1.8%20The%20Real%20Number%20System

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1.8 The Real Number System
Standard 7.NS1d, 2c, 2d
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Goals Objectives

• To understand the real number system and
  recognize differences between rational and
  irrational numbers.
• To convert between different representations of
  rational and irrational numbers.

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Activites

• Power Point notes and examples

• Small Group Card Sort

• Exit Ticket

• Homework Is it Rational?

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Real Numbers

• Real numbers consist of all the rational and
  irrational numbers.
• The rational numbers have many subsets
• Natural Numbers
• Whole Numbers
• Integers

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Natural Numbers

• Natural numbers are the set of counting numbers.
• 1, 2, 3,

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Whole Numbers

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

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Integers

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

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Rational Numbers

• Rational numbers are any numbers that can be
  expressed as a ratio (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.

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Terminating Decimals

• Terminating decimals are decimals that contain a
  finite number of digits.
• Examples
• 36.8
• 0.125
• 4.5

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Repeating Decimals

• Repeating decimals are decimals that contain a
  infinite number of digits.
• Examples
• 0.333
• 7.689689
• FYIThe line above the decimals indicate that
  number repeats.

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Irrational Numbers

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

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FYIFor Your Information

• 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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What are perfect squares?

• 22 4 (v4 2)
• 32 9 (v9 3)
• 42 16 (v16 4)
• 52 25 (v25 5)
• 62 36 (v36 6)
• 72 49 (v49 7)
• 82 64 (v64 8)

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Examples of Nonperfect Squares

• v2
• v3
• v7
• v15
• v22
• v35

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Venn Diagram of the Real Number System
Irrational Numbers
Rational Numbers
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Example

• Classify all the following numbers as natural,
  whole, integer, rational, or irrational. List
  all that apply.
• 117
• 0
• -12.64039
• -½
• 6.36
• -3

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To show how these number are classified, use the
Venn diagram. Place the number where it belongs
on the Venn diagram.
Rational Numbers
Irrational Numbers
Integers
Whole Numbers
Natural Numbers
-12.64039
0
117
-3
6.36
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Solution

• Now that all the numbers are placed where they
  belong in the Venn diagram, you can classify each
  number
• 117 is a natural number, a whole number, an
  integer, and a rational number.
• is a rational number.
• 0 is a whole number, an integer, and a rational
  number.
• -12.64039 is an irrational number.
• -3 is an integer and a rational number.
• 6.36 is a rational number.
• is an irrational number.
• is a rational number.