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10 November, 03:53

When a complex number is in the denominator, why is it necessary to multiply by the conjugate?

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  1. 10 November, 04:14
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    Here is the explanation.

    For the general complex number (a + bi), its conjugate is (a - bi).

    By definition, i² = - 1.

    Evaluate (a + bi) * (a - bi) to obtain

    (a + bi) * (a - bi) = a² - abi + abi - b²i²

    = a² - b² * (-1)

    = a² + b²

    This means that multiplying a complex number by its conjugate yields a real number.

    For this reason, it is customary to make the denominator of a complex rational expression into a real number, by multiplying the denominator by its conjugate.

    Of course, the numerator should also be multiplied by the same conjugate.

    Example:

    Simplify (2 - 3i) / (1 + 4i) into the form a + bi.

    The denominator is (1 + 4i) and its conjugate is (1 - 4i).

    Multiply the denominator by its conjugate to obtain

    (1 + 4i) * (1 - 4i) = 1² + 4² = 17.

    Also, multiply the numerator by the same conjugate to obtain

    (2 - 3i) * (1 - 4i) = 2 - 8i - 3i + (3i) * (4i)

    = 2 - 11i + 12*i²

    = 2 - 11i - 12

    = - 10 - 11i

    Therefore

    (2 - 3i) / (1 + 4i) = - (10 + 11i) / 17
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