copyright 2014 scott storla rational numbers. copyright 2014 scott storla vocabulary rational number...
TRANSCRIPT
Copyright 2014 Scott Storla
Rational Numbers
Copyright 2014 Scott Storla
Vocabulary
Rational number
Proper fraction
Improper fraction
Mixed number
Prime number
Composite number
Prime factorization
Reciprocal
Copyright 2014 Scott Storla
The Rational Numbers
Copyright 2014 Scott Storla
Copyright 2014 Scott Storla
Irrational NumbersThe real numbers which are not rational.
3 14159265358979. ...
22
73 14158. ...
355
1133 1415929. ...
104348
332153 1415926539. ...
Trying to find a rational number that’s equal to pi.
Copyright 2014 Scott Storla
Fractions
Copyright 2014 Scott Storla
Proper Fraction
In a proper fraction the numerator (top) is less than the denominator (bottom).
2
3
The value of a proper fraction will always be between 0 (inclusive) and 1 (exclusive).
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Improper Fraction
In an improper fraction the numerator (top) is greater than or equal to the denominator (bottom).
3
2
The value of an improper fraction is greater than or equal to 1.
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Prime Factorization
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Prime Number
A natural number,
greater than 1,
which has unique natural number factors 1 and itself.
Ex: 2, 3, 5, 7, 11, 13
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Composite Number
A natural number,
greater than 1,
which is not prime.
Ex: 4, 6, 8, 9, 10
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Prime Factorization
I have prime factored a composite number when
the number is written as the product of prime factors.
We say 2 2 3 is the prime factorization of 12
since the factors 2 and 3 are prime.
We don't consider 2 6 a prime factorization of 12
because 6 is not prime.
Copyright 2014 Scott Storla
Prime Factorization
To write a natural number as the product of prime factors.
Ex: 12 = 2 x 2 x 3
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Factor Rules
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Decide if 2, 3, and/or 5 is a factor of
42
310
987
4950
Copyright 2014 Scott Storla
List all positive integers between 51 and 61 inclusive.
List all prime numbers between 51 and 61 inclusive.
List all rational numbers with denominators of 1 between 110 and 120 inclusive.
List all prime numbers between 110 and 120 inclusive.
List all natural numbers between 31 and 40 inclusive.
List all prime numbers between 31 and 40 inclusive.
Copyright 2014 Scott Storla
Building a factor tree for 20
The prime factorization of 20 is 2 x 2 x 5.
20
45
22
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Property – The Commutative Property of Multiplication
English: The order of the factors doesn’t affect the product.
Example: 2 4 4 2
Note: Division is not commutative. For instance 4 2 2 4 .
5 2 3 2 2 2 3 5
Copyright 2014 Scott Storla
The Fundamental Theorem of Arithmetic
Every natural number, greater than 1, has a unique prime factorization.
Example: 20 5 4 2 10 2 2 5
Copyright 2014 Scott Storla
Procedure – To Prime Factor a Natural Number
1. Build a factor tree using the factor rules for 2,3,and 5.
2. After step 1 divide uncircled factors by the prime numbers beginning with 7 up to the square root of the number.
3. Write your prime factors in order from smallest to largest.
Copyright 2014 Scott Storla
The prime factorization of 24 is 2 x 2 x 2 x 3.
24
2 12
Find the prime factorization of 24
2 6
2 3
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The prime factorization of 315 is 3 x 3 x 5 x 7.
315
5 63
Find the prime factorization of 315
3 21
7 3
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The prime factorization of 119 is 7 x 17.
119
7 17
Find the prime factorization of 119
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The prime factorization of 495 is 3 x 3 x 5 x 11.
495
5 99
Find the prime factorization of 495
9 11
3 3
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Prime Factorization
Copyright 2014 Scott Storla
Reducing Fractions
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Property – The Associative Property of Multiplication
English: The grouping of the factors doesn’t affect the product.
Example: 2 3 4 2 3 4
Note: Division is not associative. For instance 8 4 2 8 4 2 .
Property – The Multiplicative Identity
English: 1 is the multiplicative identity. Multiplying an expression by
one results in an equivalent expression.
