2.1 – The Meaning and Properties of Fractions 2.2 – Prime Numbers, Factors, and Reducing to...

2.1 – The Meaning and Properties of Fractions

2.2 – Prime Numbers, Factors, and Reducing to Lowest Terms

Catherine Conway

Math 081

Definitions in 2.1

Fraction – Any number that can be put into the form (sometimes written as a/b), where a and b are numbers and b cannot be zero.

In the fraction, a and b are called terms of the fraction, where a is called the numerator and b is called the denominator.

Example: name the numerator and denominator

a. b. c. d.

Definition

A proper fraction is a fraction in which the numerator is less than the denominator.

An improper fraction is a fraction in which the numerator is greater than or equal to the denominator.

Example: Determine which is a proper fraction and which is an improper fraction.

a. c. b. d. e. =

Definition

Equivalent – Fractions that represent the same number. Equivalent may look different but they have the same value when reduced.

Example: the following are equivalent fractions

a. b. c. d. e.

Property 1 for fractions

If a, b, and c are number and b and c are not zero, then it is always true that

If the numerator and the denominator are multiplied by the same nonzero factor, the result is equivalent to the original.

Example: Write 4/7 as an equivalent fraction with denominator of 42.

a.

¿𝟐𝟒𝟒𝟐

Property 2 for fractions

If a, b, and c are number and b and c are not zero, then it is always true that

If the numerator and the denominator are divided by the same nonzero factor, the result is equivalent to the original.

Example: Write 48/56 as an equivalent fraction with denominator of 7.

a.

¿𝟔𝟕

The number “1” and fractions

1. When the denominator of a fraction is 1

If we let a represent any number, then for any number a.

2. When the numerator and the denominator of a fraction are the same nonzero number. If we let a represent any number, then for any number a.

Example: Simplify each expression.a. b. c. d.

a. 72 b. c. d.

Comparing Fractions

Comparing fractions are used to see which fraction is larger or smaller when they have the same denominator.

Example: Write each fraction as an equivalent fraction with the denominator 30. Then write then in order from smallest to greatest.

a. b. c. d.

¿𝟐𝟑𝟎

¿𝟐𝟓𝟑𝟎

¿𝟐𝟏𝟑𝟎

¿𝟏𝟓𝟑𝟎

a. d. c. b.

Application

Go to page 152 #69, 71, 73

𝟒𝟓

𝟐𝟗𝟒𝟑

𝟏 ,𝟏𝟐𝟏𝟏 ,𝟕𝟗𝟏

Definition in 2.2

Prime Numbers – Any whole number greater than 1 that has exactly two divisors – itself and 1. ( number is a divisor of another number if it divides it without remainder)

Composite Number – Any whole number greater than 1 that is not a prime number. A composite number always has at least one divisor other than 1 and itself.

Example:

a. 81 b. 21 c. 19 d. 108

composite composite compositeprime

Prime and Composite Numbers

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Prime Factorization

Prime Factorization is where you write the composition number using prime factors.

Example: 108108

2 54

6 9

2 3 3 3

2 · 2 · 3 · 3 · 3 = 22 · 3 3

150

15 10

2 5

2 · 3 · 5 · 5 = 2 · 3 · 5 2

3 5

Lowest Terms

A fraction is said to be in lowest terms if the numerator and the denominator have no factors in common other than the number 1.

Example: see page 159 #26, 34, 37, 39, 48

Reduce each fraction to lowest terms

= = =

= =

Go to page 160 #65, 67, 69 (Application)

= =

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