| Notable Properties of Specific Numbers |
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1.324717957244746025960908854...
The ratio between successive terms of the Perrin and Padovan sequences. It is the only real solution to the simple cubic equation x3=x+1.
This is 3/4 times the square root of π. It is gamma(5/2), and is sometimes also called (3/2)!, the factorial of 3/2.
The square root of 2, √2, also sometimes called "Pythagoras' constant".
The continued fraction for √2 is:
√2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... )))
yielding the series of rational approximations:
| 1 | 3 | 7 | 17 | 41 | 99 | 239 | a | c | e | ||
| 1 , | 2 , | 5 , | 12 , | 29 , | 70 , | 169 , | . . . | b , | d , | f |
where each new fraction e/f is computed from the previous ones by the formulas:
e = 2c + a = c + 2d
f = 2d + b = c + d
These two sequences are the Pell numbers and another similarly defined sequence discussed under the entry for 99.
If you multiply each numerator by its denominator you get the series discussed under 204.
See also 1.632526919438....
This is the highest value of the function y = x(1/x), and also the highest value of z for which the infinite power tower
zzzzz...
converges to a finite value. (The lowest such value is e-e = 0.065988...).
Euler discovered that the value xxx... converges to is the solution to the equation y1/y=x. That means that you can solve the equation y1/y=x for y by evaluating xxx... (and if it doesn't converge, then there's no solution to the equation). However, it usually takes a lot of iterations to converge and Newton's Method is faster. This iteration can also be used to find some values of x such that xx=y (see here for details).
See also 0.692200....
This is the area of the Mandelbrot set
The number has been measured statistically based on computers counting the "pixels" in the Mandelbrot set's image, but Cyril Soler conjectures that the value is precisely √(6π-1) - e, or 1.5065916514855...
1.52469996053809435992336357568842116222022362319977121984572...
It is very likely that Mills' theorem is true for the formula K2n and that this is the lowest value of K. If so, then the integer part of K2n is always prime: 2, 5, 29, 853, 727613, 529420677791, 280286254072681840639693, 78560384222095957698731679318817728959447134363, ... (Sloane's A059784). This depends on the unproven conjecture that there is always a prime between any two consecutive squares. Given the difficulty of proving that there is a prime between consecutive cubes, it seems very unlikely that anyone will ever answer this one way or the other.
The solution of xx=2. See also 2.506184... and 56.9612....
Numbers of this type are transcendental, except for cases like xx=27. Clearly, if X is an integer, XX is an integer. It is fairly easy to show that if X is not an integer, XX is irrational. These things together with the Gelfond-Schneider Theorem allow one to show that if XX is an integer, and X is not an integer, then X is transcendental.
π/2.
The precise ratio between a mile and a kilometer. Because of its proximity to phi, one can use the Fibonacci numbers as a handy mnemonic for conversion: 3 mi ≅ 5 km, 5 mi ≅ 8 km, 8 mi {~=] 13 km, etc. The approximation is better than 1/2 km all the way up to 55 mi ~= 89 km. See also 63241.077088071.
The Golden Ratio, (1+√5)/2, 1.61803398874989..., commonly designated by the Greek letter Φ. It is called the golden ratio because of its use in ancient, classical and Renaissance art and architecture. It is the ratio between the height and width of the so-called "golden rectangle", considered to be more aesthetically appealing than any other rectangle. This was probably the result of the fact that, if you take a golden rectangle and add a square to the longest edge to make a larger rectangle, that larger rectangle will have the same ratio. This is reflected in the relation:
1 + 1/Φ = Φ
or (in numbers):
1 + 1/1.618033... = 1.618033...
The ratio of two consecutive Fibonacci numbers approaches Φ as you go up to higher and higher Fibonacci numbers.
