Thinking Mathematically

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Thinking Mathematically

• Real Numbers and Their Properties

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Real Numbers
Real numbers are the union of the rational
numbers and the irrational numbers. They are the
numbers that can represent distances and are
identified geometrically with the points on a
number line.
The real number a is identified with the point on
a number line whose distance from zero is aand
whose direction from zero is given by the sign of
a.
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Subsets of the Real Numbers

• Natural Numbers 1, 2, 3, 4, 5, These
  numbers are used for counting.
• Whole Numbers 0, 1, 2, 3, 4, 5, The whole
  numbers add 0 to the set of natural numbers.
• Integers , -3, -2, -1, 0, 1, 2, 3, The
  integers add the negatives of the natural numbers
  to the set of whole numbers.

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Subsets of the Real Numbers

• Rational Numbers These numbers can be expressed
  as an integer divided by a nonzero integer a/b
  a and b are integers b does not equal zero.
  Rational numbers can be expressed as terminating
  or repeating decimals.
• Irrational Numbers This is the set of numbers
  whose decimal representations are neither
  terminating nor repeating. Irrational numbers
  cannot be expressed as a quotient of integers.

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Closure of the Real Numbers
The sum or the difference of any two real numbers
is another real number. This is called the
closure property of addition.
The product or the quotient of any two real
numbers (the denominator cannot be zero) is
another real number. This is called the
closure property of multiplication.
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Commutative Property of Addition
Two real numbers can be added in any order. This
is called the commutative property of addition.
a b b a
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Commutative Property of Addition
Two real numbers can be multiplied in any order.
This is called the commutative property of
multiplication.
a x b b x a
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Associative Property of Addition
When three real numbers are added, it makes no
difference which two are added first. This is
called the associative property of addition.
(a b) c a (b c)
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Associative Property of Multiplication
When three real numbers are multiplied, it makes
no difference which two are multiplied first.
This is called the associative property of
multiplication.
(a x b) x c a x (b x c)
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Multiplication Distributes over Addition
The product of a number with a sum is the sum of
the individual products. This is called the
distributive property of multiplication over
addition.
a x (b c) (a x b) (a x c)
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Thinking Mathematically

• Real Numbers and Their Properties