1.1: Real Numbers

Question 1

What are the individual objects in a set called?

Answer 1

Elements

Question 2

What are natural numbers?

Answer 2

Ordinary numbers we use to count things. Positive integers.

1, 2, 3, 4, 5, etc.

Question 3

What is a composite number?

Answer 3

A natural number other than 1 that is not prime

4, 6, 9, 10, etc.

Question 4

What is a prime number?

Answer 4

A natural number other than 1 that is evenly divisible only by itself and 1

Question 5

What are whole numbers?

Answer 5

Whole numbers are all natural numbers AND 0

Question 6

What are integers?

Answer 6

Integers are the whole numbers, the negative “whole numbers,” and 0

-3, -2, -1, 0, 1, 2, 3, etc.

Question 7

What are rational numbers?

Answer 7

Numbers that can be written in the form f a fraction p/q where p and q are integers and q is NOT 0

Numbers that can be written as repeating or terminating decimals

Question 8

What are irrational numbers?

Answer 8

Numbers that cannot be written as terminating or repeating decimals, like PI.

Question 9

What are real numbers?

Answer 9

Rational and irrational numbers taken together.

Question 10

What is a variable?

Answer 10

A symbol that can stand for something else like a pronoun

Question 11

What does “x is an element of the set {0, 2, 4, 6} mean?

Answer 11

It means x can be replaced by 0, 2, 4, or 6

Question 12

What is a domain?

Answer 12

A domain is the set of a variable

{0, 2, 4, 6} is the domain of X

Question 13

Define inequality

Answer 13

If a and b are two real numbers and a is to the left of b on the number line, then a is less than b. a < b

If a and b are two real numbers and a is to the right of b on the number line, then a is greater than b. a > b

Question 14

Additive inverse

Answer 14

The additive inverse, or opposite, of a number is the number that, when added to a, yields zero. The additive inverse of a is denoted by unary minus: -a

Question 15

How do we denote an additive inverse?

Answer 15

We use a unary minus

• (4) = -4
• (-4) = 4

Question 16

Define absolute value

Answer 16

The absolute value of a number is a measure of its distance from zero on a number line

-5 | = 5

Question 17

What is the relationship between an additive inverse and absolute value?

Answer 17

The absolute value of a negative number is the additive inverse of the number

-8 | = 8
-(-8) = 8

Question 18

What is the absolute value of 0?

Answer 18

0

Question 19

What is the roster method of writing a set?

Answer 19

Encloses a list of elements of the set in curly braces

Question 20

What is a finite set?

Answer 20

All elements can be listed

Question 21

What is an infinite set?

Answer 21

All elements cannot be listed

Question 22

What is a null set?

Answer 22

The set that contains no elements

{ } denotes an empty or null set

Question 23

Name three ways to represent sets

Answer 23

Set-builder notation
Roster rotation
Interval notation

Question 24

What is set-builder notation?

Answer 24

Can represent any set, but very useful for infinite number sets

{x | x > -3, x is an element of integers}

Question 25

What makes set builder notation useful?

Answer 25

We use it for sets of infinite numbers

Question 26

How do we read { x | x > -3, x is an element of integers}

Answer 26

The set of all x such that x is greater than -3

Question 27

If we see se builder notation for the following, what should we assume x is?

{x | x < 5}

Answer 27

Assume x is a real number. The text omitted “x is an element of real numbers” for brevity

Question 28

What is an interval?

Answer 28

The set of all numbers between the given numbers

Question 29

What are endpoints?

Answer 29

The two numbers that define the beginning and end of an interval

Question 30

What is a closed interval?

Answer 30

A closed interval includes both end points

{x | 0 <= 4}

Question 31

What is an open interval?

Answer 31

An open interval contains neither endpoint

{x | 0 < x < 3}

Question 32

What is a half-open interval?

Answer 32

A set that contains one endpoint but not the other

{x | -1 < x <= 2}

Question 33

Interval notation

Answer 33

The brackets or parentheses that are used to graph the set are written with the endpoints of the interval

{x | -3 <= 2}

Question 34

Can we perform operations on sets?

Answer 34

Yes

Question 35

Define union

Answer 35

The union of two sets, A U B, is the set of all elements that belong to either A or B.

A U B = {x | x E A or x E B }

A = {1, 2, 3}
B = {1, 3, 5, 7}
A U B = {1, 2, 3, 5, 7}

Union represents common elements on once, discards the other one

Question 36

Define intersection

Answer 36

The intersection of two sets, A ∩ B, is the set of all elements that are common to A and B

A = {1, 2, 3}
B = {1, 3, 5}
A ∩ B = {1, 3}