers.
Table1.3.Comparison symbols for integers
symbol
read as
\(=\)
“is equal to”
\(\ne\)
“is not equal to”
\(>\)
“is greater than”
\(\ge\)
“is greater than or equal to”
\(\lt\)
“is less than”
\(\le\)
“is less than or equal to”
The first symbol in Table 1.3 is the equality symbol, \(=\text{.}\) Two integers are equal if they are the same integer. To indicate that two integers are not equal we use the symbol, \(\ne\text{.}\)
The other symbols compare the positions of two integers on the number line. An integer is greater than another integer if the first integer is to the right of the second integer on the number line. An integer is less than another integer if the first integer is to the left of the second integer on the number line.
Example1.4.Reading \(=\text{,}\)\(\ne\text{,}\)\(\gt\text{,}\)\(\ge\text{,}\)\(\lt\text{,}\) and \(\le\).
We give examples of comparisons and how to read them.
\(2=2\) is read “2 is equal to 2.”
\(2\ne 3\) is read “\(2\) is not equal to 3.”
\(3> 2\) is read “3 is greater than 2.”
\(3\ge 2\) is read “3 is greater than or equal to 2.”
\(2\lt 3\) is read “2 is less than 3.”
\(2\le 3\) is read “2 is less than or equal to 3.”
In Checkpoint 1.5 select the correct comparison operator.
Checkpoint1.5.Comparison operators.
For each of the following choose the symbol that yields a true statement:
241
<
=
>
241
242
<
=
>
95
SubsectionOperations
Addition, negation, subtraction, and multiplication are the basic operations of integers. We write “\(+\)” for plus, “\(-\)” for minus, and “\(\cdot\)” for times.
We give some examples of statements that involve integer operations. As we do not say “is false” we mean that all of these equality statements are true.
\(2+3=5\) is read “2 plus 3 is equal to 5”
\(2+0=2\) is read “2 plus 0 is equal to 2”
\(2+(-2)=0\) is read “2 plus negative 2 is equal to 0”
\(2-2=0\) is read “2 minus 2 is equal to 0”
\(2\cdot 5=10\) is read “2 times 5 is equal to 10”
\(2\cdot(-5)=-10\) is read “2 times negative 5 is equal to negative 10”
\((-2)\cdot(-5)=10\) is read “negative 2 times negative 5 is equal to 10”
Multiplication of a natural number with an integer can be viewed as repeated addition.
Example1.7.Multiplication as repeated addition.
We give examples of multiplication viewed as repeated addition.
Again, we can use ellipses (\(\ldots\)) to represent a continuing pattern:
\begin{equation*}
100 \cdot 5 =\underbrace{5+5+\ldots+5}_{100 \text{ copies of }5}=500\text{.}
\end{equation*}
Defining the multiplication of two negative integers is more difficult, and we appeal to your previously acquired knowledge about integers for that. Recall that the product of two negative integers is positive.
Example1.8.Multiplication with negative integers.
We give examples of multiplication of integers and negative integers:
\(\displaystyle 3\cdot(-5) = -15\)
\(\displaystyle (-3)\cdot 5=-15\)
\(\displaystyle (-3)\cdot (-5)=15\)
SubsectionOrder of Operations
We use parentheses to indicate the order in which expressions should be executed. We evaluate the expressions in the innermost parentheses first and then work our way outwards.
Example1.9.Order of operations.
We give examples for order of operations. The numbers and the operations are the same; only the grouping of the expressions given by the parentheses differs.
We illustrate that the order of operations does not matter for repeated addition by computing the same sums in the order indicated by the parentheses.
\(\displaystyle ((1+2)+3)+4=(3+3)+4=6+4=10\)
\(\displaystyle 1+((2+3)+4)=1+(5+4)=1+9=10\)
\(\displaystyle (1+2)+(3+4)=3+7=10\)
Usually we write \(1+2+3+4=10\text{.}\)
In most cases we will use parentheses to indicate the order of operations. There are other conventions for implicit order of operations (see Figure 1.10). One of these conventions is that multiplication is performed before addition and subtraction. We will use this convention when we feel that the additional parentheses will make it hard to read the expressions under consideration.
In the video in Figure 1.12 we recap the operations for the integers and give a motivation for the following section.
Figure1.12.The Integers (Part 2: Operations) by Matt Farmer and Stephen Steward
