| Notable Properties of Specific Numbers |
Introduction
These are some numbers with notable properties. (Most of the less notable properties are listed here.) Other people have compiled similar lists, but this is my list — it includes the numbers that I think are important (-:
A few rules I used in this list:
Everything can be understood by a typical undergraduate college student.
If multiple numbers have a shared property, that property is described under one "representative" number with that property. I try to choose the smallest representative that is not also cited for another property.
When a given number has more than one type of property, the properties are listed in this order:
1. Purely mathematical properties unrelated to the use of base 10 (example: 137 is prime.)
2. Base-10-specific mathematical properties (example: 137 is prime; remove the "1": 37 is also prime; remove the "3": 7 is also prime)
3. Things related to the physical world but outside human culture (example: 137 is close to the reciprocal of the fine-structure constant, once thought to be exact but later found to be closer to 137.036...)
4. All other properties (example: 137 has often been given a somewhat mystical significance due to its proximity to the fine-structure constant, most famously by Eddington)
Due to blatant personal bias, I only discuss positive real numbers on this page. Negative, imaginary, complex, octonions, and so on are here.
The word "zero" is the only number name in English that can be traced back to Arabic. The word came with the symbol, at around the same time the indo-arabic numerals came to Europe.44
This is the Planck time in seconds; it is related to quantum mechanics. The best interpretation for most people is that the Planck time is the shortest measurable period of time; any two events that are separated by less than this amount of time can be considered simultaneous. See also 1.6160×10-35 and 299792458.
This is the Planck length in meters; it is related to quantum mechanics. The best interpretation for most people is that the Planck length is the smallest measurable length, or the smallest length that has any relevance to events that we can observe. See also 5.390×10-44 and 299792458.
Approximate "size" of a proton71, in meters (based on its "charge radius" of 0.875 femtometers). "Size" is a pretty vague concept for particles, and different definitions are needed for different problems. See 1040.
The fine-structure constant, as given by Gabrielse75 and Hanneke78. The previous best value was 0.0072973525700(52), given by Gabrielse, et. al.49. Before that the value was 0.007297352533(27), which for a long time could be found at CODATA; this value was affected by an error referred to in the Gabrielse, et. al. paper. There is a lot more about this number under the entry for its reciprocal.
Mean eccentricity of the Moon's orbit — the average variation in the distance of the Moon at perigee (closest point to the Earth) and apogee. Due to the influence of the Sun's gravity the actual eccentricity varies a large amount, going as low as about 0.047 and as high as about 0.070; also the ellipse precesses a full circle every 9 years (see 27.554549878). The eccentricity is greatest when the perigee and apogee coincide with new and full moon. At such times the Moon's distance varies by a total of 14 percent, and its apparent size (area in sky) varies by 30 percent when the size at apogee is compared to the size at perigee. This means that the brightness of the full moon varies by 30 percent over the course of the year. In 2004 the brightest full moon is the one on July 2nd; due to the orbit's precession the brightest full moon in 2006 is a couple months later, Oct 6th.
This change in size is a little too small for people to notice from casual observation (except in solar eclipses, when the Moon sometimes covers the whole sun but at other times produces an annular eclipse). But the eccentricity is large enough to cause major differences in the Moon's speed moving through the sky from one day to the next. When the Moon is near perigee it can move as much as 16.5 degrees in a day; when near apogee it moves only 12 degrees; the mean is 13.2. The cumulative effect of this is that the moon can appear as much as 22 degrees to the east or west of where it would be if the orbit were circular, enough to cause the phases to happen as much as 1.6 days ahead of or behind the prediction made from an ideal circular orbit. It also affects the libration (the apparent "wobbling" of the Moon that enables us to see a little bit of the far side of the moon depending on when you look).
This is the lowest value of z for which the infinite power tower
zzzzz...
converges to a finite value. (The highest such value is e(1/e) = 1.444667...; see that entry for more).
See also 0.692200....
First fraction in Conway's FRACTRAN program that finds all the prime numbers. The complete program is 17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/2, 1/7, 55/1. To "run" the program: starting with X=2, find the first fraction N/D in the sequence for which XN/D is an integer. Use this value NX/D as the new value of X, then repeat. Every time X is set to a power of 2, you've found a prime number, and they will occur in sequence: 22, 23, 25, 27, 211 and so on. It's not very efficient though — it takes 19 steps to find the first prime, 69 for the second, then 281, 710, 2375 ... (Sloane's A007547).