This ratio shows up in lots of places in mathematics. Φ or 1/Φ is the solution of the simple equations x2 = x + 1, x2 = 1 - x, 1/x = x + 1, and 1/x = x - 1. Φ, 1/Φ and Φ2 all have the same digits after the decimal point: 1.618033..., 0.618033..., 2.618033... .
It is well-known that there are exponents ab such that a and b are both rational but ab is irrational. For example, the square root of 2, √2, is irrational. But is there any pair of irrational numbers x and y such that the value xy is rational? It is really easy to show the answer is yes, without actually finding any values of x or y, and the proof involves the number 1.632526919438... = √2√2, which we'll call q.
If q is irrational, then q√2 equals (√2)2, which is 2 so we have a solution with x=q and y=√2. On the other hand, if q is rational, then the solution is x=y=√2.
Two related facts: there are pairs of rational numbers a and b such that ab=a b (see 3.160493827160...) and there are pairs of rational a and b such that ab=ba (see 15.438887358552). See also 1.559610..., 2.2.66514414269....
zeta(2), or π squared over 6. If you pick two positive integers at random, the odds of them having no common divisor are 1 in 1.644934...
The square root of π. This is the area under a bell curve of unit height and unit standard deviation. It is also equal to gamma(1/2), and is sometimes also called (-1/2)!, the factorial of -1/2.
Many properties, mostly for trivial reasons (see 3.)
The second Feigenbaum constant, commonly designated by the Greek letter alpha. (The other Feigenbaum constant is 4.6692016091....)
2.50618414558876925629294092237784727177139605213321283014...
The solution of xx=10, and notable for being close to √(2π).
It is appropriate to note here that this value is transcendental — and there is no "simple" (closed-form) way to turn xx=y into an expression for x in terms of y. Instead, the function is solved by Newton's method. For example, the value of x for xx=10 was calculated this way:
| step | action | notes |
| 1. | T = 10.0 | target value |
| 2. | X = 2.0 | first approximation of answer |
| 3. | Y = XX | calculate the function |
| 4. | dY = Y (1 + ln(X)) | derivative with respect to X |
| 5. | new X = old X + (T-Y)/dY | new approximation by Newton's method |
| 6. | go back to step 3 | repeat until accurate enough |
Despite the rather poor initial approximation X=2, it only takes 8 repetitions of steps 3 4 and 5 to get the first 14 digits correct and 10 repetitions to get 57 digits.
In general, xx=y can be solved for x in terms of y using the Lambert W function: x = eW(ln(y)). For instructions on how to calculate, see the wiki page.
The following is obsolete and needs to be merged:
And although xx=y cannot be solved, there is an iterative solution for x1/x=y (see 1.444667...) that can be adapted to solve xx=y for certain values of y:
given: xx=y, y is known, find x.
1. Substitute x=1/z: (1/z)(1/z) = y
2. Substitute y=1/w: (1/z)(1/z) = 1/w
3. Take the reciprocal of both sides: z(1/z) = w
4. If w is not within the range (e-e,e(1/e))
— approximately (0.065988, 1.444667) — then
either there is no solution to xx=y or else the solution
cannot be found this way.
5. Otherwise, find the limit of the series w, ww,
www, ...: this is z (as shown by Euler).
6. Take the reciprocal to get x.
Example: starting with xx=10, we get w=1/10. Iterating, we get ww = 0.7943282347..., www = 0.1605727204..., wwww = 0.6909192287..., and so on, converging after about 350 iterations to 0.39901297826...; the reciprocal of 0.39901297826... is 2.50618414558... . (This solution, based on work of Euler, was pointed out to me by Fantini and Kloepfer.)
The equation xy=yx is much easier to solve, as is xy=xy.
Notable for being close to the solution of xx=10.
Since July 1 1959, by international agreement, the inch has been defined to be exactly 2.54 centimeters or 0.0254 meters. See 299792458 for the definition of the meter. See also 1609.344.
