REAL NUMBERS

(as opposed to fake numbers?)

Objective

• TSW identify the parts of the Real Number System

• TSW define rational and irrational numbers

• TSW classify numbers as rational or irrational

Real Numbers

• Real Numbers are every number.

• Therefore, any number that you can find on the number line.

• Real Numbers have two categories.

What does it Mean?

• The number line goes on forever.• Every point on the line is a REAL

number.• There are no gaps on the number line.• Between the whole numbers and the

fractions there are numbers that are decimals but they don’t terminate and are not recurring decimals. They go on forever.

Real Numbers

REAL NUMBERS

-8 -5,632.1010101256849765…

61

49%

π

549.23789

154,769,852,354

1.333

Two Kinds of Real Numbers

• Rational Numbers

• Irrational Numbers

Rational Numbers

•A rational number is a real number that can be written as a fraction.

•A rational number written in decimal form is terminating or repeating.

Examples of Rational Numbers

•16•1/2•3.56

•-8•1.3333…

•- 3/4

Integers

One of the subsets of rational numbers

What are integers?

• Integers are the whole numbers and their opposites.

• Examples of integers are 6-120186-934

• Integers are rational numbers because they can be written as fraction with 1 as the denominator.

Types of Integers

• Natural Numbers(N): Natural Numbers are counting numbers from 1,2,3,4,5,................N = {1,2,3,4,5,................}

• Whole Numbers (W): Whole numbers are natural numbers including zero. They are 0,1,2,3,4,5,...............W = {0,1,2,3,4,5,..............} W = 0 + N

WHOLENumber

s

REAL NUMBERS

IRRATIONALNumbers

NATURALNumbers

RATIONALNumbers

INTEGERS

Irrational Numbers

•An irrational number is a number that cannot be written as a fraction of two integers.

•Irrational numbers written as decimals are non-terminating and non-repeating.

A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

Caution!

Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number.

Examples of Irrational Numbers

• Pi

Try this!

• a) Irrational

• b) Irrational

• c) Rational

• d) Rational

• e) Irrational66 e)

d)

25 c)

12 b)

2 a)

115

Additional Example 1: Classifying Real Numbers

Write all classifications that apply to each number.

5 is a whole number that is not a perfect square.

5

irrational, real

–12.75 is a terminating decimal.–12.75rational, real

16 2

whole, integer, rational, real

= = 24 2

16 2

A.

B.

C.

A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

State if each number is rational, irrational, or not a real number.

21

irrational

0 3

rational

0 3

= 0

Additional Example 2: Determining the Classification of All Numbers

A.

B.

not a real number

Additional Example 2: Determining the Classification of All Numbers

4 0C.

State if each number is rational, irrational, or not a real number.

Objective

• TSW compare rational and irrational numbers

• TSW order rational and irrational numbers on a number line

Comparing Rational and Irrational Numbers

• When comparing different forms of rational and irrational numbers, convert the numbers to the same form.

Compare -3 and -3.571 (convert -3 to -3.428571…

-3.428571… > -3.571

37

37

Practice

Ordering Rational and Irrational Numbers

• To order rational and irrational numbers, convert all of the numbers to the same form.

• You can also find the approximate locations of rational and irrational numbers on a number line.

Example• Order these numbers from least to

greatest. ¹/₄, 75%, .04, 10%, ⁹/₇

¹/₄ becomes 0.2575% becomes 0.750.04 stays 0.0410% becomes 0.10

⁹/₇ becomes 1.2857142…

Answer: 0.04, 10%, ¹/₄, 75%, ⁹/₇

Practice

Order these from least to greatest:

Objectives

• TSW identify the rules associated computing with integers.

• TSW compute with integers

Examples: Use the number line if necessary.

42) (-1) + (-3) =

-43) 5 + (-7) =

-2

0 5-5

1) (-4) + 8 =

Addition Rule1) When the signs are the same,

ADD and keep the sign.(-2) + (-4) = -6

2) When the signs are different,SUBTRACT and use the sign of the

larger number.(-2) + 4 = 22 + (-4) = -2

Karaoke Time!Addition Rule: Sung to the tune of

“Row, row, row, your boat”Same signs add and keep,

different signs subtract,keep the sign of the higher

number,then it will be exact!

Can your class do different rounds?

-1 + 3 = ?

1. -42. -23. 24. 4

Answer Now

-6 + (-3) = ?

1. -92. -33. 34. 9

Answer Now

The additive inverses (or opposites) of two numbers

add to equal zero.

-3Proof: 3 + (-3) = 0 We will use the additive inverses

for subtraction problems.

Example: The additive inverse of 3 is

What’s the difference between

7 - 3 and 7 + (-3) ?7 - 3 = 4 and 7 + (-3) = 4

The only difference is that 7 - 3 is a subtraction problem and 7 + (-3) is an addition problem.

“SUBTRACTING IS THE SAME AS ADDING THE OPPOSITE.”(Keep-change-change)

When subtracting, change the subtraction to adding the opposite

(keep-change-change) and then follow your addition rule.

Example #1: - 4 - (-7)- 4 + (+7)

Diff. Signs --> Subtract and use larger sign.3

Example #2: - 3 - 7- 3 + (-7)

Same Signs --> Add and keep the sign.-10

Which is equivalent to-12 – (-3)?

Answer Now

1. 12 + 32. -12 + 33. -12 - 34. 12 - 3

7 – (-2) = ?

Answer Now

1. -92. -53. 54. 9

1) If the problem is addition, follow your addition rule.

2) If the problem is subtraction, change subtraction to adding the opposite (keep-change-change) and then follow the addition rule.

Review

State the rule for multiplying and dividing integers….

If the signs are the same,

If the signs are different,

the answer will be positive.

the answer will be negative.

1. -8 * 3 What’s The

Rule?

DifferentSigns

NegativeAnswer

-24

2. -2 * -61

SameSigns

PositiveAnswer

122

3. (-3)(6)(1)

Just

take

Two

at a

tim

e

(-18)(1) -18

4. 6 ÷ (-3)

-2

5. - (20/-5) - (-4)

4

6.

408

6

68

Start inside ( ) first

7. At midnight the temperature is 8°C. If the temperature rises 4°C per hour, what is the temperature at 6 am?

How longIs it fromMidnightto 6 am?

How muchdoes the

temperaturerise each

hour?

6 hours

+4 degrees

(6 hours)(4 degrees per hour)

= 24 degrees

8° + 24° = 32°C

Add this tothe original temp.

8. A deep-sea diver must move up or down in the water in short steps in order to avoid getting a physical condition called the bends. Suppose a diver moves up to the surface in five steps of 11 feet. Represent her total movements as a product of integers, and find the product.

What does This mean?

Multiply

(5 steps) (11 feet)

(55 feet)

5 * 11 = 55