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Dividing complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy
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Dividing complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy

Dividing Complex Numbers: A Step-by-Step Guide

Introduction to Division of Complex Numbers

  • The task is to divide the complex number 6 + 3i by 7 - 5i, aiming for a result in the form of a complex number (real part + imaginary part).
  • Division can be expressed as a rational expression, where the denominator contains the complex number.

Using the Complex Conjugate

  • To eliminate the imaginary unit from the denominator, we multiply both numerator and denominator by the complex conjugate of 7 - 5i, which is 7 + 5i.
  • Multiplying by this conjugate does not change the value of the expression since it equals one.

Expanding Numerator and Denominator

  • The numerator expands through distribution:
  • 6 times 7 = 42
  • 6 times 5i = 30i
  • 3i times 7 = 21i
  • 3i times 5i = -15 (since i^2 = -1).
  • The denominator simplifies using FOIL:
  • First: 7 times 7 = 49
  • Outer: 7 times (-5i) = -35i
  • Inner: -5i times 7 = +35i (these cancel out).
  • Last: -5i times +5i = +25.

Simplifying Results

  • After simplification:
  • The numerator combines to give real part: 42 -15 =27, and imaginary part: 30 +21 =51 i.
  • The denominator results in: 49 +25 =74.

Final Formulation

  • The final result is expressed in standard form as:

[

27/74 + 51/74 i.

]

  • This representation confirms that both parts are divided by the same denominator, yielding a clear format for complex numbers.
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