1–1: Number Sets
Jan 18, 2016
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 non-repeating 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 .
Real Numbers
• Real numbers include all rational and irrational numbers.
Rational Numbers
Integers
Whole Numbers
Counting 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) B) 3 C) 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 PropertyCommutative 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 PropertyAssociative 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 PropertiesDo not change the value
1) Additive IdentityWhat do you add to get the same #?
a + 0 = a-6 + 0 = -6
2) Multiplicative IdentityWhat do you mult. to get the same #?
a • 1 = a8 • 1 = 8
Inverse PropertiesUndo an operation
1) Additive Inverse (Opposite)
• a + (-a) = 0
• 5 + (-5) = 0
2) Multiplicative Inverse
(Reciprocal)
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