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3.14159267204... = (e (φ + 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-φ√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:
The approximation 355/113 for π was known to the Chinese mathematician Zu Chongzhi around 480 AD.
The approximation to pi used by Ptolemy.
3.141851... = 3 141/994 = (3 1/7 + 3 10/71) / 2
Some say that Archimedes used a value for pi that was "between 3 1/7 and 3 10/71", rather than simply 3 1/7; this would be an improvement since 3 10/71 is closer to the true value of pi. 3 141/994 is the average of the two approximations.
An approximation to pi given by the unitless expression:
(Planck energy times
the combined fuel economy of a Toyota Prius)
divided by
(the pressure in the center of the earth times
the width of the English Channel at its narrowest point)
which is given in this xkcd cartoon. (All of the units cancel out, as you will see if you follow the links to each of these constants, and the approximation is correct to within the limits of measurement and/or experiment. I have given only 27 digits of precision, but more may be possible in the near future — watch this space! This formula is a great commentary on all the theories described at 137.035.)
22/7 (or 3+1/7), an approximation of π used in the time of Archimedes. The difference between 22/7 and π is given by the integral
∫0 to 1 [ x4 (1 - x)4 / (1 + x2) dx ]
3.160493827160... = 256/81 = 44/34 = 3+1/9+1/27+1/81
44/34 is an approximation to π used by the Egyptians, and it is also sort of close to √10, but it has a much more interesting property.
There are an infinite number of pairs of rational numbers a and b such that ab=a b. To show this, start with:
xy = x y
then define l to be the log base x of y:
l = logxy
then the first equation becomes:
xxl = x xl
which (for x>1) reduces to:
xl = l+1
Solving for x, and substituting back for y, we get:
x = (l+1)(1/l)
y = xl = ((l+1)(1/l))l = l+1
This is the general solution for xy = x y. (The same equations also happen to solve x④y=xy.)
To get both x and y to be rational numbers, define l to be 1/n for some integer n>1. Then you get:
x = (1/n+1)n = (n+1)n/nn = ((n+1)/n)n
y = (n+1)/n
x = yn
For n=1 you get x=y=2, which isn't a very interesting solution, but for n=2 you get this solution and for n=3 you get x=64/27 and y=4/3, with the product x y=xy=256/81=3.160493827160...
As n gets larger, y approaches 1 and x approaches e because
e = lim (1 + 1/n)n
See also 1.632526919438... and the solution of xy=yx.
This has sometimes been used as an approximation to π
3.375 = 27/8 = (9/4)(3/2) = (9/4) × (3/2)
The equation x④y=xy, where ④ is the lower-valued form of the hyper4 operator, can be solved easily. The formula for x④y is:
x④y = xxy-1
so we're really solving:
xxy-1 = xy
If x>1 we have:
xy-1 = y
and thus:
x = y1/(y-1)
(If you change y to l you can see the similarity to the solution of xy=yx. The same values of l and x that solve the xy=yx equation also solve x④l = xl.)
We can also express xy in terms of y:
xy =(y1/(y-1))y =yy/(y-1)
and now notice that this is equal to the product xy, because:
xy =yx =y y1/(y-1) =y(y-1+1)/(y-1)
=yy/(y-1)
Therefore, the solutions to x④y=xy also simultaneously solve xy=xy.
If 1/(y-1) is an integer then x and y will both be rational numbers; this happens when y is of the form (a+1)/a, where a is any nonzero integer. Then x is:
x = ((a+1)/a)a = (a+1)a/aa
The smallest non-trivial solution is when a=2, we get x=9/4 and y=3/2, and the value of x④y =xy =xy is 27/8=3.375.
Implicitly mentioned in the Bible in a few laces, see 1260.
Another number with zillions of properties: the smallest composite, a square, the first non-Fibonacci number, a power of 2, etc.
The trivial solution of xy=xy is for x=y=2, and xy=xy=4.
