04 real numbers
TRANSCRIPT
The Real Number System
Math 11 – College Algebra
“Education is not the filling of a bucket, but the lighting of a fire.”
- W.B. Yeats
Real Numbers
• Real numbers consist of all the rational and irrational numbers.
• The real number system has many subsets:– Natural Numbers – Whole Numbers – Integers
Natural Numbers
• Natural numbers are the set of counting numbers. {1, 2, 3,…}
Denoted by N N={1,2,3,...}
Set of Positive integers
Whole Numbers
• Whole numbers are the set of numbers that include 0 plus the set of natural numbers.
{0, 1, 2, 3, 4, 5,…}
Denoted by W W={0,1,2,3,...}
Integers
• Integers are the set of whole numbers and their opposites.
{…,-3, -2, -1, 0, 1, 2, 3,…}
Denoted by J J={-3,-2,-1,0,1,2,3,...}
Set of Positive Integers, Negative Integers and Zero
Rational Numbers
• Rational numbers are any numbers that can be expressed in the form of , where a and b are integers, and b ≠ 0. b
a
Denoted by Q
Q ={...,-52, -8/5,0,7/8,18,72,...}
Terminating Decimals
• Terminating decimals are decimals that contain a finite number of digits.
• Examples:36.80.1254.5
Repeating Decimals
• Repeating decimals are decimals that contain a infinite number of digits.
• Examples: 0.333… 7.689689…
FYI…The line above the decimals indicate that numberrepeats.
9.1
J and Q
J Q
Terminating Decimals
35
0.60
94 2.25
Consider the following rational numbers:
Nonterminating and Repeating Decimals
Consider the following rational numbers:
13
0.333...4711 4.272727...
Question
The Answer is
Irrational Numbers
2 1.41421356...
3.14159...Denoted
by H3{..., 11, 2, , 17,...}H
REAL NUMBERS
H Q
Irrational Numbers
• Irrational numbers are any numbers that cannot be expressed as .
• They are expressed as non-terminating, non-repeating decimals; decimals that go on forever without repeating a pattern.
• Examples of irrational numbers:– 0.34334333433334…– 45.86745893…– (pi)–
ba
2
ba
Venn Diagram of the Real Number System
Irrational NumbersRational Numbers
Example
• Classify all the following numbers as natural, whole, integer, rational, or irrational. (N, J, Q, H) List all that apply.
a. 117b. 0c. -12.64039…d. -½e. 6.36f. g. -3
To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn
diagram.
94
21
Rational NumbersIntegers
Whole Numbers
NaturalNumbers
Irrational Numbers
-12.64039…
117
0
6.3694
-3
The Bigger PictureNatural Numbers
(N)Positive Integers
Zero (0)
Whole Numbers (W)
Negative Integers
Integers (J) Noninteger Rational Numbers
Rational Numbers (Q)
Irrational Numbers (H)
REAL NUMBERS (R)
FYI…For Your Information
• When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non-repeating. Therefore, those numbers are always irrational.
Try This!
• Insert either or to make the statements correct
a.N__J d. {0}__Qb.Q__J e. { , , 3.5}__Hc.N__R
/
2
Try This!
• In each of the following, determine which one of the sets N,J,Q,H,R and is equal to the given set
a.J Q c. N H b.J Q d. H R
/
Try This!
• Insert either or to make the statements correct
a.N__J d. {0}__Qb.Q__J e. { , , 3.5}__Hc.N__R
/
2
Try This!
• In each of the following, determine which one of the sets N,J,Q,H,R and is equal to the given set
a.J Q c. N H b.J Q d. H R
/