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Objective: You will write, add, subtract, multiply, and divide complex numbers. The Standard Form a bi - where a and b are real numbers a = the real part of the complex number bi = the imaginary part of the complex number b = the coefficient of the imaginary number Complex Numbers Real Numbers Imaginary Numbers i 1 1 • Every real number is a complex number because a = a + 0i . • Every imaginary number is a complex number because bi = 0 + bi . Imaginary Numbers Why do we have imaginary numbers? Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions. For instance, an equation like x2 = -1 has no real solution because you cannot take the square root of -1. i 1 sadi = -1 Imaginary Numbers i=i i5 = i i2 = -1 i6 = -1 i3 = -i i7 = -i i4 = 1 i8 = 1 Write in standard form: 5 27 16 5 3 18 Write in standard form: 5 27 16 5 0 4i 5 0i 5 3 5 3i 3 18 3 3i 2 Adding & Subtracting Complex Numbers: 1. Combine like terms (treat the “i” like a variable). 2. Write the answer in the form a + bi or a – bi. Examples: 1. (3 - i) + (2 + 3i) 2. 3 - (-2 + 3i) + (-5 + i) 3. 2i – (4 – 3i) 4. -2 + -8 + 5 - -50 Examples: 1. (3 - i) + (2 + 3i) 5 2i 2. 3 - (-2 + 3i) + (-5 + i) 0 2i 3. 2i – (4 – 3i) 4 5i 4. -2 + -8 + 5 - -50 3 3i 2 Multiplying & Dividing Complex Numbers: 1. Multiplying Complex Numbers A. Distribute, Box or FOIL. B. Combine like terms. C. Write as a + bi or a – bi D. Replace all i2s with -1. 2. Divide Complex Numbers: A. Multiply fraction (both top and bottom) by the denominator’s conjugate. B. Combine like terms. C. Write as a + bi or a – bi. Examples: 4 2 3i 2 i 4 3i Examples: 4 2 3i 8 12i 2 i 4 3i 11 2i Examples (continued): 5 2i 1 i 2 3i 4 2i Examples (continued): 3 7 5 2i i 2 2 1 i 2 3i 1 4 i 4 2i 10 5