Notable Properties of Specific Numbers
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:
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 give one entry each to complex, imaginary, negative numbers and zero, devoting all the rest (27 pages) to positive real numbers. I also have a bit of an integer bias but that hasn't had such a severe effect. A little more about complex numbers, quaternions and so on, is here.
One of the square roots of i.
When I was about 12 years old, my step-brother gave me a question to pass the time: If i is the square root of -1, what is the square root of i?. I had already seen a drawing of the complex plane, so I used it to look for useful patterns and noticed pretty quickly that the powers of i go in a circle. I estimated the square root of i to be about 0.7 + 0.7i.
I can't remember why I didn't get the exact answer: either I didn't know trigonometry or the Pythagorean theorem, or how to solve multivariable equations, or perhaps was just tired of doing maths (I had clearly hit on Euler's formula and there's a good chance that contemplating the powers of 1+i would have led me all the way through base-i logarithms and De Moivre's formula to the complex exponential function).
But you don't need that to find the square root of i. All you need to do is treat i as some kind of unknown value with the special property that any i2 can be changed into a -1. You also need the idea of solving equations with coefficients and variables, and the square root of i is something of the form "a+bi". Then you can find the square root of i by solving the equation:
(a+bi)2 = i
Expand the (a+bi)2 in the normal way to get a2 + 2abi + b2i2, and then change the i2 to -1:
a2 + 2abi - b2 = i
Then just put the real parts together:
(a2-b2) + 2abi = i
Since the real coordinate of the left side has to be equal to the real coordinate of the right, and likewise for the imaginary coordinates, we have two simultaneous equations in two variables:
a2-b2 = 0
2ab = 1
From the first equation a2-b2 = 0, we get a=b; substituting this into the other equation we get 2a2 = 1, and a=±1/√2 and this is also the value of b. Thus, the original desired square root of i is a+bi = (1+i)/√2 (or the negative of this).
(This is the only complex number with its own entry in this collection, mainly because it's the only one I've had much interest in; see the "blatant personal bias" note above :-).
(This is the only imaginary number with its own entry in this collection, mainly because it stands out way above the rest in notability. In addition, non-real numbers don't seem to interest me much...)
The "first" negative number, unless you define "first" to be "lowest"
(This is the only negative number with its own entry in this collection, mainly because negative numbers do not interest me much. I suppose this is because I still think of numbers in terms of counting things like "the 27 sheep on that hill" or "the 40320 permutations of the Loughborough tower bells".)
The word "zero" is the only number name in English that can be traced back to Arabic (صِفر ʂifr "nothing", "cipher"; which became zefiro in Italian, later contracted by removing the fi). The word came with the symbol, at around the same time the western Arabic numerals came to Europe.44,105
The practice of using a symbol to hold the place of another digit when there is no value in that place (such as the 0 in 107 indicating there are no 10's) goes back to 5th-century India, where it was called shunya or Śūnyatā107.
This is the Planck time in seconds; it is related to quantum mechanics. According to the Wikipedia article Planck time, "Within the framework of the laws of physics as we understand them today, for times less than one Planck time apart, we can neither measure nor detect any change". One could think of it as "the shortest measurable period of time", and for any purpose within the real world (if one believes in Quantum mechanics), any two events that are separated by less than this amount of time can be considered simultaneous.
See also 1.416833(85)×1032.
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. This uses the CODATA 2010 value50. See also 5.390×10-44 and 299792458.
The elementary charge or "unit charge", the charge of an electron in coulombs, from CODATA 2010 values50. This is no longer considered the smallest quantum of charge, now that matter is known to be composed largely of quarks which have charges in multiples of a quantum that is exactly 1/3 this value.
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 vacuum permittivity constant in farads per meter. In older times this was called the "permittivity of free space". Due to a combination of standard definitions, notably the exact definition of the speed of light, this constant is exactly equal to 107/(4 π 2997924582).
