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 62,006 would be expressed in hexadecimal as F236₁₆. In programming, a number of notations are used to denote hexadecimal numbers, usually involving a prefix. The prefix 0x is used in C, which would denote this value as 0xF236.

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.

Representation

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.

Verbal and digital representations

Magnuson (1968)^[1]
naming method
Number	Pronunciation
Α	alpha
Β	beta
Γ	gamma
Δ	delta
Ε	epsilon
Ζ	zeta
Η	eta
Θ	theta
Ι	iota
Κ	kappa
Λ	lambda
Μ	mu
Ν	nu
Ξ	xi
Ο	omicron
Π	pi
Ρ	rho
Σ	sigma
Τ	tau
Υ	upsilon
Φ	phi
Χ	chi
Ψ	psi
Ω	omega
Α0	alphaty
5Κ	fifty-kappa
Γ01Ξ	gammaty xiteen
1ΒΛ0	betateen lambdaty
3Α7Π	thirty-alpha seventy-pi

Real numbers

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 0.1. 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 0.19 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.

n	Decimal Prime factors of: base, b = 10: 2, 5; b − 1 = 9: 3		Hexadecimal Prime factors of: base, b = 16₁₀ = 10: 2; b − 1 = 15₁₀ = F: 3, 5
n	Reciprocal	Prime factors	Positional representation (hexadecimal)	Positional representation (decimal for comparison)	Prime factors	Reciprocal
2	1/2	2	0.Θ	0.5	2	1/2
3	1/3	3	0.ΒΒΒΒ... = 0.Β	0.3333... = 0.3	3	1/3
4	1/4	2	0.4	0.25	2	1/4
5	1/5	5	0.3	0.2	5	1/5
6	1/6	2, 3	0.2A	0.16	2, 3	1/6
7	1/7	7	0.249	0.142857	7	1/7
8	1/8	2	0.2	0.125	2	1/8
9	1/9	3	0.1C7	0.1	3	1/9
10	1/10	2, 5	0.19	0.1	2, 5	1/A
11	1/11	11	0.1745D	0.09	Β	1/B
12	1/12	2, 3	0.15	0.083	2, 3	1/C
13	1/13	13	0.13B	0.076923	Δ	1/D
14	1/14	2, 7	0.1249	0.0714285	2, 7	1/E
15	1/15	3, 5	0.1	0.06	3, 5	1/F
16	1/16	2	0.1	0.0625	2	1/10
17	1/17	17	0.0F	0.0588235294117647	Θ	1/11
18	1/18	2, 3	0.0E38	0.05	2, 3	1/12
19	1/19	19	0.0D79435E5	0.052631578947368421	Κ	1/13
20	1/20	2, 5	0.0C	0.05	2, 5	1/14
21	1/21	3, 7	0.0C3	0.047619	3, 7	1/15
22	1/22	2, 11	0.0BA2E8	0.045	2, B	1/16
23	1/23	23	0.0B21642C859	0.0434782608695652173913	Ξ	1/17
24	1/24	2, 3	0.0A	0.0416	2, 3	1/18
25	1/25	5	0.0A3D7	0.04	5	1/19
26	1/26	2, 13	0.09D8	0.0384615	2, D	1/1A
27	1/27	3	0.097B425ED	0.037	3	1/1B
28	1/28	2, 7	0.0924	0.03571428	2, 7	1/1C
29	1/29	29	0.08D3DCB	0.0344827586206896551724137931	Υ	1/1D
30	1/30	2, 3, 5	0.08	0.03	2, 3, 5	1/1E
31	1/31	31	0.08421	0.032258064516129	Χ	1/1F
32	1/32	2	0.08	0.03125	2	1/20
33	1/33	3, 11	0.07C1F	0.03	3, B	1/21
34	1/34	2, 17	0.078	0.02941176470588235	2, 11	1/22
35	1/35	5, 7	0.075	0.0285714	5, 7	1/23
36	1/36	2, 3	0.071C	0.027	2, 3	1/24

