2 + 10 i 3 + 10 i ⋅ 3 – 10 i 3 – 10 i Prepare to multiply the numerator and denominator by the complex conjugate of the denominator. 2 + 10 i 3 + 10 i Rewrite the denominator in standard form. 106 109 + 10 109 i Separate the real and imaginary parts. 6 – 20 i + 30 i – 100 i 2 9 – 30 i + 30 i – 100 i 2 Multiply using the distributive property or the FOIL method. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.Ģ + 10 i 10 i + 3 Substitute 10 i for x.
Suppose we want to divide c + d i c + d i by a + b i, a + b i, where neither a a nor b b equals zero.
Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another. Note that complex conjugates have a reciprocal relationship: The complex conjugate of a + b i a + b i is a − b i, a − b i, and the complex conjugate of a − b i a − b i is a + b i. In other words, the complex conjugate of a + b i a + b i is a − b i. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. Dividing Complex Numbersĭivision of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator.
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