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5 Complex Numbers
5.1 Complex numbers in rectangular coordinates.
Unlike the HP-33S (and its ancestor HP-32SII) complex numbers are straight supported and used in
HP-42S.
There is almost nothing special to say. Just enter -1 and press √x, what are you going to have is
x: 0.0000 i1.0000
which means i.
(Just to you have an idea to do the same in HP-33S we have to do
0 ENTER 1 +/- ENTER 0 ENTER .5 CMPLX
y
x
and we will have 0 and 1 meaning i)
Despite it is possible we don't need to calculate the square root of -1 every time, to have i.
We can use ▀ COMPLEX function which take line y and line x of the stack and creates a complex
number y+ix.
Again unlike HP-33S almost all functions of HP-42S fully support complex numbers.
Example: Show that
i
2
is -1.
Solution:
0 ENTER 1 ▀ COMPLEX ▀
x
2
which gives -1.0000 i0.0000 (means -1).
5.2 Complex numbers in polar coordinates
When representing a point in
R
2
we can use any kind of coordinate system. The most more used
are the rectangular (or Cartesian system) which use the usual coordinates x and y and the polar
system which use the coordinates r and θ.
The relationship between them is
x= r cos θ, y= r sin θ and
r
2
=x
2
y
2
, tan θ=y/x.
When dealing with complex numbers we can think is real axis as being the x and the imaginary axis
as being y and then we can use also polar coordinates.
In this case i will be r=1 and θ=π/2 (90°).
To change between rectangular or polar modes use RECT and POLAR in ▀ MODES menu.