1.1: Real Numbers Flashcards
What are the individual objects in a set called?
Elements
What are natural numbers?
Ordinary numbers we use to count things. Positive integers.
1, 2, 3, 4, 5, etc.
What is a composite number?
A natural number other than 1 that is not prime
4, 6, 9, 10, etc.
What is a prime number?
A natural number other than 1 that is evenly divisible only by itself and 1
What are whole numbers?
Whole numbers are all natural numbers AND 0
What are integers?
Integers are the whole numbers, the negative “whole numbers,” and 0
-3, -2, -1, 0, 1, 2, 3, etc.
What are rational numbers?
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
What are irrational numbers?
Numbers that cannot be written as terminating or repeating decimals, like PI.
What are real numbers?
Rational and irrational numbers taken together.
What is a variable?
A symbol that can stand for something else like a pronoun
What does “x is an element of the set {0, 2, 4, 6} mean?
It means x can be replaced by 0, 2, 4, or 6
What is a domain?
A domain is the set of a variable
{0, 2, 4, 6} is the domain of X
Define inequality
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
Additive inverse
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
How do we denote an additive inverse?
We use a unary minus
- (4) = -4
- (-4) = 4
Define absolute value
The absolute value of a number is a measure of its distance from zero on a number line
-5 | = 5
What is the relationship between an additive inverse and absolute value?
The absolute value of a negative number is the additive inverse of the number
-8 | = 8
-(-8) = 8
What is the absolute value of 0?
0
What is the roster method of writing a set?
Encloses a list of elements of the set in curly braces
What is a finite set?
All elements can be listed
What is an infinite set?
All elements cannot be listed
What is a null set?
The set that contains no elements
{ } denotes an empty or null set
Name three ways to represent sets
Set-builder notation
Roster rotation
Interval notation
What is set-builder notation?
Can represent any set, but very useful for infinite number sets
{x | x > -3, x is an element of integers}
What makes set builder notation useful?
We use it for sets of infinite numbers
How do we read { x | x > -3, x is an element of integers}
The set of all x such that x is greater than -3
If we see se builder notation for the following, what should we assume x is?
{x | x < 5}
Assume x is a real number. The text omitted “x is an element of real numbers” for brevity
What is an interval?
The set of all numbers between the given numbers
What are endpoints?
The two numbers that define the beginning and end of an interval
What is a closed interval?
A closed interval includes both end points
{x | 0 <= 4}
What is an open interval?
An open interval contains neither endpoint
{x | 0 < x < 3}
What is a half-open interval?
A set that contains one endpoint but not the other
{x | -1 < x <= 2}
Interval notation
The brackets or parentheses that are used to graph the set are written with the endpoints of the interval
{x | -3 <= 2}
Can we perform operations on sets?
Yes
Define union
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
Define intersection
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}