The first Feigenbaum constant, commonly designated by the Greek letter delta. In the Mandelbrot set, it shows up as the ratio between each "circle" and the next smaller one in the series of "circles" on the real axis connected to the large cardioid. (Only the first of these, the one centered at -0.75, is actually a perfect circle.) For more information, click here.
The value is approximated to 7 digits by the formula π+arctan(eπ) (which is 4.6692019318...)
The second Feigenbaum constant is 2.5029078750....
Many properties, mostly for trivial reasons (see 3.) The third prime, a Fibonacci number, a pyramidal number, etc.
This number is the highest solution to the equation ex = gamma(x+1), and corresponds to the point where the Gamma function starts to exceed the ex function.
A perfect number. See here for more.
Also a triangular number, a factorial, composite, etc.
The smallest positive integer whose reciprocal has a pattern of more than one repeating digit: 1/7 = 0.142857142857... It is also the smallest number for which the digit sequence of 1/n is of length n-1 (the longest such a sequence can be). The next such numbers are 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, ... (Sloane's integer sequence A006883). See 17 for more on this.
Tests for divisibility by 7 are difficult. About the best one I've found is as follows (demonstrated by testing 4156726):
There is another divisibility test that works as follows:
7 is considered "lucky" by many people and given much spiritual significance. The early religious and cultural use of the 7-day week almost certainly arose from the fact that the moon goes through its 4 phases in a bit over 28 days, which divides nicely into 7 days per phase. Another, different connection between moon phases and 7 is that there are 7 easily distinguishable, visible phases of the moon: waxing crescent, first-quarter, waxing gibbous, full, waning gibbous, third-quarter, waning crescent. The new moon doesn't count because you can never see it (the sun is too bright). And there is also the well-known count of the 7 moving objects in the sky: sun, moon, Mercury, Venus, Mars, Jupiter, Saturn.
The smallest "non-trivial" cube: 8=23=2×2×2.
When specifying directions on a map, most people choose from one of these 8 directions: north, northeast, east, southeast, south, southwest, west, and northwest. These are the 8 directions a queen or king in chess can move. The knight moves in 8 directions too, but not the same 8.
In three-dimensional space there are 8 "diagonal" ways to move, corresponding to the eight "octants" you get if you divide the three-dimensional space with three mutually-perpendicular planes.
In 4-dimensional space-time, there are 8 non-diagonal directions: up, down, left, right, forward, back, future, and past.
Apart from the trivial cases of 0 and 1, 8 is the smallest number for which the sum of the digits of its cube is equal to the number: 83 = 512, 5 + 1 + 2 = 8. The largest number with this property is 27, and it is perhaps of interest that 8 and 27 are themselves cubes.
As mentioned in the 17 entry, 8 and 9 are the only pair of consecutive integers that are nontrivial integer powers of integers. See also 25.
9 is the largest single-digit number. It would also be the least frequently used digit if it were not for the tendency of businesses to set prices that end with one or more 9's. In situations where the number doesn't matter much (like street or apartment numbers) it is the least frequently used.
Because 9 is one less than the base of our number system, it is easy to see if a number is divisible by 9 by adding the digits (and repeating on the result if necessary). This process is sometimes called casting out nines. Similar processes can be developed for divisibility by 99, 999, etc. or any number that divides one of these numbers; see 11, 37 and 101 for examples.
When you were learning your multiplication tables you might have noticed that if you were dividing a 2-digit number by 9, you could check to see if the two digits add up to 9, and if they do the answer is the first digit plus 1, or 10 minus the last digit: 63 / 9 = 6 + 1 or 10 - 3. This idea can be extended to give an easy way to divide a three-digit number by 9:
1. To start with, you need to know that the number is divisible by 9:
The digits must add up to 9, 18 or 27. If they don't, subtract
enough from the 3-digit number so that the digits add up to 9 or 18
(the amount you subtract is the remainder that will be left over after
dividing by 9.)
2. Easy case: if it ends in 0, take the first two digits divided by 9, and add a 0 to the end. For example, 540 / 9 = 60, because 54 / 9 = 6 and you add a 0. You're done.
