Subsets of the Real Numbers

Sets You have seen several instances now the use of a phrase such as “suppose that 𝑥 is a real number” One way that mathematicians interpret this kind of phrase is to say that 𝑥 represents an element of a set *In mathematics, we can think of a set as a collection of things that have some properties in common; an element of the set is just something that belongs to the set *To say, “suppose that 𝑥 is a real number” means the same as saying 𝑥 is an element of the set of real numbers; but just what is in the set of real numbers?

Sets First, let’s talk about the symbols that mathematicians use to represent sets *We indicate that a collection is a set by enclosing it with braces: *For example, seeing 0,1,2,3,4,5,6,7,8,9 , you could say that this is the set of the first nine whole numbers; when you see the braces you know to begin to describe the set by saying “the set of” However, some sets have too many elements to list, and other have an infinite number of elements To represent these sets we may use set-builder notation

Set-Builder Notation *The basic pattern for set-builder notation is: {𝑥|"something about 𝑥"} This reads as “the set of numbers 𝑥 such that ‘something true about 𝑥’” *For example, to represent the set of all even numbers we could write: {𝑛|𝑛 is an even number}; we say “the set of all numbers 𝑛 such that 𝑛 is an even number” *What are three elements of this set? *What are three elements that are not in this set?

Reading Set-Builder Notation Rewrite the following sets as a complete sentence, then give two elements of the set. 𝑛 𝑛>0 𝑥 𝑥 is an odd number {𝑝|𝑝 is a prime number}

Subsets Note that the title of this lesson is “Subsets of the Real Numbers”; what exactly is a subset? *Here is the definition: If 𝐴 and 𝐵 represent any two sets, then 𝑨 is a subset of 𝑩 if every element of 𝐴 is also an element of 𝐵 Suppose that set 𝐵 is the set of all students at DRHS, and set 𝐴 is the set of all student’s in Mr. Longoria’s algebra class *Is it correct to say that 𝐴 is a subset of 𝐵? Why or why not? *Is it correct to say that 𝐵 is a subset of 𝐴? Why or why not?

The Real Numbers Recall that the properties you studied in the previous two class periods were called the Real Number Properties We can think of the real numbers as any number that can be represented by a point on the number line. However, this doesn’t tell us much about the numbers themselves One way to get to know the real numbers is by looking at some important subsets of the real numbers

The Natural Numbers *The numbers that you first learned about, probably in kindergarten, were the Natural Numbers, also called the Counting Numbers *This is the set of numbers ℕ= 1,2,3,… (the symbol ℕ is sometimes used as an abbreviation for the Natural Numbers) Note that the ellipses, … , indicate that the same pattern continues indefinitely; this set has an infinite number of elements *Which of the following are elements of ℕ? a) −4 b) 1/3 c) 101 d) 0 e) 10,321

The Integers Recall that the Inverse Property for Addition says that, for every non- zero number 𝑎, there exists a unique number, −𝑎 From this we can infer the existence of the numbers that we call the Integers *The integers include the natural numbers as a subset, and also include zero and the opposite values of the natural numbers *We can represent the set as ℤ= …,−2,−1,0,1,2,… *Which of the following are elements of ℤ? a) 0.25 b) −53 c) 21 d) 1/3 e) 0

The Rational Numbers On the number line, the natural numbers and the integers account for the points that we usually mark on the line But there are more points in between these Some of these points are accounted for by the rational numbers *The rational numbers are those numbers that can be written as 𝑎 𝑏 , where 𝑎,𝑏 are each integers and 𝑏≠0.

