Natural Numbers {1, 2, 3, 4, . . .}

Whole Numbers {0, 1, 2, 3, 4, . . .}

Integers {. . . , -3, -2, -1, 0, 1, 2, 3, . . .}

Rational Numbers {| p and q are integers and q ¹ 0 }

The set of rational numbers contains all numbers that can be written as fractions, or quotients of integers. Integers are also rational numbers since they can be represented as fractions. All decimals that repeat or terminate belong to the set of rational numbers. The following are all rational numbers:

, -, 1, -5 =, 0 =, 0.125 =, 0.6666 . . . =

Irrational Numbers {x | x is real but not rational }

The irrational numbers are nonrepeating, nonterminating decimals. They cannot be represented as the quotient of two integers. The following are all irrational numbers:

p , , -

Real Numbers {x | x corresponds to a point on the number line }

The set of real numbers consists of all the rational numbers together with all the irrational numbers.

Example

Given set A = {, -, 0, 2.9, -5, 4, -, , -7, p }, list all the elements of A that belong to the set of : a) natural numbers, b) whole numbers, c) integers, d) rational numbers, e) irrational numbers, and f) real numbers.

a) 4

b) 0, 4

c) 0, -5, 4

d), 0, 2.9, -5, 4, -, -7

e) -, , p

f) all elements of A are real numbers

Order of Operations

1. Perform operations in grouping symbols (parentheses, brackets, braces, or fraction bars). Start with the innermost and work outward.

2. Calculate powers and roots, working from left to right.

3. Perform multiplication and division in order from left to right.

4. Perform addition and subtraction in order from left to right.

Example

Use order of operations to evaluate:

a) 6(-5) – (-3)(2)

b)

c) -9 – {6 – 2[12 – (8 – 15)] – 4}

Solution:

a) 6(-5) – (-3)(2) = 6(-5) – (-3)(16) No grouping symbols; power calculated first

= -30 – (-48) Multiplication performed

= -30 + 48 Subtraction changed to addition

= 18 Addition performed

b) Begin by simplifying the numerator and denominator of fraction.

= Calculate powers first

= Perform multiplications

= Perform additions and subtractions

= Simplify

c) -9 – {6 – 2[12 – (8 – 15)] – 4} = -9 – {6 – 2[12 – (-7)] – 4} Start with innermost grouping symbol, parentheses, and subtract

= -9 – {6 – 2[19] – 4} Working outward, perform subtraction in brackets

= -9 – {6 – 38 – 4} Within braces, multiply

= -9 – {-36} Within braces, subtract

= -9 + 36 Change subtraction to addition

= 27 Add

Properties of the Real Numbers

For all real numbers a, b, and c:

1. Commutative Property for Addition: a + b = b + a

2. Commutative Property for Multiplication: ab = ba

The commutative properties state that two numbers may be added or multiplied in any order.

3. Associative Property for Addition: a + (b + c) = (a + b) + c

4. Association Property for Multiplication: a(bc) = (ab)c

For the associative properties, the order of the terms or factors remains the same; only the grouping is changed.

5. Identity Property for Addition: There is a unique real number, 0, such that a + 0 = a and 0 + a = a

The identity property for addition tells us that adding 0 to any number will not change the number.

6. Identity Property for Multiplication: There is a unique real number, 1, such that a·1 = a and 1·a = a

The identity property for multiplication tells us that multiplying any number by 1 will not change the number.

7. Inverse Property for Addition: Each nonzero real number a has a unique additive inverse, represented by –a, such that

a + (-a) = 0 and –a + a = 0

Additive inverses are called opposites.

8. Inverse Property for Multiplication: Each nonzero real number a has unique multiplicative inverse, represented by , such that and

Multiplicative inverses are called reciprocals.

9. Distributive Property: a(b + c) = ab + ac

Example

Identify the property illustrated in each statement:

a) (x + 7) + 8 = x + (7 + 8)

b) 4x + 0 = 4x

c) 10 · (x) = (10 ·)x

d) (x+ 1) · = 1

e) 4(x + 5) = 4x + 20

f) 3 · (5 · a) = 3 · (a · 5)

g) -6x + 6x = 0

h) (2 + y) + 5 = 5 + (2 + y)

i) (y + 5)(y – 3) = (y – 3)(y + 5)

j) 5 · 1 = 5

Solution:

a) Associative Property for Addition. Order of terms remains the same. Only the grouping changes.

b) Identity Property for Addition. Adding zero to something does not change it.

c) Associative Property for Multiplication. Order of factors is the same. Only the grouping changes.

d) Inverse Property for Multiplication. The product of reciprocals is 1.

e) Distributive Property.

f) Commutative Property for Multiplication. Order of the factors is changed.

g) Inverse Property for Addition. The sum of opposites is 0.

h) Commutative Property for Addition. The order of the terms is changed.

i) Commutative Property for Multiplication. The order of the factors is changed.

j) Identity Property for Multiplication. Multiplying a number by 1 does not change it.