# Complex Numbers

Cuthbert Nyack

## Notation

A complex number is written as *z* = *a* + *jb*
where *a* is the real part of *z* and *b* is the
imaginary part of *z*. *j* is the square root of -1 so
*j*^{2} = - 1. A complex number can also be seen as a vector
in a two dimensional space with axes *Re z* and *Im z*.
In this space the vector will extend from the origin to the point
(*a*, *b*). Every complex number *a* + *jb*
has a **complex conjugate** given by *a* - *jb*.
## Operations

For 2 complex numbers *y* = *a* + *jb* and
*z* = *c* + *jd* the following holds.

*y* = *z* only if *a* = *c* and *b* = *d*.
If a complex number is zero, then both its real and imaginary parts are
zero.

**Addition:-** *y* + *z* = (*a* + *c*) + *j*(*b*
+ *d*)

**Subtraction:-** *y* - *z* = (*a* - *c*) + *j*(*b*
- *d*)

**Multiplication:-** *yz* = (*a* + *jb*) (*c* + *jd*)
= *ac* + *jad* + *jbc* + *j*^{2}*bd* =
(*ac* - *bd*) + *j*(*ad* + *bc*)

**Reciprocal:-** 1/(*a* + *jb*) = (*a* - *jb*)/
[(*a* + *jb*)(*a* - *jb*)] =
(*a* - *jb*)/(*a*^{2} + *b*^{2})

**Division:-** *y/z* = (*a* + *jb*)/(*c* + *jd*) =
[(*a* + *jb*)(*c* - *jd*)]/[(*c* + *jd*)(*c* - *jd*)] =
[(*ac* + *bd*) + *j*(*bc* - *ad*)]/(*c*^{2} +
*d*^{2})

This can also be written :-
*y/z* = (*ac* + *bd*)/(*c*^{2} +
*d*^{2}) + *j*(*bc* - *ad*)/(*c*^{2} +
*d*^{2})

## Polar Form of Complex Numbers

A complex number *z* = *a* + *jb* can be written is polar form
as *z* = *r e*^{ jq} where
*r*^{2} = *a*^{2} + *b*^{2}. *r* is called
the **magnitude** or **modulus** or **absolute value** of *z*. and
q is the **phase** or argument of *z*.

## De Moivre's equation

( *a* + *jb* )^{n} =
(*r* e^{jq} )^{n} =
*r*^{n} (e^{jq} )^{n} =
*r*^{n} (e^{jnq} ) =
*r*^{n} ( cos nq
+ j sin nq)

where *r*^{2} = *a*^{2} + *b*^{2}

and q = tan^{-1}(b/a)

## Operations on complex numbers in polar form

Operations like multiplication, division,
powers and roots are very easy if complex numbers
are in polar form. Consider 3 complex numbers:-

*z* = *r e*^{ jq}

*z*_{1} = *r*_{1}
e^{ jq1}

*z*_{2} = *r*_{2}
e^{ jq2}

Multiplication:- *z*_{1}*z*_{2} =
*r*_{1}r_{2}
*e*^{ j(q1 +
q2})

Division:- *z*_{1}/*z*_{2} =
*r*_{1}/r_{2}
*e*^{ j(q1 -
q2})

Powers:- *z*^{n} = *r*^{n} e^{
jnq}

Roots:- *z*^{1/n} = *r*^{1/n} e^{
j(q +
2mpi)/n}

Logs:- *ln z* = *ln r* +
*j(q +
2mpi)*

## nth roots of unity

*1*^{1/n} = [e^{ j 2pm}]^{1/n} =
cos(2pm/n) + j sin(2pm/n)

The nth roots of unity are obtained for values of m from 0 to n - 1.

*(-1)*^{1/n} = [e^{ j (2m + 1)p}]^{1/n} =
cos(p(2m + 1)/n) + j sin(p(2m + 1)/n)

The nth roots of -1 are obtained for values of m from 0 to n - 1.

## Complex Numbers in Oscillations and Waves

For oscillations, the following relation is useful.
*e*^{ jw t} =
cos *w t* + j
sin *w t*

cos *w t* =
Re [*e*^{ jw t}]

sin *w t* =
Im [*e*^{ jw t}]

For waves, the following is useful.
*e*^{ j(kx - w t)} =
cos(*kx - w t*) + j
sin(*kx - w t*)

## Functions of a Complex Variable

A function *f*(*s*) = 1/(*s* + *a*), where *s* is a
complex number, is a function of a complex variable *s* or a complex function.
Of particular interest in complex functions is the location of the **poles**(points at
which the function goes to infinity) of the
function. In *f*(*s*) there is only one pole at *s* = - *a*.

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Copyright 1996 © Cuthbert A. Nyack.