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 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.
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.
Defining the multiplication of two negative integers is more involved, and we will not go into the reasoning here - just recall that the product of two negative integers is always positive.
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.
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 expressions where the only operation is addition by computing the same sums in the order indicated by the parentheses. This phenomenon is called the associative property of addition
In most cases, we will use parentheses to show the order of operations. Standard conventions also exist for the order of operations when parentheses are not used (see Figure 1.9). One of these is that multiplication and division are performed before addition and subtraction. We follow these standard rules whenever too many parentheses would make an expression hard to read.