Irrational numbers

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

Number	Positional representation
Number	Decimal	Hexadecimal
√2 (the length of the diagonal of a unit square)	1.414213562373095048...	1.6A09E667F3BCD...
√3 (the length of the diagonal of a unit cube)	1.732050807568877293...	1.BB67AE8584CAA...
√5 (the length of the diagonal of a 1×2 rectangle)	2.236067977499789696...	2.3C6EF372FE95...
$φ$ (phi, the golden ratio = $(1+ \sqrt 5)/2$ )	1.618033988749894848...	1.Μ0Ζ72Β2048ΡΩΛ0ΜΜ5ΒΚΦΖ30Χ...
$π$ (pi, the ratio of circumference to diameter of a circle)	3.141592653589793238462643 383279502884197169399375105...	3.243F6A8885A308D313198A2E0 3707344A4093822299F31D008...
$e$ (the base of the natural logarithm)	2.718281828459045235...	2.ΟΕΒΒΥΕΙ7ΟΔ3Β1ΛΘΩΡΝΤΓΥ6Α9...
$τ$ (the Thue–Morse constant)	0.412454033640107597...	0.Ε0Σ35ΦΚ69ΑΜ0368ΞΝΓ38ΑΦΑ8...
$γ$ (the limiting difference between the harmonic series and the natural logarithm)	0.577215664901532860...	0.ΚΜ8Φ2Η47Ν2Ξ4ΔΑ59Ο6ΨΞΡΠΤΩ...

Powers

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

2^x	Value	Value (Decimal)
2⁰	1	1
2¹	2	2
2²	4	4
2³	8	8
2⁴	Η₃₄	16_dec
2⁵	Ψ₃₄	32_dec
2⁶	1Φ₃₄	64_dec
2⁷	3Ρ₃₄	128_dec
2⁸	7Ι₃₄	256_dec
2⁹	Ζ2₃₄	512_dec
2^Α (2^10_dec)	Φ4₃₄	1024_dec
2^Β (2^11_dec)	1Ρ8₃₄	2048_dec
2^Γ (2^12_dec)	3ΙΗ₃₄	4096_dec
2^Δ (2^13_dec)	72Ψ₃₄	8192_dec
2^Ε (2^14_dec)	Ε5Φ₃₄	16,384_dec
2^Ζ (2^15_dec)	ΤΒΡ₃₄	32,768_dec
2^Η (2^16_dec)	1,ΝΞΙ₃₄	65,536_dec
2^Θ (2^17_dec)	3,ΒΔ2₃₄	131,072_dec
2^Ι (2^18_dec)	6,ΝΡ4₃₄	262,144_dec
2^Κ (2^19_dec)	Δ,ΒΙ8₃₄	524,288_dec
2^Λ (2^20_dec)	Ρ,Ξ2Η₃₄	1,048,576_dec
2^Μ (2^21_dec)	1Κ,Γ4Ψ₃₄	2,097,152_dec
2^Ν (2^22_dec)	34,Ο9Φ₃₄	4,194,304_dec
2^Ξ (2^23_dec)	69,ΕΚΡ₃₄	8,388,608_dec
2^Ο (2^24_dec)	ΓΙ,Υ5Ι₃₄	16,777,216_dec
2^Π (2^25_dec)	Π3,ΟΒ2₃₄	33,554,432_dec
2^Ρ (2^26_dec)	1Η7,ΕΝ4₃₄	67,108,864_dec
2^Σ (2^27_dec)	2ΨΕ,ΥΑ8₃₄	134,217,728_dec
2^Τ (2^28_dec)	5ΦΥ,ΟΛΗ₃₄	268,435,456_dec
2^Υ (2^29_dec)	ΒΣΠ,Ζ6Ψ₃₄	536,870,912_dec
2^Φ (2^30_dec)	ΞΜΗ,ΦΔΦ₃₄	1,073,741,824_dec
2^Χ (2^31_dec)	1,Δ8Ω,ΡΣΡ₃₄	2,147,483,648_dec
2^Ψ (2^32_dec)	2,ΡΘΩ,ΚΜΙ₃₄	4,294,967,296_dec
2^Ω (2^33_dec)	5,Κ1Ω,592₃₄	8,589,934,592_dec
2¹⁰ (2^34_dec)	Β,43Ψ,ΑΙ4₃₄	17,179,869,184_dec

^ Cite error: The named reference Magnuson-1968-01 was invoked but never defined (see the help page).

[Magnuson-1968-01-1] Cite error: The named reference Magnuson-1968-01 was invoked but never defined (see the help page).

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