Complex Numbers

Elzed represents complex numbers as a coordinate pair -- real part first, imaginary part second. There's an elegance to this notation that I find appealing, as it builds on the concept of the single-ordinate number line.

The square root of -1, for example, can be expressed as ( 0 |_ 1 ). The real part of the complex number is 0, while the imaginary part is 1. Any coordinate pair whose second ordinate is 0 doesn't stray into the imaginary plane, and so is a regular old real number. Aesthetically pleasing, no?

Elzed does not support the (perhaps) more familiar ( x + yi ) notation. This notation exists to support manual calculation--one can apply the normal rules of algebra to complex numbers expressed in this form. For example:

   ( 3 + 2i )( 4 + 1.2i )

Here we're multiplying two complex numbers. With complex numbers expressed in ( x + yi ) form, we can use the good old F.O.I.L. method to calculate the answer, like so:

   ( 3 + 2i )( 4 + 1.2i )
= (12 + 8i + 3.6i - 2.4)
= (9.6 + 11.6i)

Straightforward, yes, but the point here is that Elzed will be doing the calculating. The coordinate-based format is simple and unambiguous, and the ( x + yi ) form converts directly to the ( x |_ y ) form. If, for example, Elzed is given the same problem in the form (3 |_ 2) * (4 |_ 1.2), it returns (9.6 |_ 11.6).

Note the use of the "|_" (vertical bar, underscore) which separates the coordinates. This is the rectangular coordinate operator, and denotes a rectangular coordinate pair. Elzed also supports polar coordinates through the use of the "|/" (vertical bar, forward slash) polar coordinate operator. Using this operator, the square root of -1 is expressed as ( 0 |/ 90 ); the imaginary part is expressed as 90 degrees. When using the polar coordinate operator, be sure to observe the current angle unit setting. You may mix rectangular and polar operands in an expression.

When expressing a complex number result, Elzed will observe the current coordinate format setting and use the appropriate coordinate operator. Elzed will express the angle in polar coordinates using the current angle unit setting.