The basics

Real and imaginary

The square of a real number, positive or negative, is a positive real number.

So 2² = 4 and also (-2)² = 4

The square root of a positive number is another number.  Actually it can be either of a pair of numbers, one positive and the other negative.

But for the square root of a negative number we have to look elsewhere.  We have to invent 'imagnary numbers' built around the idea that the square root of -1 is something that we call j.

The square root of -4 is now 2j, or it could also be -2j.

Complex

What happens if we want to take the square root of 2j?  The answer is a mixture of real and imaginary numbers.  If we take the square of (1 + j), multiplying it out we see that

    (1 + j)² = 1 + 2j + j² = 1 + 2j +(-1) = 2j

By multiplying expressions out in this way we can show that the result will always be in the form

    a + jb, where a and b are real, 'ordinary numbers'.


But how do we deal with complex numbers in the denominator?  How about 1/(a + jb) ?

Each complex number (a + jb) has a 'complex conjugate,' (a - jb).

If we multiply 1/(a + jb) by (a - jb)/(a - jb) we will not change its value. 

But now we have a numerator which is

  (a - jb)

and a denomiator that is

  (a + jb)(a - jb). 

When we multiply this second term out we have

    (a + jb)(a - jb) = a²  - (jb)² = a² + b²

So

   1/(a + jb) = (a - jb)/(a² + b²) = a/(a² + b²) - j b/(a² + b²)

We can 'rationalise' any complex expression to get a real part plus an imaginary part.

The complex plane

We can plot a point in a plane to represent (a + jb) at coordinates (a, b),
i.e. real numbers are on the x-axis and imaginary numbers are on the y-axis.

To add two complex numbers you just add the real parts to get the new real part and add the imaginary parts to get the imaginary part.

But to multiply them together you have to expand the product and pick up the pieces.

The complex exponential

An important result that is not hard to prove is that

    e^jt = cos t + j sin t

(If you differentiate each side twice, the exponential is multiplied by j², the cos turns into -cos and the sin turns into -sin, so everything hangs together.)

As a result, any complex number can be represented as

    a + jb =  c e^jd

where the real number c is the 'modulus', the square root of (a² + b²), while d is the inverse tangent of (b/a), called the 'argument'.

So a rule for multiplying complex numbers could be "Multiply the moduli and add the arguments".  
But in general, converting to angles is hard work and it is easier just to expand the expressions.


Now we can consider the exponential of a complex number.  To multiply two exponentials you just add their exponents:

    2² times 2⁵ = 2⁷

But we turn this around to see that

    e^{(a + jb)} = e^a  times  e^jb = e^a(cos b + j sin b)

and a simple bit of complex arithmetic will make solving differential equations a whole lot easier.

Self test

Calculate the following.  Click on your selected answer.

1.    (1 + j)(2+j) = ?



2.    5/(2+j) = ?



3.    (1 + j)⁴ = ?