Applying Properties of Real Numbers

Sec. 1.2Sol: A.11, A.12, A.1

Rational VS. Irrational numbers

Rational Numbers• Can be written as quotients

(fractions) of integers.Ex:

• Can be written as decimals that terminate or repeat.Ex: 0.75 or -0.333

Irrational Numbers• Cannot be written as

quotients of integers.• Cannot be written as

decimals that terminate or repeat.Ex:

Real number line

• Real number lines are graphed as points on a line and increase from left to right.

Ex: Graph the real numbers on a number line. (work on Board)

Try This: Graph the numbers

Venn Diagram Key:R- Real numbersI – Irrational

numbersW – Whole

numbersQ- rational

numbersZ – integersN – Natural

numbers

Q I

Z

Z WN

• Natural Numbers: { 1, 2, 3, 4, 5,…}

• Whole Numbers: {0, 1, 2, 3, 4, 5,…}

• Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}

Properties of addition and multiplication.Let a,b, and c be real numbers

Property

Closure

Commuitative

Associative

Addition

a + b is a real number

a + b = b + a

(a +b) + c = a + (b + c)

Multiplication

ab is a real number

ab = ba

(ab)c = a(bc)

Properties of addition and multiplication.Let a,b, and c be real numbers

Property

Identity

Inverse

Addition

a + 0 = a, 0 + a = a“No changing the

Value”a + (-a) = 0

Multiplication

a(1) = a , (1) a = a

a(1/a)=1 , a≠0

Distributive property: a(b+c) = ab + ac“Give it away”

Ex: Identify the property used

1. 5 +(9 + 12)=(5 + 9)+ 122. 250(1) = 250

Try These:1. (2 × 3)× 9=2 ×(3 × 9)2. 15 + 0 = 153. 4(5 + 25) = 4(5) + 4(25)4. 1(500) = 500

Defining Subtraction and Division

Subtraction: Adding the opposite(Additive inverse ex: b→ - b)Ex: a – b = a + (-b)

Division: Multiplying by the reciprical.(Multiplicative inverse)

Ex: Use properties and definition of Properties to show that (10÷c)•c=10 where c ≠0

(10÷c)•c = 10

10•1 = 1010 = 10

GivenDef. of division

Assoc. prop. Of mult.

Inverse prop. Of Mult.Identity prop. Of

Mult.

Try these:

1. b(4 ÷ b)=02. 3x + (6 + 4x) = 7x + 6