Complex Numbers (LUX)

6.2.2.4 Complex Numbers

Complex numbers consist of two parts, called the real part and the imaginary part. A complex number that has no imaginary part is called a real number; A complex number that has no real part is called an imaginary number. Any imaginary number can be written as the product of a real number and the imaginary unit. The imaginary unit is generally denoted i in mathematical notation and j in engineering texts, and is available in LUX as read-only variable #i, and as suffix i or I on any non-imaginary number: 2*#i and 2i are two ways of expressing twice the imaginary unit. The imaginary unit is defined such that #i^2 = -1.

Complex numbers come in two data types: complex float (cfloat), and complex double (cdouble). cdouble is explicitly indicated by use of the D specifier (for double) and the I specifier (for imaginary).

Complex numbers have no natural ordering, so there is no obvious way to decide whether a complex number x is less than or greater than another complex number y. If x = a + b*#i and y = c + d*#i where a, b, c, d are real (non-complex) numbers, then we adopt the rule that x > y if a + b > c + d, and x < y if a + b < c + d. With that rule, the ordering of cfloat or cdouble numbers with their imaginary part equal to 0 (i.e., so-called purely imaginary numbers) is the same as the ordering of the corresponding float or double numbers (e.g., 1+0i GT -2+0i, just like 1 GT -2), and the ordering of complex numbers with their real part equal to 0 is similar to that of real numbers (e.g., 1i GT -2i).

However, any such rule can lead to somewhat unusual results. For real (non-complex) numbers, if both x LT y and x GT y are false, then x EQ y must be true, and vice versa. For complex numbers, that relationship does not necessarily hold anymore. For example, with the above rule, x LT y and x GT y and x EQ y are all false if x = 3+2i and y = 8-3i, because 3+2 is equal to 8-3. So, even though x is not less than y, and x is not greater than y, yet x is not equal to y.

The abovementioned ordering rule for complex numbers is used by the binary relational operators lt, le, gt, ge, and also by the binary bounding operators >, <.