The gravitational constant in cubic meters per kiogram second squared, from CODATA 2010 values50. This is one of the most important physical constants in physics, notably cosmology and efforts towards unifying relativity with quantum mechanics. It is also one of the most difficult constants to measure.
There are a few "coincidences" regarding multiples of 1/127:
e/π = 0.865255... ≈ 110/127 = 0.866141...
√3 = 1.732050... ≈ 220/127 = 1.732283...
π = 3.141592... ≈ 399/127 = 3.141732...
√62 = 7.874007... ≈ 1000/127 = 7.874015...
eπ = 23.140692... ≈ 2939/127 = 23.141732...
1/100, or "one percent".
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%, and its apparent size (area in sky) varies by 30% when the size at apogee is compared to the size at perigee. This means that the brightness of the full moon varies by 30% over the course of the year. In 2004 the brightest full moon was the one on July 2nd; due to the orbit's precession the brightest full moon in 2006 was 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
converges to a finite value. (The highest such value is e(1/e) = 1.444667...; see that entry for more).
See also 0.692200....
The numbers 12 and 24 play a central role in mathematics thanks to a series of "coincidences" that is just beginning to be understood. One of the first hints of this fact was Euler's bizarre "proof" that
1 + 2 + 3 + 4 + ... = -1/12
which he obtained before Abel declared that "divergent series are the invention of the devil". Euler's formula can now be understood rigorously in terms of the Riemann zeta function, and in physics it explains why bosonic strings work best in 26=24+2 dimensions.
Baez, at the end of his "24" lecture, indicates that the significance of 24 is connected to the fact that there are two ways to construct a lattice on the plane with rotational symmetry: one with 4-fold rotational symmetry and another with 6-fold rotational symmetry and 4×6=24. A connection between zeta(-1)=-1/12 and symmetry of the plane makes more sense in light of how the Zeta function is computed for general complex arguments. Also, the least common multiple of 4 and 6 is 12.
The fraction 1/7 is the simplest example of a fraction with a repeating decimal that has an interesting pattern. See the 7 article for some of its interesting properties.
Reader C. Lucian points out that many of the well-known constants can be approximated by multiples of 1/7:
gamma = 0.5772156... ≈ 4/7 = 0.571428...
e/π = 0.865255... ≈ 6/7 = 0.857142...
√2 = 1.414213... ≈ 10/7 = 1.428571...
√3 = 1.732050... ≈ 12/7 = 1.714285...
e = 2.7182818... ≈ 19/7 = 2.714285...
π = 3.1415926... ≈ 22/7 = 3.142857...
eπ = 23.140692... ≈ 162/7 = 23.142857...
A reader suggested to me the idea that some people might define "zillion" as "a 1 followed by a zillion zeros". This is kind of like the definition of googolplex but contradicts itself, in that no matter what value you pick for X, 10X is bigger than X.
However, this is actually only true if we limit X to be an integer (or a real number). If X is allowed to be a complex number, then the equation 10X=X has infinitely many solutions.
Using Wolfram Alpha, put in "10^x=x" and you will get:
x ≈ -0.434294481903251827651 Wn(-2.30258509299404568402)
with a note describing Wk as the "product log function", which is related to the Lambert W function (see 2.50618...). This function is also available in Wolfram Alpha (or in Mathematica) using the name "ProductLog[k, x]" where k is any integer and x is the argument. So if we put in "-0.434294481903251827651 * ProductLog[1, -2.30258509299404568402]", we get:
0.529480508259063653364... - 3.34271620208278281864... i
Finally, put in "10^(0.529480508259063653364 - 3.34271620208278281864 * i)" and get:
0.52948050825906365335... - 3.3427162020827828186... i
If we used -2 as the initial argument of ProductLog, we get 0.5294805+3.342716i, and in general all the solutions occur as complex conjugate pairs. Other solutions include x=-0.119194...±0.750583...i and x=0.787783...±6.083768...i.
