Classifying Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers.
Real Numbers presentation
Transcript of Real Numbers presentation
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REAL NUMBERS
(as opposed to fake numbers?)
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Objective
TSW identify the parts of the RealNumber System
TSW define rational and irrationalnumbers
TSW classify numbers as rational or
irrational
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Real Numbers
Real Numbers are every number.
Therefore, any number that you canfind on the number line.
Real Numbers have two categories.
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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 aredecimals but they dont terminate andare not recurring decimals. They go onforever.
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Real Numbers
REAL NUMBERS
-8 -5,632.1010101256849765
61
49%
549.23789
154,769,852,354
1.333
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Two Kinds of Real Numbers
Rational Numbers
Irrational Numbers
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Rational Numbers
A rational number is a realnumber that can be written
as a fraction. A rational number written in
decimal form is terminatingor repeating.
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Examples of Rational
Numbers16
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 theiropposites.
Examples of integers are
6
-12
0
186
-934
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Integers are rational numbersbecause they can be written as
fraction with 1 as the denominator.
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Types of Integers
Natural Numbers(N):Natural Numbers are counting numbersfrom 1,2,3,4,5,................
N = {1,2,3,4,5,................}
Whole Numbers (W):
Whole numbers are natural numbersincluding zero. They are0,1,2,3,4,5,...............W = {0,1,2,3,4,5,..............}
W = 0 + N
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WHOLE
Numbers
REAL NUMBERS
IRRATIONALNumbers
NATURALNumbers
RATIONALNumbers
INTEGERS
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Irrational Numbers
An irrational number is anumber that cannot be
written as a fraction of twointegers. Irrational numbers written as
decimals are non-terminatingand non-repeating.
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A repeating decimal may not appear torepeat on a calculator, becausecalculators show a finite number of digits.
Caution!
Irrational numberscan be written only as
decimals that do notterminate 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) Irrational
66e)
d)
25c)12b)
2a)
115
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Additional Example 1: Classifying Real
Numbers
Write all classifications that apply to eachnumber.
5 is a whole number that isnot a perfect square.
5
irrational, real
12.75 is a terminating decimal.12.75
rational, real
162
whole, integer, rational, real
= = 242
162
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
03
rational
03
= 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
40C.
State if each number is rational,irrational, or not a real number.
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Objective
TSW compare rational and irrationalnumbers
TSW order rational and irrationalnumbers on a number line
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Comparing Rational andIrrational Numbers
When comparing different forms ofrational and irrational numbers,
convert the numbers to the sameform.
Compare -3 and -3.571(convert -3 to -3.428571
-3.428571 > -3.571
3
737
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Practice
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Ordering Rational andIrrational Numbers
To order rational and irrationalnumbers, convert all of the numbers
to the same form. You can also find the approximatelocations of rational and irrationalnumbers on a number line.
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Example Order these numbers from least to
greatest./, 75%, .04, 10%, /
/ becomes 0.2575% becomes 0.75
0.04 stays 0.04
10% becomes 0.10
/ becomes 1.2857142
Answer: 0.04, 10%, /, 75%, /
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Practice
Order these from least to greatest:
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Objectives
TSW identify the rules associatedcomputing 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 = 2
2 + (-4) = -2
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Karaoke Time!Addition Rule: Sung to the tune
of Row, row, row, your boatSame 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?
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-1 + 3 = ?
1. -4
2. -2
3. 2
4. 4Answer Now
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-6 + (-3) = ?
1. -9
2. -3
3. 3
4. 9
Answer Now
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The additive inverses (oropposites) of two numbers add
to equal zero.
-3Proof: 3 + (-3) = 0
We will use the additiveinverses for subtraction
problems.
Example: The additive inverse of 3 is
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Whats the difference
between7 - 3 and 7 + (-3) ?7 - 3 = 4 and 7 + (-3) = 4
The only difference is that 7 - 3 is asubtraction problem and 7 + (-3) is anaddition problem.
SUBTRACTING IS THE SAME AS
ADDING THE OPPOSITE.
(Keep-change-change)
When subtracting change the
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When subtracting, change thesubtraction to adding the opposite (keep-
change-change) and then follow youraddition 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 + 3
2. -12 + 3
3. -12 - 34. 12 - 3
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7 (-2) = ?
Answer Now
1. -9
2. -5
3. 5
4. 9
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1) If the problem is addition, followyour addition rule.
2) If the problem is subtraction,change subtraction to adding theopposite(keep-change-change) and thenfollowthe 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 WhatsThe
Rule?
Different
Signs
NegativeAnswer
-24
2. -2 * -61
Same
Signs
Positive
Answer
122
3. (-3)(6)(1)
(-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 8C.
If the temperature rises 4C per hour,
what is the temperature at 6 am?
How long
Is it from
Midnight
to 6 am?
How much
does the
temperature
rise eachhour?
6
hours
+4
degrees
(6 hours)(4 degrees per hour)
= 24 degrees
8 + 24 = 32C
Add this to
the 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.
Multiply
(5 steps) (11 feet)
(55 feet)
5 * 11 = 55