2.66514414269022518865029724987313984827421131371465... = 2√2
This number is known as the Gelfond-Schneider constant. Along with eπ, it was shown to be transcendental by Gelfond's theorem (sometimes called the Gelfond-Schneider theorem). This theorem shows that if x is not 0 or 1 but is algebraic, and y is both algebraic and irrational, then xy is transcendental. It was proven independently by Gelfond and Schneider in 1934.
See also 1.632526919438..., 3.141592653..., 23.140692632779269005....
Khintchine's constant, the geometric mean of the terms of the continued fraction for most (but not all) real numbers. Rational numbers, roots of 3rd order polynomials and certain other classes of real numbers have patterns in their continued fraction expansion — so they do not obey this principle. However, the real numbers that do obey this average outnumber those that don't, and they outnumber by such a great extent, that if you picked a real number at "random" the probability that it would not have this average is 1/infinity.
e, base of natural logarithms, Euler's number, etc. Competes with the square root of 2 for second-most-famous non-integer after π. e is the limit of
(1 + 1/X)X
as X goes to infinity. An efficient way to calculate e goes like this:
e = 2
i = 2
f = 1
repeat until satisfied
f = f/i
e = e + f
i = i + 1
end repeat
which uses the well-known infinite series: e is the sum of the reciprocals of the factorials of all the natural numbers.
The solution of xx=27. :-)
There are so many properties of small numbers like this that it is somewhat tedious to list them all (but if you're interested look here). 3 is a prime, a triangular number, a Fibonacci number, the number of spatial dimensions of our universe, etc. etc.
There are many occurrences of a 3-times repetition in rituals, customs, stories, etc. Some examples include: the three knocks of Black Rod when calling the House of Commons to hear the address of the Sovereign at the opening of Parliament; three verses in the standard song format; the three pigs' houses (straw, wood, stone/brick) in the children's tale. The use of three (rather than two or four) is probably due to the fact that three is the minimum necessary to establish a pattern (such as a regular tempo) or to convey the impression of an ongoing sequence or succession.
3.14159264581... = 3 + 1/8 + 1/61 + 1/5020
The closest approximation to π using three terms of the greedy Egyptian fraction algorithm. It is somewhat more accurate than this one but requires a larger denominator term.
The most well-known non-integral mathematical constant, the subject of several books, etc. Pi shows up in many places you don't expect it to (for instance, see 1.644934....)
There are lots of ways calculate π. One of the most efficient goes like this:
A = 545140134; B = 13591409; C = 640320;
D = C3; E = C/12;
total = 0;
n = 0;
repeat until satisfied:
total = total + -1n
((6n)!(A n + B))/(n!3(3n)!Dn)
n = n + 1;
end repeat
π = E √C/total
Every time you repeat the loop adds more than 14 digits to the accuracy of the approximation. Take a large integer N. Round up to the nearest multiple of N-1. Then round that number up to the nearest multiple of N-2. Continue until you get to the nearest multiple of 1, and call this final value F. For sufficiently large initial values of N, the ratio N2/F comes arbitrarily close to π.
One of the famous problems from ancient times is called squaring the circle. The original problem was to use a straightedge and compass to determine the size of a square whose area is the same as that of a given circle. This is essentially equivalent to finding a precise formula for π that involves integers, the operations + - × /, and a few simple others like square root.
This problem eventually evolved into a more refined version, that of finding an integer polynomial to which π is a root. Numbers that can be expressed this way are algebraic, those that are not are transcendental. For various reasons it became important to be able to determine if certain numbers are algebraic or transcendental.
The transcendental nature of π was first shown by Lindemann in 1882. See also 1.632526919438... and 2.66514414269....
Digits of pi
The following formula allows one to compute the Nth digit (in base 16) of π without computing all the preceding digits:
π = SUM [ (4/8n+1 - 2/8n+4 - 1/8n+5 - 1/8n+6) × 16-n ]
This poem is a mnemonic for remembering the first 42 digits of π.
