This article explains, with examples, how to simplify fractions that have a complex number in the denominator. The technique used is based on the "difference of two squares" - a well-known mathematical identity, used in all math high school and college degree courses.
Difference of Two Squares
This is one of the best-known mathematical identities, and is learned at an early stage in most math careers:
(a - b) × (a + b) = a² - b²
It can be shown as follows. Expanding the left-hand side and using basic principles:
a × (a + b) - b × (a + b)
= a² + a.b - b.a - b²
= a² - b²
Complex Conjugates
Using the identity described and proved in the section "Difference of Two Squares", it is clear that it is possible to get rid of an inconvenient √(-1) in the denominator of a fraction. This is achieved by setting a = (the real part of the denominator) and b = (the complex part of the denominator).
For example, if the denominator of a fraction is (1 + i), where i = √(-1), the complex conjugate of the denominator = (1 - i). Similarly, if the denominator is (3 - 2.i), then the complex conjugate equals (3 + 2.i). In general, for a complex number (x + y.i), the complex conjugate is equal to (x - y.i).
Complex Conjugate Uses - Examples
Example 1: 1 / (1 + i)
1 / (1 + i)
= (1 - i) / [(1 + i).(1 - i)]
= (1 - i) / (1² - i²) : But i² = -1,
= (1 - i) / (1 - -1)
= (1 - i) / 2
Example 2: 1 / (3 - 2.i)
1 / (3 - 2.i)
= (3 + 2.i) / [(3 - 2.i).(3 + 2.i)]
= (3 + 2.i) / (3² - (2.i)²) : But i² = -1,
= (3 + 2.i) / (9 - -4)
= (3 + 2.i) / 13
Complex Conjugate Uses - General Case
Example 1: 1 / (x + y.i)
1 / (x + y.i)
= (x - y.i) / [(x + y.i).(x - y.i)]
= (x - y.i) / (x² - (y.i)²) : But i² = -1,
= (x - y.i) / (x² - -y²)
= (x - y.i) / (x² + y²)
This may then be broken down into simple real and imaginary parts,
= x / (x² + y²) -y.i / (x² + y²)
The last line is much easier to deal with in calculations or algebraic expressions.
Complex Conjugates Summary
The definition of a complex conjugate has been given, as well as examples of how it may be used, with the "difference of two squares", to simplify complex expressions into real and imaginary parts. Examples have been shown, and students will understand how to use complex conjugates better by trying out their own examples.
Complex Conjugates - References
This subject is covered in most high-school level math texts, as well as many college-level math and science program text-books. The first description of a standard form for imaginary numbers was in 1637, when Rene Descartes used the form (a + b.i), although the use of the letter "i" did not happen until it was used by Leonard Euler in 1777.
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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