Section 1.8
The numbers that make up the set of Real Numbers can be classified as counting numbers whole numbers integers rational numbers irrational numbers Each is said to be a subset of the real numbers.
Subsets of Real Numbers
The counting numbers are the numbers with which we count. They are the numbers 1, 2, 3, and so on. The notation we use to specify a group of numbers like this is called set notation. We use the curly braces { } to enclose the members of the set.
We express the set of Counting Numbers as: Counting numbers = {1, 2, 3,... } Which of the numbers in the following set are not counting numbers? { 3, 0,, 1, 1.5, 3} The numbers 3, 0, numbers., and 1.5 are not counting
The whole numbers include the counting numbers and the number 0. Whole numbers = {0, 1, 2,... } The set of integers includes the whole numbers and the opposites of all the counting numbers. Integers = {..., 3, 2, 1, 0, 1, 2, 3,... } When we refer to positive integers, we are referring to the numbers 1, 2, 3,.... Likewise, the negative integers are 1, 2, 3,.... The number 0 is neither positive nor negative.
The set of rational numbers is the set of numbers commonly called fractions together with the integers. The set of rational numbers is difficult to list in the same way we have listed the other sets, so we will use a different kind of notation: Rational numbers = { a and b are integers (b 0)}
If a number can be put in the form, where a and b are both from the set of integers, then the number is called a rational number. So think of a rational numbers as:
Show why each of the numbers in the following set is a rational number. { 3,, 0, 0.333..., 0.75} The number 3 is a rational number because it can be written as the ratio of 3 to 1; that is, The number 2 to 3 can be thought of as the ratio of
The number 0 can be thought of as the ratio of 0 to 1 0 1
Any repeating decimal, such as 0.333... (the dots indicate that the 3's repeat forever), can be written as the ratio of two integers. 0.333... is the same as the fraction Finally, any decimal that terminates after a certain number of digits can be written as the ratio of two integers. The number 0.75 is equal to the fraction therefore a rational number. and is
There are other numbers that can be associated with a point on the real number line, but cannot be written as the ratio of two integers. In decimal form these numbers never terminate and never repeat. They are called irrational numbers (because they are not rational): Irrational numbers = {non-repeating, nonterminating decimals}
Irrational numbers can t be written in a form that is familiar to us. Irrational numbers cannot be written as the ratio of two integers. One example of an irrational number is π. It is not 3.14. Rather, 3.14 is an approximation to π. We use this approximation because we cannot write π as a terminating decimal number.
Other representations for irrational numbers are and, in general, the square root of any number that is not itself a perfect square. Every real number is either rational or irrational. (It cannot be both!) The set of Real Numbers is the set of all rational and irrational numbers Real numbers = {all rational numbers and all irrational numbers}
Prime Numbers and Factoring
The following diagram shows the relationship between multiplication and factoring: When we read the problem from left to right, we say the product of 3 and 4 is 12. Or we multiply 3 and 4 to get 12. When we read the problem in the other direction, from right to left, we say we have factored 12 into 3 times 4, or 3 and 4 are factors of 12.
The number 12 can be factored still further: 12 = 4 3 = 2 2 3 = 2 2 3 The numbers 2 and 3 are called prime factors of 12 because neither of them can be factored any further.
Here is a list of the first few prime numbers. Prime numbers = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,... } When a number is not prime, we can factor it into the product of prime numbers. To factor a number into the product of primes, we simply factor it until it cannot be factored further.
What is the prime factorization of 210? We can use a factor tree to factor 210 into a product of primes 210 = 2 3 5 7
Factor the number 60 into the product of prime numbers. We begin by writing 60 as the product of any two positive integers whose product is 60, like 6 and 10: 60 = 6 10 We then factor these numbers: 60 = 6 10 = (2 3) (2 5) = 2 2 3 5 = 2 2 3 5
Reducing to Lowest Terms
Fractions can be reduced to lowest terms by dividing the numerator and denominator by the same number. We will use the prime factorization of numbers to help us reduce fractions with large numerators and denominators.
Reduce to lowest terms. First we factor 210 and 231 into the product of prime factors (using factor trees) Factor the numerator and denominator completely
Then we reduce to lowest terms by dividing the numerator and denominator by any factors they have in common. Divide the numerator and denominator by 3 7
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