NWZ
QIR
The Real Number System
Presented By: Prince
Objectives
Real Number
Real Number
Real Number
What does it Mean?
Real Numbers
REAL NUMBERS
-8 -5,632.1010101256849765…
61
49%
π
549.23789
154,769,852,354
1.333
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.
Real Number System Tree Diagram
Real Numbers
IntegersTerminating Decimals
Repeating Decimals
WholeNumbers
Rational Numbers
Irrational Numbers
Negative #’s
Natural #’s Zero
Non-TerminatingAndNon-RepeatingDecimals
Two Kinds of Real Numbers
Rational Numbers
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 are6-120186-934
What are integers?.......
• Integers are rational numbers because they can be written as fraction with 1 as the denominator.
Types of Integers
WHOLENumber
s
REAL NUMBERS
IRRATIONALNumbers
NATURALNumbers
RATIONALNumbers
INTEGERS
Irrational Numbers
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
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
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!
-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
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?
Thank you !!!Thank you !!!
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