4 In the loop is a set of whole numbers In the loop is a set of whole numbers. Is the sum of any two whole numbers also a whole number? Is the product of any two whole numbers also a whole number?Whole Numbers0, 1 , 2, 3, ….
5 We say that the set of whole numbers is closed under addition and multiplication because every sum or product is a whole number. Referring to our illustration we might say that we can find any sum or product of whole numbers within the whole numbers loop.
6 Let us consider subtraction Let us consider subtraction. If we subtract any two whole numbers, is the result a whole number? 3 – 1 is a whole number, but 1 – 3 = -2 which is not a whole number and therefore outside the loop.
8 Integers includes the whole numbers as well as the negatives of whole numbers. The set of integers is closed under addition, subtraction and multiplication. Every sum, difference, and product can be found inside the integers loop.
9 Now we will consider division Now we will consider division. If we divide any two integers, is the result an integer?
10 The divisions 1÷2, 3÷2, 3÷4 and many others all have quotients that are fractions and not integers. These quotients are examples of rational numbers.
12 Rational Numbers*: All numbers that can be written as a ratio of two integers. ⅗ and 37 are rational numbers. √2 and π are not rational numbers.
13 On the number line, rational numbers include the integers as well as many points between the integers.
14 Every integer can be expressed as a ratio of two integers Every integer can be expressed as a ratio of two integers. Therefore, every integer is a rational number. Only some rational numbers can be expressed as a whole number or a negative of a whole number. Therefore, only some rational numbers are integers.
15 Example: Describe each number as a whole number, an integer, or a rational number. Write every term that applies. Then graph on a number line ¾
16 Answer: -3: integer and rational number 2: whole number, integer, and rational number ¾: rational number
17 One way to express a rational number is as a fraction One way to express a rational number is as a fraction. Many different fractions can name the same number. These are examples of equivalent fractions. 4/8 3/6 2/4 1/2
18 Equivalent Fractions. : Different fractions that name the same amount
19 We can reduce fractions by removing pairs of factors that the numerator and denominator have in common. By doing this we get the fraction in simplest form. 12/18 = 4/6 = 2/3
20 We divide 12/18 by 6 to get 2/3 or the simplest form We divide 12/18 by 6 to get 2/3 or the simplest form. This means that 6 is the greatest common factor. Greatest Common Factor*: The largest whole number that is a factor of every number in the set.
21 In this lesson we will practice reducing fractions by first writing the prime factorization of the terms of the fraction. Then we remove pairs of like terms from the numerator and denominator and simplify.
22 Example: Using prime factorization, reduce 72 108
23 Answer: 72 = 2 x 2 x 2 x 3 x x 2 x 3 x 3 x 3 = 2/3 Each identical pair of factors from the numerator and denominator reduces to 1 over 1.
24 Notice that a fraction equals 1 if the numerator and denominator are equal. Multiplying by a fraction equal to 1 does not change the size of the fraction, but it changes the name of the fraction.
25 What property of multiplication are we using when we multiply by a fraction equal to 1?