Notable Properties of Specific Numbers  


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3.732050... = 2+√3 = tan(15o)

See 1.389910....

See also 3.732050....

4

Another number with zillions of properties: the smallest composite, a square, the first non-Fibonacci number, random (under the right circumstances124), a power of 2, etc.

The trivial solution of xy=xy is for x=y=2, and xy=xy=4.

4 is also the only x for which there exists a k such that hy(k, n, k) for all n (where hy(a, b, c) is the generalized hyper function).

4.236067...

The cube of the golden ratio phi, and the value of these three expressions all involving Φ and its square:

Φ2 × Φ = (Φ2}2 - (Φ)2 = (Φ2}3 / (Φ)3

Thus, Φ2 and Φ are the answer to an old algebra problem (from 78):

What two numbers are those whose product, difference of their squares, and the ratio or quotient of their cubes, are all equal to each other?

It is also equal to 1+2Φ. See also 2.618033...

4.6692016091...

(the (first) Feigenbaum constant)

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....

5

Many properties, mostly for trivial reasons (see 3.) The third prime, a Fibonacci number, a pyramidal number, etc.

5.290316093...

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.

6

A perfect number. See 496 and my largenum notes for more.

Also a triangular number, a factorial, composite, etc.

6.283185... = 2π

2 π is seen often enough that it is sometimes treated as a constant in its own right, separate from π itself (see also π/2).

There has been a grass-roots movement (beginning roughly at the time of Palais' 2001 opinion article in The Mathematical Intelligencer [167]) to replace the use of π with this constant. Palais called it "tau", but used a new "three-legged pi" symbol instead of the Greek letter τ, and described a "blight of factors of 2" throughout mathematics and physics that might be eliminated by replacing π with .

Indeed, using this constant, some formulas become a little more elegant: we have cos(x+)=cos(x) in place of the normal cos(x+2π)=cos(x); Euler's identity becomes ei=1 rather than eπ=-1 or eπ+1=0; and so on. However, "tau" is a fringe culture phenomenon, and not accepted for use in e.g. high school or university mathematics education.

6.5

According to Randall Munroe's What-If Numbers stream on Twitter, 6.5 is the "mass, in grams, of a kilometer-long strand of human hair". You'll find many similar numbers, most of them have units and/or are variable in value (example: "ounces of gold it would take to buy one ounce of 64GB MicroSD cards").

7

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 A6883). 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.

7.874007874... = √62

The square root of 62 has a remarkable digit pattern:

62 = 7.874 007874 011811 019685 034448...

The "7874" repeats 6 digits later, then we get "11811" (which is 7874×3/2) and then "19685" (which is 7874×5/2).

A similar pattern with an amazing similarity is seen in the square root of 2:

2 = 1.4 14 21 35 62 ...

Again we have "14" repeated twice, then 14×3/2 = 21, then 14×5/2 = 35.

These patterns of digits, and the recurrence of the ratios 2:2:3:5, is not a coincidence. They result from the fact that 62 times 127 squared equals 999998, which is a power of 10 minus 2; and similarly, 2 times 7 squared is 98. Note also that 62×127 = 7874, which are the first 4 digits of √62, and 2×7 = 14, the first 2 digits of √2.

For a full understanding, we start with

1/999998 = 0.000001000002000004000008000016000032000064...

a digit pattern featuring powers of 2 which is described in the 998 entry. Now it happens that 0.001000001 squared is 0.000001000002000001 (see 10201), and 0.001000001000001 squared is 0.000001000002000003...

Since 1/999998 = 0.000001000002000004... is a little bigger than this, it makes sense that its square root should be a little bigger than 0.000001000001000001:

1/999998 = 0.001000001000001500002500004375007875...

The sequence (1, 2, 4, 8, 16, ...) has become (1, 1, 1.5, 2.5, 4.375, ...) Those two 1's make the repeated digits possible.

Now instead of the square root of 1/999998, we want the square root of an integer. Since the factors of 999998 are 2×31×1272, we can multiply √1/999998 by 2×31×127, which is 7874. The 7874 goes inside the radical and becomes 78742, then most everything cancels out:

7874 √1/999998 = √7874×7874/999998 = √62×999998/999998 = √62

Therefore

62 = 7874 √1/999998 = 7874 × 0.001000001000001500002500004375007875... = 7.874007874011811019685034448812007863...

So we started with 999998, which is 2 fewer than a power of 10, and is divisible by 1272. The remaining part of 999998 is 62, and we found that the square root of 62 has the digits "7874", which is 62×127, and these digits are repeated.

