Real numbers system

47
N W Z Q IR The Real Number System Presented By: Prince

description

for 9th class maths

Transcript of Real numbers system

Page 1: Real numbers system

NWZ

QIR

The Real Number System

Presented By: Prince

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Objectives

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Real Number

Real Number

Real Number

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What does it Mean?

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Real Numbers

REAL NUMBERS

-8 -5,632.1010101256849765…

61

49%

π

549.23789

154,769,852,354

1.333

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The Real Number LineAny real number corresponds to a point on the real number line.

Order Property for Real Numbers Given any two real numbers a and b,  - if a is to the left of b on the number line, then a < b. - if a is to the right of b on the number line, then a > b.

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Real Number System Tree Diagram

Real Numbers

IntegersTerminating Decimals

Repeating Decimals

WholeNumbers

Rational Numbers

Irrational Numbers

Negative #’s

Natural #’s Zero

Non-TerminatingAndNon-RepeatingDecimals

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Two Kinds of Real Numbers

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Rational Numbers

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Examples of Rational Numbers

•16•1/2•3.56

•-8•1.3333…

•- 3/4

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Integers

One of the subsets of rational numbers

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What are integers?

• Integers are the whole numbers and their opposites.

• Examples of integers are6-120186-934

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What are integers?.......

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

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Types of Integers

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WHOLENumber

s

REAL NUMBERS

IRRATIONALNumbers

NATURALNumbers

RATIONALNumbers

INTEGERS

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Irrational Numbers

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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.

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Examples of Irrational Numbers

• Pi

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Try this!

• a) Irrational

• b) Irrational

• c) Rational

• d) Rational

• e) Irrational66 e)

d)

25 c)

12 b)

2 a)

115

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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.

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A fraction with a denominator of 0 is undefined because you cannot divide by

zero. So it is not a number at all.

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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.

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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.

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Objective

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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

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Practice

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Ordering Rational and Irrational Numbers

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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

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Practice

Order these from least to greatest:

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Objectives

• TSW identify the rules associated computing with integers.

• TSW compute with integers

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Examples: Use the number line if necessary.

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

-43) 5 + (-7) =

-2

0 5-5

1) (-4) + 8 =

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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

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Karaoke Time!

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-1 + 3 = ?

1. -42. -23. 24. 4

Answer Now

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-6 + (-3) = ?

1. -92. -33. 34. 9

Answer Now

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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

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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)

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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

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Which is equivalent to-12 – (-3)?

Answer Now

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

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7 – (-2) = ?

Answer Now

1. -92. -53. 54. 9

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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

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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.

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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

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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.

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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

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Summary

• What did you learn in this lesson?• What are some important facts to

remember about the real number system?

• Is there something within the lesson that you need help on?

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Thank you !!!Thank you !!!