chapter 1 section 1 properties of real numbers. 1-1 algebra 2 lesson 1-1 simplify. 1.–(–7.2)2.1...

Chapter 1

Section 1

Properties of Real Numbers

ALGEBRA 2 LESSON 1-1ALGEBRA 2 LESSON 1-1

Simplify.

1. –(–7.2) 2. 1 – (–3)

3. –9 + (–4.5) 4. (–3.4)(–2)

Properties of Real NumbersProperties of Real Numbers

5. –15 ÷ 3 6. – + (– )25

35

Solutions

1. –(–7.2) = 7.2

3. –9 + (–4.5) = –13.5

5. –15 ÷ 3 = –5

6. – + (– ) = = – = –125

35

–2 + (–3)5

55

2. 1 – (–3) = 1 + 3 = 4

4. (–3.4)(–2) = 6.8

Classifications of Numbers

Imaginary Numbers will be introduced later.

Real Numbers

The largest classification we will deal with Include any number that you can tell me

Ex:

Split into Rational and Irrational Numbers17

6 ,3 21,-54,

Real Numbers

Irrational Numbers Numbers that cannot be written as ratios Decimals that never terminate and never

repeat Square roots of positive non-perfect

squares Ex: √2, -√7, √(8/11),

1.011011101111011111…

Real Numbers

Rational Numbers

All the numbers that can be written as a ratio (fraction)

This includes terminating and repeating decimals.

Ex: 8, 10013, -54, 7/5, -3/25, 0, 0/6, -1.2, .09, .3333….

Real NumbersRational Numbers

Integers “Complete” numbers (no parts – fractions

or decimals) Negative, Zero, and Positive Each negative is the additive inverse (or

opposite) of the positive Ex: -543, 76, 9, 0, -34

Real NumbersRational NumbersIntegers

Whole Numbers Zero and positive integers

Ex: 0, 1, 2, 3, 4, …

Real NumbersRational NumbersIntegersWhole Numbers

Natural Numbers Also known as Counting Numbers Think of young children

Ex: 1, 2, 3, 4, 5, 6, …

Which set of numbers best describes the variable?a) The cost C in dollars for admission for n peopleb) The cost C of admission is a rational number and

the number n of people is a whole numberc) The maximum speed s in meters per second on a

roller coaster of height h in metersd) Since the speed s is calculated using a formula with a

square root, s is real (either rational or irrational). The height h is measured in rational numbers.

e) The park’s profit (or loss) P in dollars for each week w of the year

f) The profit P is a rational number and the week number w is natural.

Try This Problemp. 6 Check Understanding

The number r is the ratio of the number of adult tickets sold to the number of children’s tickets sold. Which set of numbers best describes the values of r? Which set of numbers best describes the average cost c per family for tickets?

r is a ratio, so it is a rational number; c is the cost which means a terminating decimal, so it is a rational number.

Graphing Numbers on a Number Line Make sure your number lines have zero Make them fairly accurate Label Important points

-3 -2 -1 0 21 3 4 5

0 15 20

0-120 -119

Ordering Real Numbers

Less Than <Or equal to ≤

Greater Than >Or equal to ≥

Properties of Real Numbers

Opposite or Additive Inverse – of any number a is –a The sum of opposites is 0.

Reciprocal or Multiplicative Inverse – of any number a is

Think of it as a flip Remember you may have to make a decimal into a fraction before

flipping it. Use the place (hundredths) to write a fraction. The product of reciprocals is 1.

a

1

Find the opposite and reciprocal of each number.a) .

b) -3.2

3125.16

5

32

10

2.3

1

2.3)2.3(

reciprocal

opposite

5

3

3

55

3

reciprocal

opposite

Try These Problemsp. 7 Check Understanding

State the opposite and reciprocal of each number.

a) 400

b) 4 1/5

c) -.002

d) -4/9

Opposite Reciprocal

-400 -1/400

-4 1/5 5/21

.002 -500

4/9 -9/4

Properties of Real Numbers

Property Addition MultiplicationClosure a+b is a real number ab is a real number

Commutative commute = to move

a + b = b + a Ab = ba

Associative associate = regroup

(a+b)+c = a+(b+c) (ab)c = a(bc)

Identity a+0=a,0+a=a a*1=a, 1*a=a

Inverse a+(-a)=0 a*(1/a)=1,a≠0

Distributive a(b+c) = ab + ac

Identify the propertyExample 5

Which Property is illustrated?a) 6 + (-6) = 0

a) Inverse Property of Addition

b) (-4 ∙ 1) – 2 = -4 – 2a) Identity Property of Multiplication

You MUST state Addition or Multiplication.Appropriate abbreviations:

Prop. Of Add. Or Prop. Of Mult.Comm. Assoc. Ident. Inv. Dist

Try these Problemsp. 7 Check Understanding

Which Property is illustrated?

a) (3 + 0) – 5 = 3 – 5a) Identity Property of Addition

b) -5 + [2 + (-3)] = (-5 + 2) + (-3)a)Associative Property of Addition

Absolute Value

The absolute value of a number is the distance from zero on the number line.

Its always positive. Be careful to watch for negatives outside

the absolute value bars (then the answer is negative).

Absolute Value Symbols │a│

Example 6

a) │-4 │ =

b) │0 │=

c) │-1 ∙ (-2) │=

d) │-10 │=

e) │1.5 │=

f) │0 - 3 │=

a) 4

b) 0

c) 2

d) 10

e) 1.5

f) 3