Download Sets of Real Numbers

REAL NUMBERS
(as opposed to fake numbers?)
Warm UP
Simplify the following expressions.
1.
2( z  3)  3(5  3 z )
2.
 2 x  3
2
7 z  21
4 x 2  12 x  9
Essential Question
• How can I classify real numbers?
• Why is the sum or product of
rational numbers rational?
• Why is the sum of a rational number
and an irrational number irrational?
• Why is the product of a nonzero
rational number and an irrational
number irrational?
Vocabulary Words
Real Number- a number which can be written as a
decimal.
Rational Numbers- a number expressible in the form
a/b or –a/b (fractions) including integers.
Irrational Numbers- a number not expressible in the
form a/b or –a/b and the decimal form is non
repeating and non terminating.
Integers- the numbers …-3,-2, -1, 0, 1, 2, 3 …., if 1 is
in the denominator can be written as a fraction.
Whole Number- the numbers 0, 1, 2, 3, … .
Natural Number- the numbers 1, 2, 3, … .
Two Kinds of Real Numbers
• Rational Numbers
• Irrational Numbers
Branching Real Numbers
Real Numbers
Rational
Irrational
Now let’s break these
subsets down
further!
Examples of Rational
Numbers
•16
•1/2
•3.56
•-8
•1.3333…
•- 3/4
Can you write 16 as a ratio?
Now try writing -8 as a ratio!
Examples of Irrational
Numbers
• Square roots of
non-perfect
“squares”
• Pi
17
What is the square root of 17?
What is another non-perfect square?
Integers
One of the subsets of rational
numbers
So what do integers branch off of?
What are integers?
• Integers are the whole numbers and their
opposites.
• Examples of integers are
6
-12
0
186
-934
What do you notice about these integers?
Whole Numbers
One of the subsets of rational
numbers and integers
So what do whole numbers branch off of?
Natural Numbers
One of the subsets of rational
numbers, integers and whole
numbers
So what do natural numbers branch off of?
Classify the following
Numbers
• What subset(s) do the following
numbers belong to? Remember they
could belong to more than one!
• 0
• Whole, Integers, Rational, and Real
Classify the following
Numbers (Con’t)
• 4
• Natural, Whole, Integer, Rational, Real
• -9
• Integer, Rational, Real
• Π
• Irrational, Real
Classify the following
Numbers (Con’t)
• 3/4 or 0.75
• Rational and Real
•
4
•
15
• Natural, Whole, Integer, Rational, Real
• Irrational, Real
Warm up
• Simplify
• 1.
63xy 2
• 2.
• 3.
 18  2 8  3 5



2  3 5 2 1
The Big Picture!!!
Why is the sum or product of rational numbers
rational?
• Find the sum of two rational numbers.
3 1 7
•
 
4 8 8
or
8  5  3
• Find the product of two rational numbers.
• 10 7 
70
or
7 3
21
 
13 4
52
Why is the sum of a rational number and an
irrational number irrational?
• Find the sum of the rational number and
the irrational number.
• 1. 5  3  6.732050808...
1
• 2.    3.641592654...
2
Why is the product of a non zero rational
number and an irrational number irrational?
• 1. 4  7  10.58300524...
4
• 2.
 19  .917662935...
19