User:40bus/Base34 Greek

In mathematics and computing, the Base34 Greek numeral system is a positional numeral system that represents numbers using a radix (base) of 34. Unlike the decimal system representing numbers using 10 symbols, Base34 Greek uses 34 distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9, and Greek letters "Α"–"Ω" (or alternatively "α"–"ω") to represent values from 10 to 33.

In mathematics, a subscript is typically used to specify the base. For example, the decimal value $$ would be expressed in hexadecimal as undefined. In programming, a number of notations are used to denote hexadecimal numbers, usually involving a prefix. The prefix  is used in C, which would denote this value as.

Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.

Written representation
In most current use cases, the letters A–F or a–f represent the values 10–15, while the numerals 0–9 are used to represent their decimal values.

There is no universal convention to use lowercase or uppercase, so each is prevalent or preferred in particular environments by community standards or convention; even mixed case is used. Seven-segment displays use mixed-case AbCdEF to make digits that can be distinguished from each other.

There is some standardization of using spaces (rather than commas or another punctuation mark) to separate hex values in a long list. For instance, in the following hex dump, each 8-bit byte is a 2-digit hex number, with spaces between them, while the 32-bit offset at the start is an 8-digit hex number.

Rational numbers
As with other numeral systems, the Base34 Greek can be used to represent rational numbers, although repeating expansions are common since sixteen (1016) has only a single prime factor; two.

For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as. Because the radix 16 is a perfect square (42), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.

All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.1$\overline{9}$ in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.062510 (one-sixteenth) is equivalent to 0.116, 0.0912, and 0;3,4560.

Irrational numbers
The table below gives the expansions of some common irrational numbers in decimal and hexadecimal.

Powers
Powers of two have very simple expansions in Base34 Greek. The first 35 powers of two are shown below.