This is e-π/2, which is also equal to ii. (Because eix = cos(x) + isin(x), eiπ/2=i, and therefore ii = (eiπ/2)i = ei2π/2 = e-π/2 .)
0.288788095086602421278899721929... = 1/2 × 3/4 × 7/8 × 15/16 × 31/32 × ... × 1-2-N × ...
This is an infinite product of (1-2-N) for all N. This is also the product of (1-xN) with x=1/2. Euler showed that in the general case, this infinite product can be reduced to the much easier-to-calculate infinite sum 1 - x - x2 + x5 + x7 - x12 - x15 + x22 + x26 - x35 - x40 + ... where the exponents are the pentagonal numbers N(3N-1)/2 (for both positive and negative N), Sloane's A001318.30
If you take a string of 1's and 0's and follow it by its complement (the same string with 1's switched to 0's and vice versa) you get a string twice as long. If you repeat the process forever (starting with 0 as the initial string) you get the sequence
011010011001011010010110...
and if you make this a binary fraction 0.0110100110010110...2 the equivalent in base 10 is 0.41245403364..., and is called the Thue-Morse constant or the parity constant. Its value is given by a ratio of infinite products:
4 K = 2 - PRODUCT[22n-1] / PRODUCT[22n]
= 2 - (1 × 3 × 15 × 255 × 65535 × ...)/(2 × 4 × 16 × 256 × 65536 × ...)
0.567143290409783872999968662210355549753815787186512508135131...
This is the Omega constant, which satisfies each of these simple equations (all equivalent):
| ex = 1/x | x = ln(1/x) = - ln(x) | |
| e-x = x | -x = ln(x) | |
| x ex = 1 | x+ln(x) = 0 | |
| x1/x = 1/e | x/ln(x) = -1 | |
| x-1/x = (1/x)(1/x) = e | ln(x)/-x = 1 |
Thus it is sort of like the golden ratio. In the above equations, if e is replaced with any number bigger than 1 (and "ln" by the corresponding logarithm) and you get another "Omega" constant. For example:
if 2x=1/x, then x=0.6411857445...
if πx=1/x, then x=0.5393434988...
if 4x=1/x, then x=1/2
if 10x=1/x, then x=0.3990129782...
if 27x=1/x, then x=1/3
if 10000000000x=1/x, then x=1/10
This is the Euler-Mascheroni constant, commonly designated by the Greek letter gamma. It is defined in the following way. Consider the sum:
Sn = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... + 1/n
The sequence starts 1, 1.5, 1.833333..., 2.083333..., etc. As n approaches infinity, the sum approaches ln(n) + gamma.
Here are some not-particularly-significant approximations to gamma:
1/(√π - 1/25) = 0.5772159526...
gamma = 0.5772156649...
1/(1+ 1/√10)2 = 0.5772153925...
The golden ratio (reciprocal form): see 1.618033....
This is the lowest point in the function y = xx. See also 1.444667...
You can create a long string of 1's and 0's by using "substitution rules" and iterating from a small starting string like 0 or 1. If you use the rule:
0 → 1
1 → 10
and start with 0, you get 1, 10, 101, 10110, 10110101, 1011010110110, ... where each string is the previous one followed by the one before that (Sloane's A036299 or A061107). The limit of this is an infinite string of 1's and 0's which you can make this into a binary fraction: 0.1011010110110...2, you get this constant (0.709803... in base 10) which is called the Rabbit Constant. It has some special relationships to the Fibonacci sequence:
Value of x such that x=cos(x), using radians as the unit of angle. You can find the value with a scientific calculator just by putting in any reasonably close number and hitting the cosine key over and over again. Here are a few more digits: 0.7390851332151606416553120876738734040134117589007574649656...26
0.8507361882018672603677977605320666044113994930...
Decimal value of the "regular paperfolding sequence" 1 1 0 1 100 1 1100100 1 110110001100100 1 1101100111001000110110001100100 ... converted to a binary fraction. This sequence of 1's and 0's gives the left and right turns as one walks along a dragon curve. It is the sum of 82k/(22k+2-1) for all k≥0, a series sum that gives twice as many digits with each additional term.
The minimum value of the Gamma function, the continuous analogue of the factorial function. This is gamma(1.461632...).
This is 1/2 of the square root of π. It is gamma(3/2), and is sometimes also called (1/2)!, the factorial of 1/2.
Catalan's constant, which can be defined by
G = ∫(0,1) [ arctan(x) / x dx ]
or
G = 1 - 1/32 + 1/52 - 1/72 + 1/92 - ...
