Complex Numbers
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Written by Bryce Ringwood   

Well, its back to the slog. Complex numbers are really the most important part of this maths review, since they are used a great deal in developing and calculating electrical and electronic quantities - and even for visualising what's going on. We will be looking at impedance in one of the articles. This is in itself a complex quantity.

I'm going to begin with complex algebra. Then I will apply the algebraic rules to complex numerical quantities. We will also look at representing complex numbers geometrically. Along the way we will look at applying computers and calculators to ease our pain.

Some calculators have complex arithmetic built-in. Maybe check for this before you buy. I discovered my SHARP has this facility as a result of doing these articles. (After 10+ years of ownership I'm finally reading the manual. Never trust a manual that's bigger than the thing its meant for)

A complex number looks like this:

{\displaystyle {\bf N}= a + ib} where {a } and{   b} are real numbers and i =\sqrt{-1} I have represented the complex number in bold capitals. I don't like to put a line over the top of a character because I get confused with vectors. Vector arithmetic is something else, which I don't intend covering here - except for that brief essay in solving equations. 

Complex numbers can be added, subtracted, multiplied , divided etc. just like ordinary numbers.:

\displaystyle{(a +  ib ) + ( c + id)  = (a + c) + i(b + d)}
\displaystyle{(a +  ib ) - ( c + id)  = (a - c) + i(b - d) }
\displaystyle{(a +  ib ) \times ( c + id)  = (a c - bd) + i(b c  + ad)  }
\displaystyle{{{a + ib}\over{c + id}} = {{(a + ib)(c - id)}\over{(c^2 + d^2)}} }
They can also have roots, since if: \displaystyle{ {\bf N}^2 = {\bf Z}} then  \displaystyle{{\bf N }=\sqrt{\bf Z}} (I have to tell you, its a s*d to calculate).

When we divided two complex numbers, we multiplied the denominator by its complex conjugate.  If we have a complex number \displaystyle{{\bf N}} then its complex conjugate is often written \displaystyle{{\bf N*}}.

If

\displaystyle{{\bf N}={a + ib}}   then \displaystyle{{\bf N*}={a - ib}}
and so:
\displaystyle{{\bf NN*}={a^2 +b^2}}

Complex numbers are sometimes split into the real and imaginary parts, using the following arcane notation:

\displaystyle{{\Re (\bf N)} = a}
and
\displaystyle{{\Im (\bf N)} = b}
in our example.

In real life, you might be faced with sorting out the real and imaginary parts from a very complicated piece of algebra, like the square root problem (or  worse!).

Finally, some texts would write our example for \displaystyle{\bf N} as just a pair of "coordinates", thus: \displaystyle{(a,b)}. Again, I'm not that happy with doing that because it may be confused with other forms of notation.


Representing Complex Numbers Geometrically

Just as we can imagine real numbers extending from +infinity to -infinity along a horizontal line, so imaginary numbers can be represented as extending along an imaginary axis at right angles to it. This idea lends itself to a "complex plane". 

Lets look at the way some real, imaginary and complex numbers might be represented:

 

 

In the above image:

\displaystyle{ {\bf Z_1}=3 + i4}
\displaystyle{ {\bf Z_2}=\pi +i0}
\displaystyle{ {\bf Z_3}=0 +i5}
\displaystyle{ {\bf Z_4}=-2 -  i}
\displaystyle{ {\bf Z_5}=-2.5 + i4.2}

By looking at this picture, you can understand that the magnitude of a complex number is given by good old Pythagoras' theorem:

\displaystyle{ {\bf |Z_1|} = ({3^2} + {4^2})^{0.5} = 5 }

This lends itself to another way of representing complex numbers.


