2. 1 – The Meaning and Properties of Fractions 2 2.1 – The Meaning and Properties of Fractions 2.2 – Prime Numbers, Factors, and Reducing to Lowest Terms Catherine Conway Math 081

Fraction – Any number that can be put into the form 𝒂 𝒃 (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 Definitions in 2.1 a. 𝟓 𝟔 b. 𝒙 𝟑 c. 𝟕 𝟐 d. 𝟒 𝟏

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. Definition a. 𝟓 𝟓 b. 𝟕 𝟏 c. 𝟐 𝟑 d. 𝟒 𝟑 = 𝟔 𝟏 e. 𝟔

Equivalent – Fractions that represent the same number 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 Definition a. 𝟐 𝟑 b. 𝟔 𝟗 c. 𝟐𝟎 𝟑𝟎 d. 𝟏𝟖 𝟐𝟕 e. 𝟏𝟐 𝟏𝟖

Property 1 for fractions If a, b, and c are number and b and c are not zero, then it is always true that 𝒂 𝒃 = 𝒂 ∙ 𝒄 𝒃 ∙ 𝒄 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. Property 1 for fractions a. 𝟒 𝟕 = 𝟒 𝒙 𝟔 𝟕 𝒙 𝟔 = 𝟐𝟒 𝟒𝟐

Property 2 for fractions If a, b, and c are number and b and c are not zero, then it is always true that 𝒂 𝒃 = 𝒂 ÷ 𝒄 𝒃 ÷ 𝒄 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. Property 2 for fractions a. 𝟒𝟖 𝟓𝟔 = 𝟒𝟖 ÷ 𝟖 𝟓𝟔 ÷ 𝟖 = 𝟔 𝟕

The number “1” and fractions 1. When the denominator of a fraction is 1 If we let a represent any number, then 𝒂 𝟏 =𝒂 for any number a. When the numerator and the denominator of a fraction are the same nonzero number. If we let a represent any number, then 𝒂 𝒂 =𝟏 for any number a. Example: Simplify each expression. The number “1” and fractions a. 𝟕𝟐 𝟏 b. 𝟐𝟖 𝟐𝟖 c. 𝟖𝟏 𝟐𝟕 d. 𝟑𝟔 𝟏𝟐 a. 72 b. 𝟏 c. 𝟑 d. 𝟑

Comparing Fractions = 𝟐 𝟑𝟎 = 𝟐𝟓 𝟑𝟎 = 𝟐𝟏 𝟑𝟎 = 𝟏𝟓 𝟑𝟎 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. Comparing Fractions a. 𝟏 𝟏𝟓 = 𝟐 𝟑𝟎 b. 𝟓 𝟔 = 𝟐𝟓 𝟑𝟎 c. 𝟕 𝟏𝟎 = 𝟐𝟏 𝟑𝟎 d. 𝟏 𝟐 = 𝟏𝟓 𝟑𝟎 a. 𝟏 𝟏𝟓 d. 𝟏 𝟐 c. 𝟕 𝟏𝟎 b. 𝟓 𝟔

Go to page 152 #69, 71, 73 Application 𝟒 𝟓 𝟐𝟗 𝟒𝟑 𝟏,𝟏𝟐𝟏 𝟏,𝟕𝟗𝟏

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 Definition in 2.2 composite composite prime composite

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 and Composite Numbers

Prime Factorization is where you write the composition number using prime factors. Example: 108 Prime Factorization 108 150 2 54 15 10 6 9 3 5 2 5 2 3 3 3 2 · 3 · 5 · 5 = 2 · 3 · 5 2 2 · 2 · 3 · 3 · 3 = 22 · 3 3

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. Lowest Terms

Example: see page 159 #26, 34, 37, 39, 48 Reduce each fraction to lowest terms = 𝟑 𝟓 = 𝟓 𝟑 = 𝟏𝟏 𝟕 = 𝟓 𝟑 = 𝟑𝟗 𝟓𝟓

Go to page 160 #65, 67, 69 (Application) 𝟕𝟎 𝟐𝟏𝟎 = 𝟕 𝟐𝟏 = 𝟏 𝟑 𝟏 𝟖 𝟏𝟐 𝟑𝟐 = 𝟑 𝟖