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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91         Multiplying complex numbers

randRange( -5, 5 )        randRange( -5, 5 )        randRange( -5, 5 )        randRange( -5, 5 )                    "\\color{" + ORANGE + "}{" + A_REAL + "}"                            "\\color{" + ORANGE + "}{" + A_IMAG + "}"                            "\\color{" + BLUE + "}{" + B_REAL + "}"                            "\\color{" + BLUE + "}{" + B_IMAG + "}"                "\\color{" + ORANGE + "}{" + complexNumber( A_REAL, A_IMAG ) + "}"        "\\color{" + BLUE + "}{" + complexNumber( B_REAL, B_IMAG ) + "}"        ( A_REAL * B_REAL ) - ( A_IMAG * B_IMAG )        ( A_REAL * B_IMAG ) + ( A_IMAG * B_REAL )

Multiply the following complex numbers:

(A_REP) \cdot (B_REP)

Complex numbers are multiplied like any two binomials.

First use the distributive property:

\qquad (A_REP) \cdot (B_REP) =                        (A_REAL_COLORED \cdot B_REAL_COLORED) + (A_REAL_COLORED \cdot B_IMAG_COLOREDi) +                        (A_IMAG_COLOREDi \cdot B_REAL_COLORED) + (A_IMAG_COLOREDi \cdot B_IMAG_COLOREDi)

Then simplify the terms:

\qquad (A_REAL * B_REAL) + (A_REAL * B_IMAGi) +                        (A_IMAG * B_REALi) + (A_IMAG * B_IMAG \cdot i^2)

Imaginary unit multiples can be grouped together.

\qquad                        A_REAL * B_REAL + (A_REAL * B_IMAG + A_IMAG * B_REAL)i + negParens( ( A_IMAG * B_IMAG ) + "i^2" )

After we plug in i^2 = -1, the result becomes                                            A_REAL * B_REAL + (A_REAL * B_IMAG + A_IMAG * B_REAL)i - negParens( A_IMAG * B_IMAG )

The result is simplified:                                            (A_REAL * B_REAL - A_IMAG * B_IMAG) + (ANSWER_IMAGi) = complexNumber( ANSWER_REAL, ANSWER_IMAG)

The real part of the result is ANSWER_REAL and the imaginary part is ANSWER_IMAG.

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