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The Number System The Complex Number System and Operations with Numbers

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Repeating Decimals Repeating decimals are decimals that contain a infinite number of digits. Examples: 0.333… … FYI…The line above the decimals indicate that number repeats.

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Terminating Decimals Terminating decimals are decimals that contain a finite number of digits. Examples:

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The Complex Number System All numbers in the world Represented by Complex Number System Real Numbers Imaginary Numbers

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The Complex Number System Complex Number System Real Numbers RationalIntegers Whole Numbers Natural Numbers Irrational Imaginary Numbers

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Imaginary numbers are all the numbers that deal with the square root of a negative number and contain the letter i in it. Example: You will learn more about these numbers in Algebra 2

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Real Numbers Real numbers consist of all numbers that can be represented on a number line. Represented by

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Complex Number System Real Numbers RationalIrrational Imaginary Numbers

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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: – … – … – (pi) –

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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. Represented by Examples:

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Complex Number System Real Numbers RationalIntegers Whole Numbers Natural Numbers Irrational Imaginary Numbers

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Integers Integers are the set of whole numbers and their opposites. {…,-3, -2, -1, 0, 1, 2, 3,…} Represented by

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Whole Numbers Whole numbers are the set of numbers that include 0 plus the positive numbers. {0, 1, 2, 3, 4, 5,…} Represented by

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Natural Numbers Natural numbers are the set of counting numbers. {1, 2, 3,…} Represented by or

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Venn Diagram of the Complex Numbers Irrational Numbers Rational Numbers Complex Numbers Imaginary NumbersReal Numbers Integers Whole Numbers Natural Numbers

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Example Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that apply. a.117 b.0 c … d.-½ e.6.36 f. -3

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FYI…For 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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Properties of Real Numbers Property Addition Multiplication Commutative a+b = b+a ab = ba Associative(a+b)+c = a+(b+c) (ab)c = a(bc) Identitya + 0 = aa1 = a Inversea + (-a) = 0 a = 1 Opposite Reciprocal Distributive Property a(b + c) = ab + ac

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Examples of Properties Name the property displayed: (x – 5) = (-2 + x) – 5 2. (-2) ( -½ ) = (4 – 5) = (4 – 5)2 4. x(y – w) = xy – xw

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Order of Operations 1.Parenthesis/Grouping Symbols 2.Exponents 3.Multiplication and Division – left to right 4.Addition and/or Subtraction – left to right

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Grouping Symbols Grouping symbols include parenthesis, braces, brackets, numerators and denominators of fractions and underneath a radical or inside absolute value symbols.

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Examples – Using Order of Operations Evaluate the following: (12 + 8) ÷ (2 + 11) ÷ 5

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