Binary Systems
Binary coding systems of complex numbers, i. e. systems with the digits, are of practical interest. Listed below are some coding systems (all are special cases of the systems above) and codes for the numbers –1, 2, -2, . The standard binary (which requires a sign) and the negabinary systems are also listed for comparison. They do not have a genuine expansion for .
| radix | twins and triplets 1 | ||||
| 0.1 = 1.0 = 1 | |||||
| 0.01 = 1.10 = | |||||
| 2 | 0.0011 = 11.1100 = | ||||
| 0.010 = 11.001 = 1110.100 = | |||||
| 2 | 1.011 = 11.101 = 11100.110 = | ||||
| 0.0011 = 11.1100 = | |||||
| 1 the underline marks the period | |||||
| 2 infinite non-repeating sequence | |||||
As in all positional number systems with an Archimedean absolute value there are some numbers with multiple representations. Examples of such numbers are shown in the right column of the table.
If the set of digits is minimal, the set of such numbers has a measure of 0. This is the case with all the mentioned coding systems.
Read more about this topic: Complex Base Systems
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“The only people who treasure systems are those whom the whole truth evades, who want to catch it by the tail. A system is just like truth’s tail, but the truth is like a lizard. It will leave the tail in your hand and escape; it knows that it will soon grow another tail.”
—Ivan Sergeevich Turgenev (1818–1883)