Archive Magazine CD 1995/Archive Magazine CD 1995.iso/discs/pipeline/4_10/ComplexNos/ReadMe

Text File | 1991-06-09 | 5.6 KB | 109 lines

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%C%Complex Numbers
%C%by Gerald L Fitton
%C%9th June 1991

Complex Numbers
Complex numbers are part of most GCE A level maths courses as well as
being obligatory for BTEC Engineers and many other courses.  The
paragraphs below might prove of particular interest to those of you
teaching complex numbers and who need to generate interesting and
instructive numerical examples.  If you are an educationalist then you
will be interested to know that I have found that getting students to
use actual numbers gives them a much better 'feel' for what is going on
than when they manipulate algebraic formulae.  'Hands on' learning is
particularly effective for the practical engineer, the teaching (or are
they called learning?) objectives are grasped much more quickly.  I
have received comments such as "I've never understood complex numbers
before" from mature engineers (with a decade of field work behind them)
after only a couple of hours of entering actual numbers into a
spreadsheet such as an extended version of the spreadsheet Complex01
described below.

First an introduction to complex numbers.

I have yet to find a hand calculator which will let me find the square
root of -4 or the logarithm of -1 even though, in the domain of complex
numbers both of these exist.  Perhaps the most famous complex number is
the square root of -1.  Sqr(-1) has two answers.  Mathematicians use
the symbol i and Engineers use j (because they use i for electric
current - well, that's what I've been told) for the positive square
root of -1.  The other square root of -1 is -i.  I prefer to say that
i * i = -1 rather than talk about i being the square root of -1.

Complex numbers can be considered to have two parts, a Real part and an
Imaginary part.  These may be visualised as the x any y coordinates of
a point on a two dimensional sheet of graph paper.  A complex number
such as (3 + 4i) is said to have a Real (x ) part of 3 and an Imaginary
(y) part of 4 and may be plotted as x and y coordinates on the so
called Argand Diagram (named after its inventor). After addition and
subtraction, perhaps the simplest thing that can be done with a complex
number is multiplication.  For example the square of (3 + 4i) is
(3 + 4i)(3 + 4i) which becomes 9 + 24i + 16i2.  Now, remember that i2
is really -1 and you get 9 +  24i - 16 as the answer.   This can be
simplified to -7 + 24i, a Real part of -7 and an Imaginary part of +24.
I think that a better way of looking at complex numbers is as pairs of
Real numbers for which the symbol i is used as a separator and, for
which, i * i = -1.

As an example I shall show you how to raise a complex number to any
power, even a complex power (later, try to find exp(-i*PI) - it
evaluates to -1).  The spreadsheet application I have called Complex01
has, as input, two complex numbers called z and w and I find z^w.
Screen01, is a screenshot showing the sheet Complex01 being used with
z = i and w = 2 to find i^2 = -1.  The intermediate steps are to find
the logarithm of z , multiply the logarithm by w and then use the
exponential function to find the inverse logarithm.  For those of you
more familiar with Real numbers, try out the formulae given in text
form in cell A13 of Screen01, z^w=e(w*ln(z)), on your calculator (using
a positive Real for z and a Real for w) and convince yourself that it
works.  Multiplication (as we have seen) can be used on complex
numbers; the two other very basic functions are exponentiation (exp)
and its inverse, the logarithmic function (ln - not log).  The
formulae for evaluating these functions as functions of complex numbers
are in the cells of the sheet.

All these 'clever' formulae (eg for ln and exp) are in the cell block
B11C13.  They appear as text in Screen02.  If you want to follow
through this tutorial then either type them in as expressions or load
the file Complex01 from the Archive monthly disc.

When you have Complex01 spreadsheet you can show that Real powers of
negative Real numbers work out correctly.   Screen03 is a snapshot of
the spreadsheet correctly evaluating (-2)^3 = -8.  The intermediate
results show that ln(z) has an Imaginary part which, to 4 decimal
places, is 3.1416.  Do you recognise this number? Try using the
spreadsheet to prove that ln(-1) = i*p by entering -1 into B8 (the Real
part of z).

Screen04 is a shot that shows that (1 + i)^4 = -4.  You can work this
out by using the usual algebraic multiplication formulae (or the
binomial expansion) and replacing i2 with -1 whenever it occurs.

Division of complex numbers is executed by multiplying by a reciprocal.
Division (or finding reciprocals) is a common GCE A level problem which
is solved numerically by using the value w = -1.   Put w = 0.5 to find
the principal square root; the second root is a bit harder to find but
it can be deduced from the principal root.

If you have an interest in complex numbers then please write and let me
know what sort of numerical examples you would like to see in
spreadsheet format and I'll see what I can do for you.  On the July
1991 PipeLine disc I have included more common functions of complex
variables such as the trigonometrical and hyperbolic functions (and
their inverses) so that you can use them, get numerical results (only
the principal values) and see how the functions are implemented.

I have found that electrical engineers particularly get highly
addicted to this spreadsheet and find it a most worthwhile learning
experience. I would like to hear from anyone who has done (or wants) a
complex numerical integration application.