Example:
3 31
4 4
Copyright 2014 Scott Storla
Reducing Fractions
A fraction is reduced when the numerator and denominator have no common factors other than 1.
6
10
2 3
2 5
3
15
3
5
32
2 5
5
2
2
3
Copyright 2014 Scott Storla
Reducing Fractions
A fraction is reduced when the numerator and denominator have no common factors other than 1.
6
10
2 3
2 5
3
5
6
12
2 3
2 2 3
1
1
2
2 3
2 5 3
15
1
Copyright 2014 Scott Storla
Procedure – Reducing Fractions
1. Prime factor the numerator and denominator.
2. Reduce common factors.
3. Find the product of the factors in the numerator and the product of the factors in the denominator.
No “Gozinta” method allowed
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No “Gozinta” (Goes into) method allowed
84
210
42
105
14
35
2
5
2
5
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No “Gozinta” (Goes into) method allowed
2
2
2
3 2
x x
x x
1x
1x
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Simplify using prime factorization
18 2 3 3
24 2 2 2 3
2 3 3
2 3 2 2
31 1
2 2
3
4
18
24
2 3 3
2 2 2 3
3
4
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18
24
2 3 3
2 2 2 3
3
4
Simplify using prime factorization
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90
63
2 3 3 5
3 3 7
10
7
Simplify using prime factorization
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120
45 2 2 2 3 5
3 3 5
8
3
Reduce using prime factorization
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126
234
2 3 3 7
2 3 3 13
7
13
Reduce using prime factorization
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168
315
2 2 2 3 7
3 3 5 7
8
15
Reduce using prime factorization
Copyright 2014 Scott Storla
Reducing Fractions
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Multiplying Fractions
Copyright 2014 Scott Storla
using prime factorizationMultiply
51 1 1
14
2
2
6 25
15 28
5
14
3 5
5 5 2 32 2 7
3
3
5
5 5
2 7
6 25
15 28 150
420
Copyright 2014 Scott Storla
Procedure – Multiplying Fractions
1. Combine all the numerators, in prime factored form, in a single numerator.
2. Combine all the denominators, in prime factored form, in a single denominator.
3. Reduce common factors
4. Multiply the remaining factors in the numerator together and the remaining factors in the denominator together.
Copyright 2014 Scott Storla
6 25
15 28
2 3 5 5
3 5 2 2 7
5
14
Multiply using prime factorization
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5 5
253 3
3 5 15 69 25
Multiply using prime factorization
2 3
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3 7 2 2 7 2 3 5 18 5 7
21 28 30 2 3 3 5 7 1
28
Multiply using prime factorization
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1 3 2 3 5 1 7 5
2 7 7 2 3
3 30 35
2 49 6
75
14
Multiply using prime factorization
3 30 55
2 49 6
5514
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Dividing Fractions
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Reciprocal
The reciprocal of a number is a second number which when multiplied to the first gives a product of 1.
The reciprocal of is because
7 11
1 7
3 21
2 3
3
2 2
3
The reciprocal of is because 7 1
7
Copyright 2014 Scott Storla
3
10
10
3
10
310
3
1
Procedure – Dividing Fractions
1. To divide two fractions multiply the fraction in the numerator by the reciprocal of the fraction in the denominator.
6
53
10
2 3 2 5
5 3
6 105 3
1
6
5
10
3 4
Copyright 2014 Scott Storla
Procedure – Dividing Fractions
1. To divide two fractions multiply the fraction in the numerator by the reciprocal of the fraction in the denominator.
6
53
10
6 10
5 3
2 3 2 55 3
4
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10181524
2 5 1 2 2 2 3
2 3 3 3 5
8
9
Divide using prime factorization
10 24
18 15
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355014
7 5
2 5 5 2 7
1
20
Divide using prime factorization
35 1
50 14
14
1
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1216
9
1 2 2 2 2 3 3
2 2 3
12
Divide using prime factorization
16 9
1 12
1
16
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59
143
21
15 7 3
3 3 2 7
1
6
Divide using prime factorization
5 3 73 3 3 5
143
79
143
7 3
9 14
6115
Copyright 2014 Scott Storla
Dividing Fractions