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The Number System

The Number System. The Complex Number System and Operations with Numbers. Repeating Decimals. Repeating decimals are decimals that contain a infinite number of digits. Examples: 0.333… 7.689689… FYI…The line above the decimals indicate that number repeats. Terminating Decimals.

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The Number System

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  1. The Number System The Complex Number System and Operations with Numbers

  2. Repeating Decimals • Repeating decimals are decimals that contain a infinite number of digits. • Examples: • 0.333… • 7.689689… FYI…The line above the decimals indicate that number repeats.

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

  4. The Complex Number System • All numbers in the world • Represented by ℂ

  5. Imaginary Numbers 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

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

  7. Irrational Numbers • Irrational numbers are any numbers that cannot beexpressed as . • They are expressed as non-terminating, non-repeatingdecimals; decimals that go on forever without repeating a pattern. • Examples of irrational numbers: • 0.34334333433334… • 45.86745893… • (pi)

  8. 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 ℚ • Examples:

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

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

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

  12. Venn Diagram of the Complex Numbers ComplexNumbers Real Numbers Imaginary Numbers Rational Numbers Integers Irrational Numbers WholeNumbers NaturalNumbers

  13. 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

  14. 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.

  15. Properties of Real Numbers Property Addition Multiplication Commutative a+b= b+aab= ba Associative (a+b)+c = a+(b+c)(ab)c = a(bc) Identity a + 0 = a a•1 = a Inverse a + (-a) = 0 a = 1 Opposite Reciprocal Distributive Property a(b + c) = ab + ac

  16. Examples of Properties Name the property displayed: • -2 + (x – 5) = (-2 + x) – 5 2. (-2) ( -½ ) = 1 3. 2(4 – 5) = (4 – 5)2 4. x(y – w) = xy – xw

  17. Order of Operations • Parenthesis/Grouping Symbols • Exponents • Multiplication and Division – left to right • Addition and/or Subtraction – left to right

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

  19. Examples – Using Order of Operations Evaluate the following: • 22(12 + 8) 5 2. 52 ÷ (2 + 11) 3. 7 • 12 + 30 ÷ 5

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