Algebra Know-It-ALL

Identity, Grouping, and Signs

Let’s review how signs work in multiplication and division. Then we’ll proceed to the

three major laws that govern the interplay between multiplication, division, addition,

and subtraction.

Notation for multiplication

When you want to multiply two numbers, you can use the familiar “times sign” and put the

numerals for the factors on either side. This symbol (×) looks like a tilted cross or a letter “x.”

Another symbol you’ll often see is the small, elevated dot (·). When a number is multiplied

by a variable, or when a variable is multiplied by another variable, you’ll see their symbols run

together without any space between. Parentheses are placed around complicated expressions

when they are multiplied by each other.

• When you see 3 × 7, it means 3 times 7.


• When you see −3·7, it means −3 times 7.


• When you see − 4 a, it means −4 times a.


• When you see ab, it means a times b.


• When you see abc, it means a times b times c.


• When you see a(b−c), it means a times (b−c).


• When you see (a+b)(c+d), it means (a+b) times (c+d).

Notation for division

In this book, we’ll use the forward slash (/) to indicate division. In arithmetic, you some-

times see the dash with two dots (÷), but that’s rarely used in algebra. When expressions are

complicated, the dividend (the number you want to divide) can be placed on top of a long

horizontal line, and the divisor (the number you divide by) is placed underneath. As with

multiplication, parentheses are placed around complicated expressions when they are divided

by each other.

• When you see 8/2, it means 8 divided by 2.


• When you see −4/a, it means −4 divided by a.


• When you see a/(−4), it means a divided by −4.


• When you see a/b, it means a divided by b.


• When you see a/(b−c), it means a divided by (b−c).


• When you see (a+b)/(c+d), it means (a+b) divided by (c+d).

The identity element

You can multiply or divide any integer by 1, and it won’t change the value. Because multiply-

ing or dividing by 1 always gives you the same number again, 1 is called the multiplicative

identity element. (For some reason I’ve never heard it called the “divisive identity element,” but

technically this term is okay.) For any integer a

a 1 =a

70 Multiplication and Division