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The Complex Number System
Matthew Smith, Dubbo High School
Now that we have learnt what an imaginary number is, we should get straight just exactly what a complex number is and what the rules for working with complex numbers are.
The name "complex number" is just the name given to a number that can exist with both a real part and an imaginary part.
For example the complex number 3 + 4i has the real component 3 and the imaginary component 4i. Complex numbers in general are often referred to as something like
z = x + iy
All this means is that the complex number z is made up of a real component (in this case the x) , and an imaginary component (i.e the y). The x's and y's aren't anything special to the complex number system, they just represent real numbers. A real number multiplied by i is imaginary and a complex number is any number with an imaginary component.
The real component of the complex number z is expressed as Re(z) and the imaginary component is expressed as Im(z). For example if z = 6 + 2i, then Re(z) = 6 and Im(z) = 2.
Equality
Two complex numbers are equal if and
only if their real parts are equal and their imaginary parts
are equal.
For example: if a + ib = 5 + 6i, then a = 5 and b = 6.
Addition of complex numbers
To add two or more complex numbers, the
procedure is to simply add the real components and add the
imaginary components. The result of the addition is a complex
number itself.
Examples:
(2 + 3i) + (6 + 8i) =
(2 + 6) + (3 + 8)i
=
8 + 11i
(3 - 4i) + (2 - 5i) =
(3 + 2) + (-4 - 5)i
=
5 - 9 i
Multiplication of complex numbers
Multiplication of complex numbers is
similar to expanding binomial quadratics. The real component of
the first complex number has to be multiplied by both the real
and the imaginary components of the second complex number. The
result of this will be a complex number which has to be added
to the result of the imaginary component of the first complex
number being multiplied by the real and imaginary components of
the second.
Examples:
Realisation of the denominator
The last part of this lesson will be looking at changing the form of complex numbers so that they actually can be read as having one real part and one imaginary part. Does this sound confusing ? Well let's look at an example of what I mean.
The complex number z (shown above) has real and imaginary
parts but does not look like the complex numbers we have seen
so far because it is expressed as a fraction. For a lot of
operations on complex numbers it is often necessary for us to
have the complex number with only 1 real part and 1 imaginary
part (or as textbooks put it x + iy).
The process which allows us to change this division of two
complex numbers into one complex number is called realising
the denominator.
Examples:
Express each complex number in the form x +
iy.
As you can see from example 1, if the denominator is purely imaginary I can just multiply by 1 in the form of i over i. If the denominator is a complex number with both real and imaginary components I can multiply by the complex conjugate which will make the denominator real.
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