Antenna Analyzer AIM4170 page 38
Appendix 2 – Complex Numbers
A complex number has two parts: a real part that we are accustomed to using for most
everyday problems, and an imaginary part. The imaginary part was introduced to handle
the square root of negative numbers. In ordinary circumstances, any number squared is
positive, so it seemed unreasonable for a negative number to have a square root. This was
resolved by defining a special value called “the square root of minus one”. This is
usually symbolized by “i” in math books and by “j” in engineering books. Using “j”
avoids confusion in an engineering context with the symbol “i” that is usually used for
current.
Complex numbers came into use about 500 years ago for solving algebraic equations,
including the familiar second order equation: ax^2 + bx + c = 0.
(note: the symbol x^2 means “the value of x squared”= x times x.)
Let’s look at a specific example: x^2 – x – 2 = 0.
In this case the coefficients are: a = 1, b = -1, c = -2
The solutions using the quadratic equation are:
x = [ - b + SQRT( b*b – 4ac) ] / 2a
and
x = [ - b - SQRT( b*b – 4ac) ] / 2a
Inserting the coefficients of the equation, we get:
x = [1 + SQRT(1 + 8)]/2 = 2
and
x = [1 – SQRT(1+8)]/2 = -1
Now, if we go back and insert x = 2 into the equation, the equation is equal to zero and
we also get zero by plugging in x = -1.
There is no problem here since we didn’t have to worry about the square root of a
negative number.
A small change of one coefficient changes the mathematical problem considerably, as we
will see now:
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