The number of letters in each word corresponds to a digit, with
10-letter words for '0' digits. It was written in 1995 by Michael
Keith based on "The Raven" by Edgar Allen Poe:
3.1 Poe, E. -
415 Near A Raven
926 Midnights so dreary,
535 tired and weary,
8979 Silently pondering volumes extolling
32384 all by-now obsolete lore.
62643 During my rather long nap
383 the weirdest tap!
27950 An ominous vibrating sound disturbing
288 my chamber's antedoor.
419716 "This", I whispered quietly, "I ignore".
If you like this kind of thing, Mr Keith finished "The Raven" and went on to write several other modified versions of well-known bits of literature, then made an entire story around it, which he calls Cadaeic Cadenza. It has about 3800 words corresponding to the first 3834 digits of π!
3.14159267204... = (e (υ + 6))/5 - 1
One of many close approximations to π. This one achieves the closest possible approximation using 5 single digits or constants and 4 operations. Here are some others: with three constants: √(4e-1) = 3.14215...; with four constants: ln((72 - e)/2) = 3.1415998...; with six constants (sort of): (4-υ√e)×(2/√7+1) = 3.141592653650... and √3+√2+1/(ln(7)-63) = 3.1415926535515... . (If this sort of thing amuses you, you might want to check out my inverse equation solver; the command line for this problem is: ries 3.1415926535897932 -Ox -x -NpSCL .)
3.1415269531... = 3 + 1/13 + 1/17 + 1/173
The numbers 13, 17 and 173 have been found together in ancient Egyptian records without explanation, but it is conjectured that they express an Egyptian fraction approximation to π. If so, it is a much better approximation than those attributed to Archimedes and Ptolemy.
The approximation to π given by the ratio 355 / 113. This approximation is admired mainly because it is "efficient": the fraction has 6 digits but the approximation gives you about 71/2 digits of π. It's also kind of easy to remember, if you remember the denominator first (because then you get "113, 355" which is a very simple pattern).
Along with 22/7, 355/113 is a member of the continued-fraction series for π. Continued fraction approximations can be used to get "optimal" rational approximations for any number. By "optimal" we mean the closest approximation that is possible with a given-size numerator and denominator. The continued fraction approximations for π are: 3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913, ...
To generate a continued-fraction series, you just keep truncating the fraction and taking the reciprocal:
3.141592653589... = 3 + 0.141592653589... = 3 + 1/7.062513305931...
7.062513305931... = 7 + 1/15.996594406685...
15.996594406685... = 15 + 1/1.003417231013...
1.003417231013... = 1 + 1/292.634591014395...
292.634591014395... = 292 + 1/1.575818089...
This gives you the terms of the continued fraction: 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, ... (Sloane's A001203). You use these integers to make the actual continued fraction for π:
π = 3+1/(7+1/(15+1/(1+1/(292+1/...))))
The fact that 355/113 is an "efficient" approximation (as described above) is because the next term in the series (292) is relatively high. On average, continued-fraction series have much smaller terms most of the time (see Khintchine's constant). The beginning of the π series is an exception.
If you stop a continued fraction series at any point and reduce to a simple fraction, you get an approximation:
3+1/(7+1/(15+1/(1))) = 3+1/(7+1/(16)) = 3+1/(113/16) = 3+16/113 = 355/113
If you want to generate a whole series of numerators and denominators,
use an algorithm like this:
(given: X = number to be converted)
N = 1
D = 0
N1 = 0
D1 = 1
repeat until satisfied:
let XI = integer part of X
let X = fractional part of X
let X = 1 / X
let N2 = N1
let N1 = N
let N = N1 * XI + N2
let D2 = D1
let D1 = D
let D = D1 * XI + D2
print "New approximation: ", N, "/", D
end repeat
The approximation 355/113 for π was known to the Chinese mathematician Zu Chongzhi around 480 AD.
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