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1–2: Properties of Real Numbers
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 expressed as a/b, where both a and b are integers and b 0.
• Includes common fractions, terminating decimals, repeating decimals, and integers.
• They do not include non-repeating decimals, such as .
Irrational Numbers
• Those numbers that cannot be expressed as a ratio of two integers
• Includes non-terminating, non-repeating decimals and special numbers, such as π and
Real Numbers•Real numbers include all rational and irrational numbers.
Rational Numbers
Integers
Whole Numbers
Counting Numbers
Irrational Numbers
Ponder the statements...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
Classifying Numbers
Properties of Real Numbers • Closure Property • Commutative Property • Associative Property • Identity Property • Inverse Property • Distributive Property • Properties of Equality
Closure Property• When you combine any two numbers in a
set, the answer is part of the set.• For example, 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 • Learn more
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!• Additive Identity – When you add zero to
any number, the result is the same numbera + 0 = a -6 + 0 = -6
• Multiplicative Identity – When you multiply a number by one, the result is the same number
a • 1 = a 8 • 1 = 8
Inverse Properties
• Undo an operation• Additive Inverse – when you add a number
and its opposite, the result is 0
a + (-a) = 0 5 + (-5) = 0• Multiplicative Inverse – when you multiply
a number and its reciprocal, the result is 1
Distributive property
Distributive property of multiplication with respect to either addition or subtraction.• a(b + c) = ab + bc
• 3(4 - 7) = 3(4) - 3(7)
• 3(2x + 4) = 3(2x) + 3(4) = 6x + 12
• Reflexive
a = a • Symmetric
If a = b, then b = a • Transitive
If a = b and b = c, then a = c
Properties of Equality
More info…• Real Numbers (mathisfun)• Properties of Real Numbers (regentsprep)• Math Properties (purplemath)• Properties of Equality (hotmath)• Glossary of Properties (dr.math/mathforum)