Using Polar Coordinates

Sometimes, complex numbers are represented as polar co-ordinates, as in the following diagram:

Then the number \displaystyle {\bf Z_1 }can be represented as

\displaystyle {r  cos\theta + i  r  sin\theta}
and
\displaystyle{{\bf|Z_1|}= r}

Since in our example \displaystyle{a = r cos\theta}
and  \displaystyle{b = r sin\theta}
  then

 \displaystyle{{tan^{-1}\theta} = {a\over b}}

When we move on to the section on differential calculus, we will see that we can simplify complex arithmetic by using Euler's number. Something to look forward to.laugh

Actually, it just simplifies things for us humans. A computer can perform complex arithmetic very well just using the relations I provided. I didn't provide the square root function, because its a mess to wade through. Using the polar form, its much easier to comprehend. BUT computers like to add multiply and divide, so if you are a computer, you had better skip this next bit.


De Moivre's Theorem (or is it a Formula ?)

In order to follow this section, you may need to revise the small section on trigonometry.

Abraham de Moivre was borm in 1667 and died in 1754. He was a contemporary of Sir Isaac Newton, Edmund Halle and many of the "scientific intelligentsia" of the 17th and 18th centuries. I suppose we would call tem "geeks" nowadays. When de Moivre was getting on a bit, he noted his "slowing down with aging process" and correctly deduced the day he would die. I personally don't intend to repeat this scientific endeavour, but I will pass on to you Mr de Moivre's formula/theorem relating the powers of complex numbers.:

\displaystyle{(r  cos\theta + i  r  sin\theta )^{n} = r  cos (n\theta ) + i  r  sin(n\theta )}

Technically "n" must be an integer. Although the formula "works" for non-integral values, it will clearly not give you all the roots. The theorem can be proved without using calculus by assuming the result is true for n = some arbitrary value k.

\displaystyle{(r  cos\theta + i  r  sin\theta )^{k} = r  cos (k\theta ) + i  r  sin(k\theta )}

Now we increment k by 1

\displaystyle{(r  cos\theta + i  r sin\theta )^{k+1} = (r  cos (k\theta ) + i  r  sin(k\theta )(r  cos \theta + i  r  sin\theta )}

hence:

\displaystyle{(r   cos\theta + i   r   sin\theta  )^{k+1} = (r  cos  (k\theta )cos\theta  - sin(k\theta )sin\theta  + i   ( sin\theta cos(k\theta )  + cos\theta sin(k\theta ) )  =}
\displaystyle{ (r  cos ((k+1)\theta ) + i  r  sin((k+1)\theta )}

If the result is true for k, then it is true for k+1.

Now the result is true for k= 1 (try it) - therefore it must be true for all real integers. (1,2,3 ...)


Automating the result

Addition:

- Have a go at this yourself

Subtraction:

This too

Multiplication:

 and this!

Division:

10 PRINT "enter x real:",
20 INPUT X
30 PRINT "enter x imag:",
40 INPUT IX
50 PRINT "enter y real:",
60 INPUT Y
70 PRINT "enter y imag",
80 INPUT IY
90 REM division
100 MODULUS=Y*Y + IY*IY
110 A=(X*Y + IX*IY)/MODULUS
120 B=(IX*Y -IY*X)/MODULUS
130 PRINT "re =",A,"im = ",B
140 END

Square Root (Principal Value only!)

10 PRINT "enter x real:",
20 INPUT X
30 PRINT "enter x imag:",
40 INPUT IX
50 PRINT "enter y real:",
60 INPUT Y
70 PRINT "enter y imag",
80 INPUT IY
90 REM division
100 MODULUS=Y*Y + IY*IY
110 A=(X*Y + IX*IY)/MODULUS
120 B=(IX*Y -IY*X)/MODULUS
130 PRINT "re =",A,"im = ",B
140 END

Rectangular to Polar

See if you can do this yourself

Polar to Rectangular

and this too.

Check your results with the complex number calculator. Even if you have a calculator that does complex numbers (and, I guess you have, now), give these problems a try.


Summary

This section is a brief summary of complex number operations. Advanced theory uses some calculus and has been omitted here. Complex numbers can be operated on by the normal addition, multiplication etc. operations, including exponentiation.

  • They can be represented in rectangular or polar coordinates
  • They are used to represent electrical quantities
  • In an identity, the real and imaginary parts can be equated.

Were you paying attention? - Calculate the square root of i.

 

 

 

 

 

 

 

 
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