Real number system

1
The Real Number System
2
Natural Numbers

• Natural numbers are the set of counting numbers
  starting with 1 going up forever.
• 1, 2, 3,

They are called counting numbers because when
you think back, these are the numbers you learned
when learning to count and very often the numbers
were linked to numbers of things...1 apple, 2
bananas..
3
Whole Numbers

• Whole numbers are the set of numbers that include
  0 plus the set of natural numbers.
• 0, 1, 2, 3, 4, 5,

After you could count, someone explained the
concept of zero Because you dont always have
stuff to count.
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Integers

• Integers are the set of whole numbers and their
  opposites.
• ,-3, -2, -1, 0, 1, 2, 3,

We also needed numbers that were actually less
than zero. .
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Rational Numbers

• Rational numbers are any numbers that can be
  expressed in the form of , where a and b are
  integers, and b ? 0.
• They can always be expressed by using terminating
  decimals or repeating decimals.

We also needed numbers to deal with parts of a
whole.
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Terminating Decimals

• Terminating decimals are decimals that contain a
  finite number of digits.
• Examples
• 368/10 36.8
• 1/8 0.125
• 9/2 4.5
• 7/1 7

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Repeating Decimals

• Repeating decimals have a last digit or a pattern
  of repeating digits that go on forever.
• Examples
• 0.333 is the same as 1/3
• is the same as 5/3
• 7.689689 is the same as 7682/99

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Irrational Numbers

• Irrational numbers are any numbers that cannot be
  
• expressed as
• They are expressed as non-terminating,
• non-repeating decimals decimals that
• go on forever without repeating a pattern.
• Examples of irrational numbers
• (cant be expressed as a fraction)
• 0.34334333433334
• 45.86745893
• 1.414213562.
• 3.141592654

When taking the square root of any number that is
not a perfect square, the resulting decimal will
be non-terminating and non-repeating. Therefore,
those numbers are always irrational.
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Real Numbers

• Real numbers consist of all the rational and
  irrational numbers.
• The real number system has many subsets
• Natural Numbers
• Whole Numbers
• Integers

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Venn Diagram of the Real Number System
Irrational Numbers
Rational Numbers
Integers
Whole Numbers
Natural Numbers
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• Classify all the following numbers as natural,
  whole, integer, rational, or irrational. List
  all that apply going from least restrictive to
  most restrictive.
• 117
• 0
• -12.64039
• -4/2
• 6.36
• -3
• 4/9

Rational Numbers,
Integers,
Whole Numbers,
Natural Numbers
Rational Numbers,
Integers,
Whole Numbers,
Irrational Numbers,
Rational Numbers,
Integers,
Rational Numbers,
Irrational Numbers,
Rational Numbers,
Integers,
Rational Numbers,
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Place the number where it belongs on the Venn
diagram. Note Under rational numbers, place
the number in the inner most box
that applies
Rational Numbers
Irrational Numbers
Integers
Whole Numbers
-12.64039
Natural Numbers
117
0
-3
-4/2
6.36
4/9

1. 117 b. 0 c. -12.64039 d. -4/2 e. 6.36
   f. g. -3 h. 4/9

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Closure
A set is closed (under an operation) if the
operation on two elements of the set produces
another element of the set.  If an element
outside the set is produced, then the operation
is not closed. 
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Closure

• Are Natural Numbers closed under the operation of
  Addition?
• N N N N N N N N N
• 3 2 5 1 4 5 73 37 110

When two natural numbers are added together, the
sum will always be natural
So, we say natural numbers are closed under
addition
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Closure

• Are Natural Numbers closed under the operation of
  Multiplication?
• N N N N N N N N N
• 3 2 6 1 4 4 73 37 2701

When two natural numbers are multiplied together,
the sum will always be natural
So, we say natural numbers are closed under
multiplication
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Closure

• Are Natural Numbers closed under the operation of
  Subtraction?
• N N N N N I N N
  N
• 3 2 1 1 4 3 73 37
  36

When two natural numbers are subtracted , the
difference will not always be natural
So, we say natural numbers are not closed under
subtraction
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Closure

• Are Natural Numbers closed under the operation of
  division?
• N / N Rat N / N Rat N / N
  Rat
• 3 / 2 1.5 1 / 4 .25 73 / 37 1.972

When two natural numbers are divided , the
quotient will not always be natural
So, we say natural numbers are not closed under
division
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Adding Rational and Irrational

• What kind of number do you get when you add a
  rational and a irrational?
• Rat Ir Ir
• Will this always be true? Why?

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Multiply Rational and Irrational

• What kind of number do you get when you multiply
  a rational and a irrational?
• Rat Ir Ir
• Will this always be true? Why?