The Rational Numbers *In set-builder notation we can write this as ℚ= 𝑟 𝑟= 𝑎 𝑏 , where 𝑎,𝑏 are integers and 𝑏≠0 *The integers are a subset of the rational numbers because any integer can be written as a fraction over 1 For example, 1 1 =1, −1 1 =−1, 2 1 =2, and so on

The Rational Numbers Rational numbers can also be written in decimal form *A decimal number is a rational number if: The decimal terminates The decimal is infinite, but it has a repeating pattern In both cases it is possible to rewrite either a terminating decimal number or a repeating decimal number as a fraction *Examples of terminating decimals are: 0.5= 1 2 , 0.1= 1 10 , 1.125= 9 8 *Examples of repeating decimals are: 0. 3 = 1 3 , 0. 15 = 5 33 , 2. 6 = 8 3

The Irrational Numbers Remember that the rational numbers were described as “some” of the numbers between the integers There are also numbers between the integers that are not rational, or irrational numbers We can think of the irrational numbers as “all the rest” *It isn’t as easy to describe the irrational numbers except as those numbers that cannot be written as 𝑎 𝑏 , where 𝑎,𝑏 are integers and 𝑏≠0 But there are many different kinds of irrational numbers

The Irrational Numbers *For algebra 2, the irrational numbers we will work with most are the square roots of numbers that are not perfect squares We will look into this more closely later, but the square root of a non- negative number 𝑎 is expressed as 𝑎 The symbol 𝑎 is, itself, a number that, if you multiply it by itself, gives 𝑎 as a result So, for example, 4 =2 because 2⋅2= 2 2 =4; 9 =3 because 3⋅ 3= 3 2 =9

The Irrational Numbers 𝒏 𝒏 𝟐 1 1 =1 2 4 4 =2 3 9 9 =3 16 16 =4 5 25 25 =5 6 36 36 =6 7 49 49 =7

The Irrational Numbers Some of the perfect squares are 1, 4, 9, 16, 25, 36, 49, and so forth It is the square roots of the numbers between these that are not perfect square The square roots of these numbers are irrational numbers *Some examples: 2 ≈1.414 3 ≈1.732 5 ≈2.236 *Remember that irrational numbers cannot be written out!

The Irrational Numbers *As decimals, irrational numbers have an infinite decimal part that has no repeating pattern The number 𝜋 is another irrational number You may have used an approximation, 𝜋≈3.14 The number itself cannot be written, which is why we use the Greek letter 𝜋 to represent the entire number However, you may have used the approximation 𝜋≈3.14 in previous classes

The Real Numbers *The real numbers are the set made by joining the rational and irrational numbers into one set Note, however, that the two subsets, the rational numbers and the irrational numbers, have no numbers in common This means that every number is either rational or irrational; it cannot be both!

The Real Numbers

Examples Choose the letter representing the smallest set of which the given set is a subset. Remember that A is a subset of B if every element of A is also found in B. 0, 1 2 ,1.55,−7 is a subset of__________? −𝜋, 3 , − 5 is a subset of __________? 3, 7, −100 is a subset of __________? 2 , 1 3 , −5, 0. 7 is a subset of __________?

Examples Choose the letter representing the smallest set of which the given set is a subset. Remember that A is a subset of B if every element of A is also found in B. 0, 1 2 ,1.55,−7 is a subset of the rational numbers −𝜋, 3 , − 5 is a subset of the irrational numbers 3, 7, −100 is a subset of the integers 2 , 1 3 , −5, 0. 7 is a subset of the real numbers

Examples Suppose that x is a number as described below. Determine which set(s) it must be an element of. x is a number with an infinite decimal part with no repeating pattern. Therefore, x is an element of _______________ x is a negative number with no decimal part. Therefore, x is an element of _______________ x is a number that can be written as the ratio of two integers. Therefore, x is an element of _______________

Examples Suppose that x is a number as described below. Determine which set(s) it must be an element of. x is a number with an infinite decimal part with no repeating pattern. Therefore, x is an element of the irrational numbers and the real numbers x is a negative number with no decimal part. Therefore, x is an element of the integers, the rational numbers, the real numbers x is a number that can be written as the ratio of two integers. Therefore, x is an element of the rational numbers, the real numbers

Concentrate!