Ch 1.1 Warm Up Problems

Objectives:

- understand/use properties & classifications of real numbers

163

1x 3154 x

))((4 xx 84 x 416 x

Properties of Real Numbers

The properties of real numbers

allow us to manipulate expressions

and equations and find the values

of a variable.

Number Classification

• Natural numbers are the counting numbers. • Whole numbers are natural numbers and zero. • Integers are whole numbers and their opposites.• Rational numbers can be written as a fraction.• Irrational numbers cannot be written as a

fraction.• All of these numbers are real numbers.

Number Classifications

Subsets of the Real Numbers

I - IrrationalZ - Integers

W - Whole

N - Natural

Q - Rational

Classify each number -1 real, rational, integer

real, rational, integer, whole, natural

real, irrational

real, rational

real, rational, integer, whole

real, rational

6

1

2

-2.2220

Properties of Real Numbers

Commutative Property• Think… commuting to work.• Deals with ORDER. It doesn’t matter

what order you ADD or MULTIPLY.

• a+b = b+a• 4 • 6 = 6 • 4

Properties of Real Numbers

Associative Property• Think…the people you associate

with, your group.• Deals with grouping when you

Add or Multiply.• Order does not change.

Properties of Real Numbers

Associative Property

• (a + b) + c = a + ( b + c)

• (nm)p = n(mp)

Properties of Real Numbers

Additive Identity Property• s + 0 = s

Multiplicative Identity Property • 1(b) = b

Properties of Real Numbers

Additive Inverse Property• a + (-a) = 0

Multiplicative Inverse Property

• a∙( ) = 1, a 0a

1

Distributive Property

• a(b + c) = ab + ac

• (r + s)9 = 9r + 9s

Properties of Real Numbers

• 5 = 5 + 0• 5(2x + 7) =

10x + 35• 8 • 7 = 7 • 8• 24(2) = 2(24)• (7 + 8) + 2 = 2 +

(7 + 8)

Additive Identity

Distributive

Commutative

Commutative

Commutative

Name the Property

Name the Property

• 7 + (8 + 2) = (7 + 8) + 2• 1 • v + -4 = v + -4

• (6 - 3a)b = 6b - 3ab• 4(a + b) = 4a + 4b

• Associative• Multiplicative

Identity• Distributive • Distributive

Homework #1

P. 975: 1-10P. 6: 1-5, 9-13, 17-37 odd, 39-44, 57-60, 66, 69, 72, 75, 78

Properties of Real Numbers

Reflexive Property

• a + b = a + b

The same expression is written on both sides of the equal sign.

Properties of Real Numbers

• If a = b then b = a

• If 4 + 5 = 9 then 9 = 4 + 5

Symmetric Property

Properties of Real NumbersTransitive Property

• If a = b and b = c then a = c

• If 3(3) = 9 and 9 = 4 +5, then 3(3) = 4 + 5

Properties of Real Numbers

Substitution Property

• If a = b, then a can be replaced by b.

• a(3 + 2) = a(5)

Name the property• 5(4 + 6) = 20 + 30• 5(4 + 6) = 5(10)• 5(4 + 6) = 5(4 + 6)• If 5(4 + 6) = 5(10) then

5(10) = 5(4 + 6)• 5(4 + 6) = 5(6 + 4)• If 5(10) = 5(4 + 6) and

5(4 + 6) = 20 + 30 then 5(10) = 20 + 30

• Distributive• Substitution• Reflexive• Symmetric

• Commutative• Transitive

Solving Equations

• To solve an equation, find replacements for the variables to make the equation true.

• Each of these replacements is called a solution of the equation.

• Equations may have {0, 1, 2 … solutions.

}

}

Solving Equations

• 3(2a + 25) - 2(a - 1) = 78

• 4(x - 7) = 2x + 12 + 2x

1 3 5 1 3772 4 6 4 6

x x x

Solving Equations

• Solve: V = πr2h, for h

• Solve: de - 4f = 5g, for e