Mathematical Expressions 1-1
A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.
An expression, such as 3 x n, that involves a variable is called a variable expression.
Expressions, such as 3 x 2, that name a certain number are called numerical expressions
When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab
In numerical expressions for products a multiplication symbol must be used to avoid confusion.
9 x 7 may be written as 9 ∙ 7
When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3 x n, then 3 x n has the value 3 x 6, or 18
Properties of Addition and Multiplication 1-2
The set of numbers 1, 2, 3, 4, 5, … is called the counting numbers.
If 0 is included in the set, we obtain the set of whole numbers
0, 1, 2, 3, 4, …
The order in which two whole numbers are added or multiplied does not change their sum or their product.
For example, 3 + 4 = 7 and 4 + 3 = 7
3 x 4 = 12 and 4 x 3 = 12
Commutative Properties
For any whole numbers a and b, a + b = b + a
For any whole numbers a and b, a x b = b x a
C+ for Commutative Property of addition
Cx for Commutative Property of multiplication
Operations indicated within parentheses in any expression are done before the others.
(9 x 2) x 5 = 18 x 5 = 90
9 x (2 x 5) = 9 x 10 = 90
The order in which we multiply whole numbers has no effect on the product
(9 x 2) x 5 = 9 x (2 x 5)
Associative Properties
For any whole numbers a, b, and c,
(a + b) + c = a + (b + c)
For any whole numbers a, b, and c, (a x b) x c = a x (b x c)
A+ for Associative Property of Addition
Ax for Associative Property of Multiplication
The numbers 0 and 1 have special properties.
6 + 0 = 6 and 0 + 6 = 6
Since the sum of any whole number and 0 is the same whole number, 0 is called the identity element for addition
Addition Property of Zero
For any whole number a,
a + 0 = a and 0 + a = a
It’s the Identity Property of Addition also know as the Additive Identity
Id+
6 x 1 = 6 and 1 x 6 = 6
1 is the identity element for multiplication, since the product of any whole number and 1 is the same whole number.
Multiplication Property of One
For any whole number a,
a x 1 = a and 1 x a = a
It’s the Identity Property of Multiplication also know as the
Multiplicative Identity
Idx
6 x 0 = 0 and 0 x 6 = 0
illustrate an important property of 0—the product of any whole number and 0 is 0
Multiplication Property of Zero
For any whole number a,
a x 0 = 0 and 0 x a = 0
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