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Complex Numbers

Complex Numbers. Imaginary Numbers. REAL NUMBERS {x | x is a rational or an irrational number}. Irrational Numbers , 8, -13. Rational Numbers 1/2 –7/11, 7/9, .33 9, {p/q | p & q are integers, q 0}. Integers {…-2, -1, 0, 1, 2, 3…}. Whole Numbers {0,1,2,3,4…}.

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Complex Numbers

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  1. Complex Numbers Imaginary Numbers REAL NUMBERS {x | x is a rational or an irrational number} Irrational Numbers , 8, -13 Rational Numbers 1/2 –7/11, 7/9, .33 9, {p/q | p & q are integers, q 0} Integers {…-2, -1, 0, 1, 2, 3…} Whole Numbers {0,1,2,3,4…} Natural Numbers {1,2,3,4…}

  2. Operations on Real Numbers Addition/Subtraction 3 + 5 = 8 -3 + 5 = 2 -3 – 5 = -8 [ THINK MONEY] 6 – 2 – 3 + 8 – 10 = -1 Multplication/Division 3 • 5 = 15 -3 • 5 = -15 -3 • (-5) = 15 2 Like Signs (+ answer) -3 • 5 • -3 = 45 -3 • -5 • -3 = - 45 2 Unlike Signs (- answer) 6/3 = 2 6/(-3) = -2 (-6)/(-3) = 2 Absolute Value | 4 | = 4 |-4| = 4 [ALWAYS POSITIVE] -|4 | = -4 - |-4| = -4 -(-|-4|) = 4 [AFFECTS INSIDE ONLY!!] ** Be sure you can perform all operations with fractions too!

  3. Properties of Real Numbers Commutatative Property Addition: a + b = b + a 13 + 7 = 7 + 13 3x + 5 = 5 + 3x Multiplication: ab = ba (7)(10) = (10)(7) (4a)(5x+8) = (5x+8)(4a) Associative Property Addition: (a + b) + c = a + (b + c) 3 + (8 + x) = (3 + 8) + x Multiplication: (a • b) • c = a • (b • c) -2 (3x) = (-2 • 3) • x = -6x Distributive Property a(b + c) = ab + ac 5 (x + 2) = 5x + (5)(2) = 5x + 10 a(b – c) = ab - ac 5 (x – 2) = 5x + (5)(-2) = 5x – 10 5x + 4x = (5 + 4)x = 9x Identity Property a + 0 = a 0 + a = a 5 + 0 = 5 7x + 0 = 7x a • 1 = a 1 • a = a 13x • 1 = 13x Inverse Property (Additive Inverse & Multiplicative Inverse) a + (-a) = 0 (-a) + a = 0 3x + (-3x) = 3x – 3x = 0 a • (1/a) = 1 (1/a) • a = 1 (1/7) • 7 = 1 Multiplication Property of Zero : a • 0 = 0 0 • a = 0

  4. Introductory Exponents 82 =8 • 8 = 64 24 = 2 • 2 • 2 • 2 = 16 -52 = - 25 (-5)2 = 25 Exponents of 1 Zero Exponents Anything to the 1 power is itself Anything to the zero power = 1 51 = 5 x1 = x (xy)1 = xy 50 = 1 x0 = 1 (xy)0 = 1 Careful with Negatives! Square Roots 25 = 5 since 52 = 25 -25 = -5 -25 is not a real number

  5. Order of Operations Groupings Exponents Multiplication & Division from Left to Right Addition & Subtraction from Left to Right Wait! What happend to Sally? I saw her in the book!?!

  6. Calculator Skills for Chapter 1 • Add, Subtract, Multiply, Divide • Exponents/Powers • Square Root • Absolute Value [Math][Num][Abs] • Negative Button – Try | -5|

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