If you have a 2n × 2n checkerboard and a supply of 2 n2 dominoes that are just large enough to cover two squares of the checkerboard, how many ways are there to cover the whole board with the dominoes? For large n, the answer is closely approximated by
f'n = e4 G n2 / π
Many properties, mostly for trivial reasons (see 3.)
Apéry's constant, or zeta(3). If you pick three positive integers at random, the odds of them having no common divisor are 1 in 1.202056... It is called Apéry's constant because in 1979 Apéry surprised the mathematical community by showing that it is irrational.
The cube root of two, the subject of a Greek legend.
On the island of Delos, the birthplace of Apollo and Artemis, was an oracle. At one time, when there was a plague in Athens, an emissary went to the oracle to ask what to do. The oracle replied that the plague would cease if the altar to Apollo were doubled in size. The altar was a perfect cube, one cubit on each side. So the people of Athens built a new altar that was a cube two cubits long on each side. But the plague continued, and when asked, the oracle explained that the new altar was eight times as large as the old one. So they built an altar that was two cubits long in one direction, but one cubit in the other two directions, but the plague continued, the altar was no longer a cube. Apollo was only to be pleased by an exact cube that was twice the volume of the original — 3√2 cubits on each side.(From Conway and Guy, The Book of Numbers page 190, paraphrased.)
4 / π, the value of the rather pretty continued fraction:
4 / π = 1 + ( 1 / ( 2 + ( 32 / ( 2 + ( 52 / ( 2 + ( 72 / ( 2 + ... ))))))
By contrast, the simple continued fraction for π (where all the numerators are 1) has no pattern:
π = 3 + 1 / ( 7 + 1 / (15 + 1 / (1 + 1 / (292 + 1 / (1 + ...)))))
1.28242712910062263687534256886979172776768892732500...
This is the Glaisher-Kinkelin constant, related to the Barnes G-function and the K-function. It appears in these approximations of the G-function and the K-function:
K(n+1) ≅ A nn2/2+n/2+1/12e-n2/4
G(n) ≅ (e1/12/A) nn2/2-1/12(2{pi|)n/2e-3n2/4
This is the fractal dimension of the "Apollonian packing", a rather pretty fractal made from circles. The precise value is a little difficult to compute; the most digits I have seen is 1.305686729, but that's probably only accurate up to the "7". To make the fractal, start with three circles that are just touching (tangent to) each other, and with none of the circles inside each other. Now there is exactly one way to add a 4th circle in the space between the three circles so that the new circle is tangent to all three. Then you can add three more smaller circles in the spaces between the 4th circle and two of the original three. As you continue this process indefinitely, you get an Apollonian packing.
The lowest value of Mills' constant assuming the Riemann Hypothesis is true. In 1947 W. H. Mills proved that there is a constant K such that, for all values of n, K3n is a prime number plus a fraction less than 1. The first 100 digits of the constant are: 1.3063778838 6308069046 8614492602 6057129167 8458515671 3644368053 7599664340 5376682659 8821501403 7011973957...
Here is what you get for the first few values of n:
K3 = 2.229494...
K32 = 11.082031...
K33 = 1361.000001...
K34 = 2521008887.000000...
K35 = 16022236204009818131831320183.000000...
Each of these numbers is the cube of the previous one, and when the fraction is removed the resulting integer is prime. It is pretty easy to see that Mills' theorem seems to be true, simply because there are so many primes. For example, start with 3: 33 is 27. There are several primes between 27 and 43=64, of which the first is 29. 293 is 24389 and 303 is 27000 — there are even more primes to choose from this time. Choosing the first available prime each time, we get the sequence 3, 29, 24391, 14510715208481, 3055388613462301256452407743005777548691, ... The constant K in this case would be approximately 3055388613462301256452407743005777548691(1/243)=1.45375086254... In a similar manner, starting with 5 we get 127, 2048413, 8595132382702380079, 634976584256084664026852011723442922433087739799461233111, ... No matter what prime one starts with, there are plenty of primes to choose from each time and therefore plenty of possible values for Mills' constant.
The difficulty in proving this for certain comes from the fact that it is difficult to prove that there is a prime between any two consecutive cubes. So far that has only been proven for primes up to 106000000000000000000.
See also 1.52469996....
(c) 1996-2009 Robert P. Munafo.
email me
more info
This work is licensed under a Creative Commons Attribution 2.5 License. Details here
Back to my main page
s.13