3. Otherwise, take the first digit, followed by the last digit subtracted from 10. For example, 477 / 9 gives 43: a 4, followed by a 3 which is 10 - 7. (This obviously only works if the 3-digit number is a multiple of 9 to start with, which is why you had to subtract the remainder in step 1.)
4. If the result from step 3 is less than or equal to the first two digits of the original number, add 10 to get the answer. Since 43 is smaller than 47 (the first two digits of 477) we need to add 10 to get 53.
Another example: 819 / 9: Step 3 gives 81, but 81 is equal to the first two digits of 819 so we add 10 to get the answer, 91.
This division technique is part of my method for testing divisibility by 27.
π squared. In certain ancient cultures it was believed (or assumed for convenience) that π was the square root of 10.
10 is both a triangular and tetrahedral number. 10 is also composite, semiprime, etc.
Number of fingers on a typical human. 10 has many other cultural properties resulting from that, or indirectly through other cultural properties (the use of 10 as our base is a cultural phenomenon).
There are three ways to test for divisibility by 11.
The first, and more commonly known, is to alternately add and subtract digits starting from the right. For example, to test the number 1234 you would compute 4-3+2-1. The original number is a multiple of 11 if and only if the answer is a (positive, negative or zero) multiple of 11 (in this case we get 2, so the answer is no).
Another method is to add digits in groups of two starting (again) from the right, and repeat the process if necessary, until you get 2 identical digits (multiple of 11) or something else (not a multiple of 11). To test 51381 this way, we'd add 5+13+81 to get 99, which is two identical digits, so 51381 is a multiple of 11.
However, the third method is the most useful, because it also gives the value of the quotient. It works by repeatedly subtracting the last digit from the remaining digits: 5138-1=5137, 513-7=506, 50-6=44, 4-4=0. If this process results in 0, the original number was divisible by 11, and the sequence of last-digits gives the quotient: The last digits were 1, 7, 6 and 4 so 51381/11=4671.
The word "eleven" does not fit into the pattern of the numnbers 13 to 19; the original word in Old English probably meant "one left over".
The main reason why so many things are grouped in 12's (inches, months, donuts, hours) is because 12 can be divided evenly in more different ways than any other number of its size: It's divisible in 4 non-trivial ways (2, 3, 4 and 6). The next record-setter is 24 (hours in a day; case of beer), which is divisible in 6 different ways. Other popular division numbers like 60 (minutes, seconds) and 360 (angular degrees) are also factorization record-setters.
Numbers that set records for number of factors are sometimes called "highly composite" numbers (Sloane's A002182). Here are the record-setters, arranged in a way that helps illustrate a couple points mentioned later:
| 1 | |||||||
| 10080 | |||||||
| 1081080 | |||||||
| 110880 | |||||||
| 12 | 120 | ||||||
| 1260 | |||||||
| 1441440 | |||||||
| 15120 | |||||||
| 166320 | |||||||
| 1680 | |||||||
| 180 | |||||||
| 2 | |||||||
| 20160 | |||||||
| 2162160 | |||||||
| 221760 | |||||||
| 24 | 240 | ||||||
| 2520 | 25200 | ||||||
| 27720 | 277200 | ||||||
| 2882880 | |||||||
| 332640 | |||||||
| 36 | 360 | 3603600 | |||||
| 4 | |||||||
| 4324320 | |||||||
| 45360 | |||||||
| 48 | |||||||
| 498960 | |||||||
| 5040 | 50400 | ||||||
| 55440 | 554400 | ||||||
| 6 | 60 | ||||||
| 6486480 | |||||||
| 665280 | |||||||
| 720 | 720720 | 7207200 | |||||
| 7560 | |||||||
| 83160 | |||||||
| 840 | |||||||
| 8648640 |
Some interesting things to note:
The word "twelve" does not fit into the pattern of the numnbers 13 to 19; the original word in Old English probably meant "two left over".
The word dozen comes (through French) from Latin duodecem ("two-ten")44 and thus is more similar to thirteen than most people realize.
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