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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.

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Real numbers

REAL NUMBERS

(as opposed to fake numbers?)


Objective
Objective

  • TSW identify the parts of the Real Number System

  • TSW define rational and irrational numbers

  • TSW classify numbers as rational or irrational


Real numbers1
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
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 numbers2
Real Numbers

REAL NUMBERS

154,769,852,354

1.333

-5,632.1010101256849765…

-8

61

π

49%

549.23789


Two kinds of real numbers
Two Kinds of Real Numbers

  • Rational Numbers

  • Irrational Numbers


Rational 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

Examples of Rational Numbers


Integers

Integers

One of the subsets of rational numbers


What are integers
What are integers?

  • Integers are the whole numbers and their opposites.

  • Examples of integers are

    6

    -12

    0

    186

    -934



Types of integers
Types of Integers fraction with 1 as the denominator.

  • 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


REAL NUMBERS fraction with 1 as the denominator.

NATURAL

Numbers

WHOLE

Numbers

IRRATIONAL

Numbers

INTEGERS

RATIONAL

Numbers


Irrational numbers
Irrational Numbers fraction with 1 as the denominator.

  • 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.


Caution! fraction with 1 as the denominator.

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

Irrational numberscan 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 fraction with 1 as the denominator.

Examples of Irrational Numbers


Try this
Try this! fraction with 1 as the denominator.

a) Irrational

b) Irrational

c) Rational

d) Rational

e) Irrational


16 2 fraction with 1 as the denominator.

4 2

= = 2

Additional Example 1: Classifying Real Numbers

Write all classifications that apply to each number.

A.

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

5

irrational, real

B.

–12.75

–12.75 is a terminating decimal.

rational, real

16 2

C.

whole, integer, rational, real


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


0 3 cannot divide by zero. So it is not a number at all.

= 0

Additional Example 2: Determining the Classification of All Numbers

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

A.

21

irrational

0 3

B.

rational


Additional Example 2: D cannot divide by zero. So it is not a number at all.etermining the Classification of All Numbers

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

4 0

C.

not a real number


Objective1
Objective cannot divide by zero. So it is not a number at all.

  • TSW compare rational and irrational numbers

  • TSW order rational and irrational numbers on a number line


Comparing rational and irrational numbers
Comparing Rational and Irrational Numbers cannot divide by zero. So it is not a number at all.

  • 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

3

7

3

7


Practice
Practice cannot divide by zero. So it is not a number at all.


Ordering rational and irrational numbers
Ordering Rational and Irrational Numbers cannot divide by zero. So it is not a number at all.

  • 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
Example cannot divide by zero. So it is not a number at all.

  • Order these numbers from least to greatest.

    ¹/₄, 75%, .04, 10%, ⁹/₇

    ¹/₄ becomes 0.25

    75% becomes 0.75

    0.04 stays 0.04

    10% becomes 0.10

    ⁹/₇ becomes 1.2857142…

    Answer: 0.04, 10%, ¹/₄, 75%, ⁹/₇


Practice1
Practice cannot divide by zero. So it is not a number at all.

Order these from least to greatest:


Objectives
Objectives cannot divide by zero. So it is not a number at all.

  • TSW identify the rules associated computing with integers.

  • TSW compute with integers


Examples use the number line if necessary

1) (-4) + 8 = cannot divide by zero. So it is not a number at all.

Examples: Use the number line if necessary.

4

2) (-1) + (-3) =

-4

3) 5 + (-7) =

-2


Addition rule
Addition Rule cannot divide by zero. So it is not a number at all.

1) 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 = 2

2 + (-4) = -2


Karaoke time
Karaoke Time! cannot divide by zero. So it is not a number at all.

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?


Answer Now cannot divide by zero. So it is not a number at all.

-1 + 3 = ?

  • -4

  • -2

  • 2

  • 4


Answer Now cannot divide by zero. So it is not a number at all.

-6 + (-3) = ?

  • -9

  • -3

  • 3

  • 9


The additive inverses or opposites of two numbers add to equal zero
The cannot divide by zero. So it is not a number at all.additive inverses(or opposites) of two numbers add to equal zero.

-3

Proof: 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
What’s the difference between cannot divide by zero. So it is not a number at all.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 cannot divide by zero. So it is not a number at all.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 cannot divide by zero. So it is not a number at all.

Which is equivalent to-12 – (-3)?

  • 12 + 3

  • -12 + 3

  • -12 - 3

  • 12 - 3


Answer Now cannot divide by zero. So it is not a number at all.

7 – (-2) = ?

  • -9

  • -5

  • 5

  • 9


Review cannot divide by zero. So it is not a number at all.

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.


State the rule for multiplying and dividing integers…. cannot divide by zero. So it is not a number at all.

If the signs are the same,

If the signs are different,

the answer will be negative.

the answer will be positive.


Different cannot divide by zero. So it is not a number at all.

Signs

Negative

Answer

What’s

The

Rule?

1. -8 * 3

4. 6 ÷ (-3)

-24

-2

Start inside ( ) first

2. -2 * -61

5. - (20/-5)

122

- (-4)

Same

Signs

Positive

Answer

4

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

(-18)(1)

6.

-18

Just take

Two at a time

68


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

How much

does the

temperature

rise each

hour?

How long

Is it from

Midnight

to 6 am?

+4 degrees

6 hours

(6 hours)(4 degrees per hour)

Add this to

the original temp.

= 24 degrees

8° + 24° = 32°C


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.

Multiply

What

does

This

mean?

(11 feet)

(5 steps)

(55 feet)

5 * 11 = 55


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