In light of the fact that the -illion numbers are all powers of 1000, another reader suggested that one should do the above starting with 10(3X+3)=X. This leads to similar results, with one of the first roots being:
-0.88063650680345718868... - 2.10395020077170002545... i
The first fraction in Conway's FRACTRAN program ( page 147) 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 A7547).
This is e-π/2, which is also equal to i i. (Because eix = cos(x) + isin(x), eiπ/2=i, and therefore i i = (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 A1318.30
0.329239474231204... = acosh(sqrt(2+sqrt(2+4))/2) = ln(2+√3)/4
This is Gottfried Helms' Lucas-Lehmer constant "LucLeh"; see 1.38991066352414... for more.
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
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 × ...)
The odds of losing a game of chance. Flip a coin: if you get heads, your score increases by π, if you get tails, your score diminishes by 1. Repeat as many times as you wish but if your score ever goes negative, you lose. Assuming the player keeps playing indefinitely (motivated by the temptation of getting an ever-higher score), what are the odds of losing?
The answer is given by a series sum: 1/2 + 1/25 + 4/29 + 22/213 + 140/217 + 969/221 + 7084/225 + 53820/229 + 420732/234 + ..., (numerators in Sloane's A181784) which adds up to 0.5436433121...
More on my page on sequence A181784.
See also 368.
This is the Omega constant, which satisfies each of these simple equations (all equivalent):
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:
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 A36299 or A61107). 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:
- In the iteration described above, the number of digits in each string is the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, ...
- Expressed as a continued fraction, the constant is 0 + 1/(20 + 1/(21 + 1/(21 + 1/(22 + 1/(23 + 1/(25 + 1/(28 + ...))))))) where the exponents of 2 are the Fibonacci numbers.
- If you take all the multiples of the Golden Ratio 0.618033 and round them down to integers, you get 1, 3, 4, 6, 8, 9, 11, 12, ...: These numbers tell you where the 1's in the binary fraction are.
If you leave off the first two binary digits (10) you get 110101101101011010110110101..., the bit pattern generated by a Turing machine at the end of the Turing machine Google Doodle. As a fraction (0.1101011...) it is 0.8392137714451.
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
This is 0.1101011011010110101101101011011010110101101101011010110110... in binary, and is the slightly different version of the Rabbit constant generated by a Turing machine Google Doodle from June 2012. More digits: 0.8392137714451652585671495977783023880500088230714420678280105786051...
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.
This is Gamma(5/4), or "the factorial of 1/4". While some Gamma function values, like 0.886226... and 1.329340..., have simple formulas involving just π to a rational power, this one is a lot more complcated. It is π to the power of 3/4, divided by (√2+4√2), times the sum of an infinite series for an elliptic function.
Catalan's constant, which can be defined by:
G = ∫(0,1) [ arctan(x) / x dx ]
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
This is the cube root of (5√27 - 5√2). Bill Gosper discovered the following identity, which is remarkable because the left side only has powers of 2 and 3, but the right side has a power of 5 in the denominator 108:
(5√27-5√2)(1/3) = (5√8 5√9 + 5√4 - 5√2 5√27 + 5√3) / 3√25
or in his original form:
(3(3/5)-2(1/5))(1/3) = (- 2(1/5)3(3/5) + 2(3/5)3(2/5) + 3(1/5) + 2(2/5) ) / 5(2/3)
See also 1.554682...
Quick index: if you're looking for a specific number, start with whichever of these is closest: 0.065988... 1 1.618033... 3.141592... 4 12 16 21 24 29 39 46 52 64 68 89 107 137.03599... 158 231 256 365 616 714 1024 1729 4181 10080 45360 262144 1969920 73939133 4294967297 5×1011 1018 5.4×1027 1040 5.21...×1078 1.29...×10865 1040000 109152051 101036 101010100 footnotes Also, check out my large numbers and integer sequences pages.