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Real Numbers and Properties
Objective: The students will be able to classify real numbers
and recognize different properties that exist with real numbers.
M11.A.1.3.1 – Locate/identify irrational numbers at the approximate location on
a number line.
M11.A.1.3.2 – Compare and/or order any real numbers (rational and irrational
may be mixed).
Key Question
• What is the difference between rational
and irrational numbers and how else can
numbers be classified?
Classifying Numbers
Set – a collection of objects
• Real Numbers – all numbers are in this set (rational OR irrational)
SUBSETS
• Rational Numbers – can be written as a fraction or a decimal that
stops or repeats
Examples:
½
-7.34
0.11
• Irrational Numbers – decimals that never stop and never repeat
Examples:
17
π
 21
Are All Radicals Irrational?
a
Radicand – number under
the radical
No…If the radicand is a perfect square, the radical is rational.
Perfect Squares
Common Perfect Squares
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
• Whenever you take the square root (radical) of
a perfect square, you get a number that is
rational. The square root undoes squaring a
number.
42  16 and 16  4
Classifying Numbers
• Natural Numbers – “counting numbers”
1, 2, 3, 4, 5, …
• Whole Numbers – includes 0
0, 1, 2, 3, 4, 5, …
• Integers – positive or negative whole number or zero
“dashes on the number line”
{…-3, -2, -1, 0, 1, 2, 3, …}
Real Numbers
Rational Numbers
Integers
Whole Numbers
Natural Numbers
Irrational Numbers
Real Numbers
Rational
Integers
Whole
Natural
Irrational
Identify the sets into which each number belongs
1.
-7
5.
22
2.
1/8
6.
0.91
3.
π
7.
5
4.
0
8.
27
3
5+7=7+5
•
Commutative Property of Addition:
•
Commutative Property of Multiplication:
•
Associative Property of Addition:
•
Associative Property of Multiplication:
•
Additive Identity Property:
•
Multiplicative Identity Property:
•
Additive Inverse Property:
•
Multiplicative Inverse Property:
4 3
 1
3 4
•
Zero Property of Multiplication:
30=0
32=23
5 + (4 + 2) = (5 + 4) + 2
2  (3  5) = (2  3)  5
5+0=5
81=8
12+ (-12) = 0
Tips to Help you Remember
• Commutative Property:
When you commute, you move from one place to another. In math,
can move numbers for addition and multiplication.
• Associative Property:
When you associate with your peers, you are often in groups. In
math, can group numbers in any order for addition and
multiplication.
• Identity Property:
Your identity is who you are. In math, identity properties allow a
number to get back to itself.
• Inverse Property:
The inverse properties look at opposites.
Name that Property!
1.
5+0=5
2.
9+3=3+9
3.
0.8 + (-0.8) = 0
4.
(4 + 2) + 1 = 4 + (2 + 1)
Name that Property!
2 2

3 3
5.
1
6.
13(x + y) = 13x + 13y
7.
6
1
1
6
8.
(ab)c = a(bc)
9.
9(2) = 2(9)
Fill in the Blank
1.
15 + ___ = 12 + 15
2.
9(8 – 3) = (___  8) – (___  3)
3.
3  (4  2) = (3  4)  ___
4.
4  ___ = 1
5.
5 + ___ = 0