Imaginary Numbers

Since imaginary and complex numbers arise so frequently in engineering and science, it is important to know how to manipulate them.

A complex number, z, can be expressed as the sum of real and imaginary parts,

`z=a+ib`

where a and b and real-valued functions

The complex conjugate, z*, can be expressed,

`z^(**)=a-ib`

The modulus (or magnitude), |z|,

`|z|^2=z^(**)z=(a+ib)(a-ib)=a^2+b^2`

Note, that `z^(**)z!=z^2`, ie., whereas `|z|` is always real-valued, `z^n` is not, ie., `z^2=a^2+2ib-b^2`

Complex numbers can be visualised using an Argand diagram. We plot the real component along the x-axis and the imaginary component along the y-axis by convention,

As can be visualised on an Argand Diagram, there are two additional important forms that can be used to represent a complex number: as an exponential or as a sum of sinusoids.

For example, we can express the individual components, `Re(z)` and `Im(z)` trigonometrically. Mapping the complex number onto the Argand diagram, we can with a magnitude and an angle, express,

`a=|z|cos(theta)` and `b=|z|sin(theta)`

Hence,

`z=|z|cos(theta)+i|z|sin(theta)=|z|[cos(theta)+isin(theta)]`

Which leads onto an alternative means to express the modulus (magnitude),

`|z|=sqrt(a^2+b^2)`

Lastly, but extrordinarily usefully, through the relationship between sinusoids and the exponential function, a complex number can also be expressed in terms of its real-valued modulus and a phase angle,

`z=|z|exp(itheta)`