Transcript Number Sets Powerpoint

1–1: Number Sets
Counting (Natural) Numbers:
{1, 2, 3, 4, 5, …}
Whole Numbers
{0, 1, 2, 3, 4, 5, …}
Integers
{…–3, –2, –1, 0, 1, 2, 3 …}
Rational Numbers
• All numbers that can be
represented as a/b, where both
a and b are integers and b  0.
• Includes:
• Common fractions
• Terminating decimals
• Repeating decimals
• Integers
• They do not include nonrepeating decimals, such as .
Irrational Numbers
Numbers that are defined as
those that cannot be expressed
as a ratio of two integers.
These include non-terminating,
non-repeating decimals.
Irrational numbers also include
special numbers and ratios,
such as  and 2 .
Real Numbers
• Real numbers include all
rational and irrational
numbers.
Counting
Numbers
Whole
Numbers
Integers
Rational Numbers
Irrational Numbers
Ponder the thought...
True or False?
• All whole numbers are integers.
• All integers are whole numbers.
• All natural numbers are real numbers.
• All irrational numbers are real numbers.
Classify each of the following
numbers using all the terms that
apply: natural (counting), whole,
integer, rational, irrational, and
real.
A)
8
5
B) 3
C)
14
D) –7
Properties of Real Numbers
•
•
•
•
•
•
•
Closure Property
Commutative Property
Associative Property
Identity Property
Inverse Property
Distributive Property
Properties of Equality
Closure Property
The answer is part of the set.
When you add or multiply real numbers, the
result is also a real number.
a + b is a real number
a x b is a real number
Commutative Property
Commutative means that the order
does not make any difference.
a+b=b+a
a•b=b•a
Examples
4+5=5+4
2•3=3•2
The commutative property does not
work for subtraction or division.
Associative Property
Associative means that the grouping
does not make any difference.
(a + b) + c = a + (b + c)
(ab) c = a (bc)
Examples
(1 + 2) + 3 = 1 + (2 + 3)
(2 • 3) • 4 = 2 • (3 • 4)
The associative property does not work
for subtraction or division.
Identity Properties
Do not change the value
1) Additive Identity
What do you add to get the same #?
a+0=a
-6 + 0 = -6
2) Multiplicative Identity
What do you mult. to get the same #?
a•1=a
8•1=8
Inverse Properties
Undo an operation
1) Additive Inverse (Opposite)
• a + (-a) = 0
• 5 + (-5) = 0
2) Multiplicative Inverse
(Reciprocal)
1
a·
= 1
a
1
4· =1
4
The distributive property of
multiplication with respect to
addition (or subtraction).
• a(b + c) = ab + bc
• 3(4 + 7) = 3(4) + 3(7)
• 3(2x + 4) = 3(2x) + 3(4) = 6x + 12
Properties of Equality
• Reflexive
a=a
• Symmetric
If a = b, then b = a
• Transitive
If a = b and b = c, then a = c