#
Complex Numbers

You can carry out complex arithmetic by setting the number of I-direction
cells to exactly two, and checking the Complex box in the matrix page of
the Options dialog.

If the number of I-cells is set to some other value you are not allowed
to leave the dialog with Complex checked. If you set the number of
J-cells greater than one you can work with a one-dimensional array of complex
numbers.

With complex numbers enabled, the column headings in the display are
"x" and "iy" to indicate the real and imaginary parts of each number.
On a desktop PC, you may want to re-configure the keypads to a landscape
layout for complex number arithmetic, because of the extra width needed
for the display. You can then adjust the size of the display so that
scroll bars are not needed.

You can perform simple arithmetic on complex numbers, or perform exponentiation
and logarithm functions and transcendental functions. To enter numbers
select the real or imaginary parts as you would array elements as described
in the section on arrays and matrices.
If you select the real or imaginary part of a complex number, the arithmetic
logic acts on the whole of the complex number, not just the part selected.

You can carry out "normal" calculations on real numbers as normal, simply
by setting the imaginary part to zero. Of course in certain cases
this may give rise to a complex number, for example, famously:

Enter 1, then change its sign ("**+/-**"). Result:

Now take the square root. Result:

Some examples are given below:

###
Basic complex arithmetic.

The arithmetic functions act in the same way as for real numbers.
Complex addition is the same as array addition, but multiplication and
division are slightly more complicated. For example:

(x_{1 }+ i.y_{1})(x_{2} + i.y_{2})
= x_{1}x_{2} + i.x_{1}y_{2} + i.x_{2}y_{1}
+ i ^{2}.y_{1}y_{2}

= (x_{1}x_{2} -y_{1}y_{2}) + i.(x_{1}y_{2}
+ x_{2}y_{1})

###
Complex plane.

To convert from real and imaginary components to modulus and argument in
the complex plane (Argand diagram), use the normal

**r-p** and

**p-r**
buttons. These are obtained by using the shift key when they replace
the "+" and "-" arithmetic buttons. Once converted, the calculator
does not "know" the values are in polar co-ordinates, and you must convert
back to cartesian form to continue calculations. The argument
will be expressed in degrees, radians or gradients depending on the angle
mode selected.

Example:

Visualizing complex numbers in the complex plane is a powerful way of
thinking about the real and imaginary components of numbers.
The behaviour of arithmetic operations can be grasped more easily by considering
the geometric equivalents in the complex plane. For example, to take
the square root of a complex number, take the square root of the modulus
and divide the argument by two. You can verify this by using the
calculator to take the square root of various numbers and converting them
to polar co-ordinates.

###
Logarithmic functions.

Logarithms, powers and roots work in the same way as for real numbers.

###
Trigonometric functions.

Trigonometric functions work in the same way as for real numbers.