We saw a sequence (1, 1, 1.5, 2.5, 4.375, ...). These numbers come from the Taylor series expansion of √1/(1-2x), which is:

1/(1-2x) = 1 + x + 3x2/2 + 5x3/2 + 35x4/8 + 63x5/8 + ...

1/(1-2x) is related to √1/999998 in a simple way: if x=10-6, then √1/(1-2x) = √1/(1-0.000002) = √1/0.999998. This has the same digits as √1/999998, but with the decimal point shifted 3 places over.

Because √999998 is close to √1000000 = 1000, the fraction 1000/127 is very close to √62. This is similar to the coincidence 10/7 ≈ 2, and see 1/127.

The coefficients of the Taylor series (the infinite polynomial) are the central binomial coefficients (OEIS sequence A000984) divided by the powers of 2: (1/1, 2/2, 6/4, 20/8, 70/16, 252/32, 924/64, 3432/128, 12870/256, 48620/512, ...). After reducing to simplest form, the numerators are OEIS sequence A001790 (1, 1, 3, 5, 35, 63, 231, 429, 6435, 12155, ...), and the denominators are OEIS sequence A060818, the highest powers of 2 that divide the factorial of N (1, 1, 2, 2, 8, 8, 16, 16, 128, 128, 256, ...).

Square Root Digits in Other Bases

This all generalizes to other number bases, not just base 10. Whenever a multiple of a square plus 2 equals an exact power of the base, there is an integer whose square root has a block of repeating digits when displayed in that base. Here are some of the more remarkable examples in other bases:

In base 2:
  √28 = 101.0100 1010100 11111111 ...
  √1820 = 101010.1010100 1010101010100 1111111111111 1010101010100 1110101010100 ...
  √116508 = 101010101.0101010100 1010101010101010100 1111111111111111111 1010101010101010100 1110101010101010100 1011111111111111111 0001111111111111110 ...
  √167772 = 110011001.100110011000 110011001100110011001 001100110011001100101 111111111111111111111 011111111111111111111 010011001100110011000 ...
  √85598 = 100100100.1001001001000 1001001001001001001000 1101101101101101101101 0110110110110110110110 0111111111111111111111 0111111111111111111111 0011111111111111111110 ...
In base 3:
  √3 = 1.2 012 021 222 ...
In base 4:
  √1820 = 222.2221111110333333311111032222211333330133331033331...
  √116508 = 11111.11110222222222133333333322222222213111111102333333332033333332...
In base 5:
  √1735 = 131.31 13131 22222 13131 21211 44444 .....
In base 6:
  √276 = 24.34 02434 04053 ...
In base 7:
  √977423 = 2611.43464 0261143464 0421565362 ...
In base 8:
  √85598 = 444.444422222221555555533333331777777677777771777777...
In base 16:
  √167772 = 199.998CCCCC999997FFFFEFFFFF4CCCC...

Here are some base 23 examples, in which almost every pair of digits adds up to 'M' (which is the highest digit in base 23):

In base 23:
  √16448427 = 7F7.F7 7F7F7 BBBBB 7F7F7 ACACA 5H5H5 ...
  √2434958270224475 = 7F7F7F.7F7F7 7F7F7F7F7F7 BBBBBBBBBBB 7F7F7F7F7F7 ACACACACACA 5G5G5G5G5G5 74747474746 BBBBBBBBBBA ...

8

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.

8.794143

A modern value of the "solar parallax", in arc-minutes. This is the angle subtended by the radius of the Earth at the mean distance from the Earth to the Sun. Measuring this angle was the purpose of the Venus transit observations in the 17th through 19th centuries used to establish the length of the astronomical unit and thus the scale of the solar system. See 149597870691 for more.

9

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.

Casting Out Nines

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.

9.869604...

π squared. In certain ancient cultures it was believed (or assumed for convenience) that π was the square root of 10.

10

(ten, the most common base)

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).

11

The number of dimensions in a superstring theory that unifies general relativity with the other benefits of string theories. 11-D superstring theory dates back to Cremmer, Julia and Scherk in 1978.

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 numbers 13 to 19; the original word in Old English probably meant "one left over".

Ours go to 11!

12

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.

Highly Composite Numbers

Numbers that set records for number of divisors are sometimes called "highly composite" numbers (Sloane's A2182). 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:

For more about highly-composite numbers, see 840, 45360, 720720, 3603600, 245044800, 278914005382139703576000, 2054221614063184107682218077003539824552559296000 and 457936×10917.

A long-standing and fairly famous problem in mathematics (the "kissing number problem") involved proving that one cannot arrange more than 12 spheres of equal size so as to touch a central sphere. It seems pretty obvious if you try it, but there is also a fair amount of space between the spheres and one could perhaps in theory work out some sort of asymmetrical arrangement that would allow a 13th sphere to be added. It was eventually proven that 12 really is the limit.

