# College Algebra

**1 Laws of algebra **

**1.1. Terminology and Notation .** In this section we review the notations

used in algebra. Some are peculiar to these notes. For example the notation

A := B indicates that the equality holds by definition of the notations in-

volved. (See for example Paragraph **1.2** which follows.) Two other
notations

which will become important when we solve equations are => and
.

The notation P => Q means that P implies Q i.e. "If P, then Q". For

example, x = 2 => x^{2} = 4. (Note however that the converse satement

x^{2} = 4 => x = 2 is not always true since it might be that x = -2.) The

notation P Q means P => Q and Q => P, i.e.
"P if and only if

Q". For example 3x - 6 = 0 x = 2. The notations => and

are explained more carefully in Paragraphs **8.2** and **8.3** below.

**1.2. Implicit Multiplication.** In mathematics the absence of an operation

symbol usually indicates multiplication: ab mean a × b. Sometimes a dot

is used to indicate multiplication and in computer
languages an asterisk is

often used.

ab := a · b := a * b := a× b

**1.3. Order of operations . **Parentheses are used to indicate the order

of doing the operations: in evaluating an expression with parentheses the

innermost matching pairs are evaluated first as in

There are conventions which allow us not to write the parentheses. For

example, multiplication is done before addition

ab + c means (ab) + c and not a(b + c).

and powers are done before multiplication:

ab^{2}c means a(b^{2})c and not (ab)^{2}c:

In the absence of other rules and parentheses , the left most operations are

done first.

a - b - c means (a - b) - c and not a - (b - c):

The long fraction line indicates that the division is done last:

means (a + b)/c and not a + (b/c):

In writing fractions the length of the fraction line indicates which fraction is

evaluated first:

means a/(b/c) and not (a/b)/c,

means (a/b)/c and not a/(b/c).

The length of the horizontal line in the radical sign indicates the order of

evaluation:

means
and not :

means
and not :

**1.4. The Laws of Algebra. **There are four fundamental
operations which

can be performed on numbers.

1. Addition. The** sum **of a and b is denoted a + b.

2. Multiplication. The **product **of a and b is denoted ab.

3. Reversing the sign. The **negative** of a is denoted -a.

4. Inverting. The **reciprocal **of a (for a ≠ 0) is denoted by a^{-1} or by

.

These operations satisfy the following laws.

Associative | |

Commutative | |

Identity | |

Inverse | |

Distributive |

The operations of **subtraction** and **division**
are then defined
by

All the rules of calculation that you learned in
elementary school follow from

the above fundamental laws. In particular, the Commutative and Associative

Laws say that you can add a bunch of numbers in any order and similarly

you can multiply a bunch of numbers in any order. For example,

(A+B)+(C+D) = (A+C)+(B+D), (A · B) (C · D) = (A · C) (B · D).

**1.5.** Because both addition and multiplication satisfy the commutative, as-

sociative, identity, and inverse laws, there are other analogies:

These identities are proved in the **Guided Exercises.
**(An**
identity** is an

equation which is true for all values of the variables which appear in it.)

**1.6.** Here are some further identities which are proved using the distributive

law.

These are also proved in the **Guided Exercises.
**

**1.7.**The following

**Zero-Product Property**will be used to solve equations.

pq = 0 p = 0 or q = 0 (or both). |

Proof: If p = 0 (or q = 0) then pq = 0 by (i) in Paragraph
**1.6 .** Conversely,

if p ≠ 0 then q = p^{-1}pq = p^{-1}0 = 0.

**definition 1.8.** For a natural number n and any number a the
nth **power**

of a is

The zeroth power is

and negative powers are defined by

**1.9. **The following laws are easy to understand when m and
n are integers.

In Theorem** 4.1 **below we will learn that these laws also hold whenever a

and b are positive real numbers and m and n are any real numbers, not just

integers.

**2 Kinds of Numbers
**

**2.1.**We distinguish the following different kinds of numbers.

• The

**natural numbers**are 1, 2, 3 ...

• The

**integers**are... - 3,-2,-1, 0, 1, 2, 3...

• The

**rational numbers**are ratios of integers like 3/2, 14/99, -1/2.

• The **real numbers** are numbers which have an infinite
decimal expan-

sion like

•The **complex numbers** are those numbers of form z = x + iy
where

x and y are real numbers and i is a special new number called the

**imaginary unit** which has the property that

Every integer is a rational number (because n = n/1),
every rational number

is a real number (see Remark **2.3 **below), and every real number is a complex

number (because x = x + 0i). A real number which is not rational is called

**irrational.**

**2.2.** Each kind of number enables us to solve equations that the previous

kind couldn't solve:

•
The solution of the equation x + 5 = 3 is x = -2 which is an integer

but not a natural number.

•
The solution of the equation 5x = 3 is which is a rational number

but not an integer.

•
The equation x^{2} = 2 has two solutions . The number
is a

real number but not a rational number. (See Remark** 2.5 **below.)

•
The equation x^{2} = 4 has two real solutions x = ±2 but the equation

z^{2}= -4 has no real solutions because the square of a nonzero real

number is always positive. However it does have two complex solutions,

namely z = ±2i.

We will not use complex numbers until Section **20** but may refer to them

implicitly as in

The equation x^{2} = -4 has no (real) solution.

**Remark 2.3**. It will be proved in Theorem **24.6** that a real
number is rational

if and only if its decimal expansion eventually repeats periodically forever as

in the following examples:

**Remark 2.4.** Unless the decimal expansion of a real number
is eventually

zero, as in :, any finite part of the decimal expansion is close

to, but not exactly equal to, the real number. For example 1.414 is close to

the square root of two but not exactly equal:

If we compute the square root to more decimal places we
get a better ap-

proximation, but it still isn't exactly correct:

(1.4142135623730951)^{2} = 2.00000000000000014481069235364401.

**Remark 2.5.** Here is a proof that the square root of 2 is irrational. If it

were rational there would be integers m and n with

By canceling common factors we may assume that m and n
have no common

factors and hence that they are not both even. Now m^{2} = 2n^{2} so m^{2 } is even

so m is even, say m = 2p. Then 4p^{2} = (2p)^{2} = m^{2} = 2n^{2} so 2p^{2} = n^{2}
so n^{2}

is even so n is even. This contradicts the fact m and n are not both even.

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