Basic Definitions

It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called `` The Complex Numbers." In this amazing number field every algebraic equation in z with complex coefficients


has a solution. To prove this fact we need Liouville's Theorem, but to get started using complex numbers all we need are the following basic rules.

Rules of Complex Arithmetic

  1. displaymath151

  2. Every complex number has the ``Standard Form''


    for some real a and b.

  3. For real a and b,


  4. displaymath165

  5. displaymath167

Click on EXERCISES for some practice using these rules.


Notice that rules 4 and 5 state that we can't get out of the complex numbers by adding (or subtracting) or multiplying two complex numbers together. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a,b,c and d, can you find two other real numbers x and y so that


As an exercise in using rules 1 through 5, multiply both sides of the above equation by c + di and then solve for x and y to prove that the answer to our question is yes.( Click on help for a solution.)

OK, so we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

Notice that


We say that c+di and c-di are complex conjugates. To simplify a complex fraction, multiply the numerator and the denominator by the complex conjugate of the denominator.

Examples of division




Real and Imaginary Parts

If z= a+bi is a complex number and a and b are real, we say that a is the real part of z and that b is the imaginary part of z and we write



Find tex2html_wrap_inline213 and tex2html_wrap_inline215 . (Solution)

Complex Conjugates

If z=a +bi is a complex number with real part a and imaginary part b, then we denote the complex conjugate of z by tex2html_wrap_inline229 .


Write tex2html_wrap_inline231 in standard form.



Prove that tex2html_wrap_inline261 for any pair of complex numbers and similarly tex2html_wrap_inline263 .



Prove that tex2html_wrap_inline287 for any integer n.


Modulus of a Complex Number

The magnitude or modulus of a complex number z is denoted |z| and defined as


Notice that tex2html_wrap_inline241 . (proof.)


Prove that tex2html_wrap_inline243 . (Solution)

[Algebra] [Complex Variables]
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Author: Michael O'Neill

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