# NUMERICAL AND ALGEBRAIC EXPRESSIONS

In arithmetic, we use symbols such as 4, 8, 17, and π to
represent

numbers. We indicate the basic operations of addition, subtraction,

multiplication, and division by the symbols +, -, ·, and ÷, respectively. Thus

we can formulate specific **numerical expressions**. For example, we can

write the indicated sum of eight and four as 8 + 4.

In algebra, the concept of a **variable** provides the basis for
generalizing.

By using x and y to represent any number, we can use the expression

x + y to represent the indicated sum of any two numbers. The x and y in

such an expression are called variables, and the phrase x + y is called an

**algebraic expression**. We commonly use letters of the alphabet such as

x, y, z, and w as variables. The key idea is that they represent numbers;

therefore, as we review various operations and properties pertaining to

numbers, we are building the foundation for our study of algebra.

Many of the notational agreements made in arithmetic are extended to

algebra with a few slight modifications. The following chart summarizes the

notational agreements pertaining to the four basic operations.

Operation |
Arithmetic |
Algebra |
Word phrase |

Addition | 4 + 6 | x + y | The sum of x and y |

Subtraction | 7 - 2 | w - z | The difference of w and z |

Multiplication | 9 · 8 | a·b, a(b), (a)b, (a)(b) or ab |
The product of a and b |

Division | or | The quotient of c and d |

As we review arithmetic ideas and introduce algebraic
concepts, it is

convenient to use some of the basic vocabulary and symbols associated

with sets. A **set** is a collection of objects, and the objects are called

**elements **or **members** of the set. In arithmetic and algebra, the
elements

of a set are often numbers. To communicate about sets, we use set

braces, { }, to enclose the elements (or a description of the elements), and

we use capital letters to name sets. For example, we can represent a set

A, which consists of the vowels of the alphabet, in these ways:

A = {vowels of the alphabet} Word description, or

A = {a, e, i, o, u} List or roster description

If we write {1, 2, 3, . . .} the set begins with the
counting numbers, 1, 2, and

3. The three dots indicate that it continues in a like manner forever; there is

no last element. A set that consists of no elements is called the **null set**

(written ).

Two sets are said to be equal if they contain exactly the same elements.

For example, {1, 2, 3} = {2, 1, 3} because both sets contain the same

elements; the order in which the elements are written doesn’t matter. The

slash mark through the equality symbol denotes “not equal to.” Thus, if A =

{1, 2, 3} and B = {1, 2, 3, 4}, we can write A ≠ B, which we read as “set A is

not equal to set B.”

**Simplifying Numerical Expressions**

Let us simplify some numerical expressions that involve the set of **whole**

**numbers** – that is, the set {0, 1, 2, 3,…}. Keep in mind that when we

simplify numerical expressions, the operations should be performed in the

order listed below:

**Order Of Operations**

1. Perform the operations inside the parentheses and brackets and

above and below each fraction bar. Start with the innermost

inclusion symbol.

2. Perform all multiplications and divisions in the order in which they

appear from left to right

3. Perform all additions and subtractions in the order in which they

appear from left to right.

**EXAMPLE 1.** Simplify

**Solution:**

a.)

b.)

c.)

**EXAMPLE 2.** Simplify

**Solution:**

a.)

b.)

c.)

d.)

**EXERCISE 3.** Work problems 1, 3, 7, 11, 15, 19 and
25 on page 8 of the

textbook.

**Simplifying Algebraic Expressions**

We can use the concept of a variable to generalize from numerical

expressions to algebraic expressions. Each of the following is an example

of an algebraic expression.

An algebraic expression takes on a numerical value
whenever each

variable in the expression is replaced by a specific number. For example, if

x is replaced by 9 and z by 4, the algebraic expression x - z becomes the

numerical expression 9 - 4, which simplifies to 5. We say that x - z **has a
value** of 5 when x equals 9 and z equals 4. The value of x - z, when x

equals 25 and z equals 12, is 13. The general algebraic expression x - z

has a specific value each time x and z are replaced by numbers. Consider

the next examples, which illustrate the process of finding a value of an

algebraic expression. The process is often referred to as evaluating

**algebraic expressions.**

**EXAMPLE 4.**Find the value of 3x + 2y when x is replaced with 5 and y

by 17.

**Solution:**

**EXAMPLE 5.**Find the value of 12a - 3b when a = 5 and b = 9.

Solution:

Solution:

**EXAMPLE 6.**Evaluate 4xy + 2xz - 3yz when x = 8, y = 6 and z = 2.

**Solution:**

**EXAMPLE 7.** Evaluate
when c = 12 and d = 4.

Solution:

**EXAMPLE 8. **Evaluate (2x + 5y)(3x - 2y) when x = 6 and y = 3.

**Solution:**

**EXERCISE 9. **Work problems 35, 37, 41 and 49 on
pages 8 - 9 of the

textbook.