Download - Real numbers system
![Page 1: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/1.jpg)
NWZ
QIR
The Real Number System
Presented By: Prince
![Page 2: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/2.jpg)
Objectives
![Page 3: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/3.jpg)
Real Number
Real Number
Real Number
![Page 4: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/4.jpg)
What does it Mean?
![Page 5: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/5.jpg)
Real Numbers
REAL NUMBERS
-8 -5,632.1010101256849765…
61
49%
π
549.23789
154,769,852,354
1.333
![Page 6: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/6.jpg)
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.
![Page 7: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/7.jpg)
Real Number System Tree Diagram
Real Numbers
IntegersTerminating Decimals
Repeating Decimals
WholeNumbers
Rational Numbers
Irrational Numbers
Negative #’s
Natural #’s Zero
Non-TerminatingAndNon-RepeatingDecimals
![Page 8: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/8.jpg)
Two Kinds of Real Numbers
![Page 9: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/9.jpg)
Rational Numbers
![Page 10: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/10.jpg)
Examples of Rational Numbers
•16•1/2•3.56
•-8•1.3333…
•- 3/4
![Page 11: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/11.jpg)
Integers
One of the subsets of rational numbers
![Page 12: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/12.jpg)
What are integers?
• Integers are the whole numbers and their opposites.
• Examples of integers are6-120186-934
![Page 13: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/13.jpg)
What are integers?.......
• Integers are rational numbers because they can be written as fraction with 1 as the denominator.
![Page 14: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/14.jpg)
Types of Integers
![Page 15: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/15.jpg)
WHOLENumber
s
REAL NUMBERS
IRRATIONALNumbers
NATURALNumbers
RATIONALNumbers
INTEGERS
![Page 16: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/16.jpg)
Irrational Numbers
![Page 17: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/17.jpg)
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.
![Page 18: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/18.jpg)
Examples of Irrational Numbers
• Pi
![Page 19: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/19.jpg)
Try this!
• a) Irrational
• b) Irrational
• c) Rational
• d) Rational
• e) Irrational66 e)
d)
25 c)
12 b)
2 a)
115
![Page 20: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/20.jpg)
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.
![Page 21: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/21.jpg)
A fraction with a denominator of 0 is undefined because you cannot divide by
zero. So it is not a number at all.
![Page 22: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/22.jpg)
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.
![Page 23: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/23.jpg)
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.
![Page 24: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/24.jpg)
Objective
![Page 25: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/25.jpg)
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
![Page 26: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/26.jpg)
Practice
![Page 27: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/27.jpg)
Ordering Rational and Irrational Numbers
![Page 28: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/28.jpg)
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
![Page 29: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/29.jpg)
Practice
Order these from least to greatest:
![Page 30: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/30.jpg)
Objectives
• TSW identify the rules associated computing with integers.
• TSW compute with integers
![Page 31: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/31.jpg)
Examples: Use the number line if necessary.
42) (-1) + (-3) =
-43) 5 + (-7) =
-2
0 5-5
1) (-4) + 8 =
![Page 32: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/32.jpg)
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
![Page 33: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/33.jpg)
Karaoke Time!
![Page 34: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/34.jpg)
-1 + 3 = ?
1. -42. -23. 24. 4
Answer Now
![Page 35: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/35.jpg)
-6 + (-3) = ?
1. -92. -33. 34. 9
Answer Now
![Page 36: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/36.jpg)
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
![Page 37: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/37.jpg)
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)
![Page 38: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/38.jpg)
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
![Page 39: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/39.jpg)
Which is equivalent to-12 – (-3)?
Answer Now
1. 12 + 32. -12 + 33. -12 - 34. 12 - 3
![Page 40: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/40.jpg)
7 – (-2) = ?
Answer Now
1. -92. -53. 54. 9
![Page 41: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/41.jpg)
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
![Page 42: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/42.jpg)
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.
![Page 43: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/43.jpg)
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
![Page 44: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/44.jpg)
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.
![Page 45: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/45.jpg)
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
![Page 46: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/46.jpg)
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?
![Page 47: Real numbers system](https://reader033.vdocuments.mx/reader033/viewer/2022061613/557c8e7dd8b42adc268b47a5/html5/thumbnails/47.jpg)
Thank you !!!Thank you !!!