Related to the kissing problem is the sphere packing problem, which is determining how to fit the greatest number of equal-sized spheres into a given space. The familiar arrangement (seen in fruit stands for example) has each sphere touching 12 others; this was also proven to be the best possible. The fruit-stand arrangement is called a "face-centered cubic lattice packing". The typical stack of fruit in a fruit stand (usually a tetrahedron or a square pyramid) does not suggest the notion of "face-centered cubic", however they are actually equivalent (in the case of the tetrahedron stack, by rotating the entire lattice). To describe this with coordinates: spheres of radius √2/2 can be placed at every point (x,y,z) such that each of the three coordinates is an integer and the sum x+y+z is even; in such an arrangement each sphere touches 12 others. (For example, the sphere centered at the origin (0,0,0) touches four centered at (0,±1,±1), another four at (±1,0,±1) and four more at (±1,±1,0)).

A cuboctahedron is a solid with 12 vertices and 14 sides (six squares and eight triangles). This shape, along with octahedra, can be used to completely fill space in all directions; each cuboctahedron touches six other cuboctahedra. If you ignore the octahedra, the cuboctahedra line up in a cube-like grid arrangement. When this is done with cuboctahedra of just the right size, the space-filling arrangement places the centers of the cuboctahedra at every position (x,y,z) such that all three coordinates are even integers, and places the vertices at every position (x,y,z) such that one coordinate is even and the other two odd; these are exactly the same coordinates as the just-described face-centered cubic lattice sphere packing.

The word "twelve" does not fit into the pattern of the number-names 13 "thirteen" to 19 "nineteen"; 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.

13

To test a number (example 746245952673) for divisibility by 13:

Another method goes like this:

14

A square pyramidal number, and also composite.

A Rhombic dodecahedron is a three dimensional figure with 14 vertices positioned like the 8 vertices of a cube combined with the 6 vertices of an octahedron that is suitably scaled (so that all 12 of its rhomboid faces are planar). Rhombic dodecahedra can be used to completely fill space in all directions; each shares a face with 12 neighbors. The centers of the rhombic dodecahedra then coincide with the centers of spheres placed in a close packing in a symmetric, regular repeating lattice pattern with each sphere touching 12 others, a "face-centered cubic" lattice. The 4-dimensional analogue of the rhombic dodecahedron is the 24-cell.

14 is the lowest base with 'easy' divisibility tests for 5 different primes, assuming that the casting out 11's method is considered 'easy'.

In base 10 there are fairly easy tests for divisibility by the prime numbers 2, 3, 5 and 11. The lowest base with easy tests for four primes is base 6 (2, 3, 5 and 7). In base 14, you can test for divisibility by 2, 3, 5, 7 and 13.

The record-setters for this property (bases with high numbers of testable primes) are shown here. The primes in bold are tested just by looking at the last digit; the primes in the plain font are tested by the digit-addition technique (casting out 9's); the primes in italic are tested by alternate addition and subtraction (see the entry on 11 for a description):

base N prime divisors
2 2 2, 3
4 3 2, 3, 5
6 4 2, 3, 5, 7
14 5 2, 3, 5, 7, 13
34 6 2, 3, 5, 7, 11, 17
154 7 2, 3, 5, 7, 11, 17, 31
286 8 2, 3, 5, 7, 11, 13, 19, 41
714 9 2, 3, 5, 7, 11, 13, 17, 23, 31
5796 10 2, 3, 5, 7, 11, 17, 19, 23, 31, 61
18446 11 2, 3, 5, 7, 11, 13, 17, 23, 31, 43, 401
90364 12 2, 3, 5, 7, 11, 13, 19, 29, 31, 41, 53, 331
304590 13 2, 3, 5, 7, 11, 13, 17, 19, 23, 41, 53, 71, 821
1013726 14 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 41, 43, 103, 139
5617821 15 2, 3, 5, 7, 11, 13, 17, 29, 31, 37, 41, 43, 101, 107, 137
23738714 16 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 41, 47, 97, 127, 157, 173
123702370 17 2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 53, 73, 83, 137, 151, 797
556876530 18 2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 37, 43, 67, 79, 101, 109, 167, 241
3760113566 19 2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 53, 61, 67, 71, 73, 131, 853, 929

Notice that all of these except 5617821 are even.

If you want the record-setters for all divisors (not just prime divisors), check the entry for 29. If you don't think the casting out 11's method should count, see the entry for 66.


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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.


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