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 25,220 would be expressed in hexadecimal as 628416. 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 0x6284
.
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
editWritten representation
editIn 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
editNumber | 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
editRational numbers
editAs 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 | ||||
---|---|---|---|---|---|---|
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
editThe table below gives the expansions of some common irrational numbers in decimal and hexadecimal.
Number | Positional representation | |
---|---|---|
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+√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
editPowers of two have very simple expansions in Base34 Greek. The first 35 powers of two are shown below.
2x | Value | Value (Decimal) |
---|---|---|
20 | 1 | 1 |
21 | 2 | 2 |
22 | 4 | 4 |
23 | 8 | 8 |
24 | Η34 | 16dec |
25 | Ψ34 | 32dec |
26 | 1Φ34 | 64dec |
27 | 3Ρ34 | 128dec |
28 | 7Ι34 | 256dec |
29 | Ζ234 | 512dec |
2Α (210dec) | Φ434 | 1024dec |
2Β (211dec) | 1Ρ834 | 2048dec |
2Γ (212dec) | 3ΙΗ34 | 4096dec |
2Δ (213dec) | 72Ψ34 | 8192dec |
2Ε (214dec) | Ε5Φ34 | 16,384dec |
2Ζ (215dec) | ΤΒΡ34 | 32,768dec |
2Η (216dec) | 1,ΝΞΙ34 | 65,536dec |
2Θ (217dec) | 3,ΒΔ234 | 131,072dec |
2Ι (218dec) | 6,ΝΡ434 | 262,144dec |
2Κ (219dec) | Δ,ΒΙ834 | 524,288dec |
2Λ (220dec) | Ρ,Ξ2Η34 | 1,048,576dec |
2Μ (221dec) | 1Κ,Γ4Ψ34 | 2,097,152dec |
2Ν (222dec) | 34,Ο9Φ34 | 4,194,304dec |
2Ξ (223dec) | 69,ΕΚΡ34 | 8,388,608dec |
2Ο (224dec) | ΓΙ,Υ5Ι34 | 16,777,216dec |
2Π (225dec) | Π3,ΟΒ234 | 33,554,432dec |
2Ρ (226dec) | 1Η7,ΕΝ434 | 67,108,864dec |
2Σ (227dec) | 2ΨΕ,ΥΑ834 | 134,217,728dec |
2Τ (228dec) | 5ΦΥ,ΟΛΗ34 | 268,435,456dec |
2Υ (229dec) | ΒΣΠ,Ζ6Ψ34 | 536,870,912dec |
2Φ (230dec) | ΞΜΗ,ΦΔΦ34 | 1,073,741,824dec |
2Χ (231dec) | 1,Δ8Ω,ΡΣΡ34 | 2,147,483,648dec |
2Ψ (232dec) | 2,ΡΘΩ,ΚΜΙ34 | 4,294,967,296dec |
2Ω (233dec) | 5,Κ1Ω,59234 | 8,589,934,592dec |
210 (234dec) | Β,43Ψ,ΑΙ434 | 17,179,869,184dec |