Copyright 2014 Scott Storla Rational Numbers. Copyright 2014 Scott Storla Vocabulary Rational number...

$: Copyright 2014 Scott Storla Rational Numbers. Copyright 2014 Scott Storla Vocabulary Rational number Proper fraction Improper fraction Mixed number Prime.$
Copyright 2014 Scott Storla

Rational Numbers


Vocabulary

Rational number

Proper fraction

Improper fraction

Mixed number

Prime number

Composite number

Prime factorization

Reciprocal


The Rational Numbers


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.


Fractions


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).


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.


Prime Factorization


Prime Number

A natural number,

greater than 1,

which has unique natural number factors 1 and itself.

Ex: 2, 3, 5, 7, 11, 13


Composite Number

A natural number,

greater than 1,

which is not prime.

Ex: 4, 6, 8, 9, 10


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.


Prime Factorization

To write a natural number as the product of prime factors.

Ex: 12 = 2 x 2 x 3


Factor Rules


Decide if 2, 3, and/or 5 is a factor of

42

310

987

4950


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 natural numbers between 31 and 40 inclusive.



Building a factor tree for 20

The prime factorization of 20 is 2 x 2 x 5.

20

45

22


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


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


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.


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



315

5 63


3 21

7 3


The prime factorization of 119 is 7 x 17.

119

7 17




495

5 99


9 11

3 3


Prime Factorization


Reducing Fractions


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


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


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


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


No “Gozinta” (Goes into) method allowed

84

210

42

105

14

35

2

5

2

5


No “Gozinta” (Goes into) method allowed

2

2

2

3 2

x x

x x

1x

1x


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


18

24

2 3 3

2 2 2 3

3

4



90

63

2 3 3 5

3 3 7

10

7



120

45 2 2 2 3 5

3 3 5

8

3

Reduce using prime factorization


126

234

2 3 3 7

2 3 3 13

7

13



168

315

2 2 2 3 7

3 3 5 7

8

15



Reducing Fractions


Multiplying Fractions


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


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.


6 25

15 28

2 3 5 5

3 5 2 2 7

5

14

Multiply using prime factorization


5 5

253 3

3 5 15 69 25


2 3


3 7 2 2 7 2 3 5 18 5 7

21 28 30 2 3 3 5 7 1

28



1 3 2 3 5 1 7 5

2 7 7 2 3

3 30 35

2 49 6

75

14


3 30 55

2 49 6

5514


Dividing Fractions


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


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


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


10181524

2 5 1 2 2 2 3

2 3 3 3 5

8

9

Divide using prime factorization

10 24

18 15


355014

7 5

2 5 5 2 7

1

20


35 1

50 14

14

1


1216

9

1 2 2 2 2 3 3

2 2 3

12


16 9

1 12

1

16


59

143

21

15 7 3

3 3 2 7

1

6


5 3 73 3 3 5

143

79

143

7 3

9 14

6115


Dividing Fractions

Copyright 2014 Scott Storla Rational Numbers. Copyright 2014 Scott Storla Vocabulary Rational number...

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