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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.

3.141666... = 3+⁸/₆₀+³⁰/_60²

The approximation to pi used by Ptolemy.

3.142857...

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 a^b=a b. To show this, start with:

x^y = x y

then define l to be the log base x of y:

l = log_xy

then the first equation becomes:

x^xl = x x^l

which (for x>1) reduces to:

x^l = l+1

Solving for x, and substituting back for y, we get:

x = (l+1)^(1/l)
y = x^l = ((l+1)^(1/l))^l = l+1

This is the general solution for x^y = x y. (The same equations also happen to solve x_④y=x^y.)

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=x^y=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 x^y=y^x.

3.162277660168... = √10

This has sometimes been used as an approximation to π

3.375 = ²⁷/₈ = (9/4)^(3/2) = (9/4) × (3/2)

The equation x_④y=x^y, where _④ is the lower-valued form of the hyper4 operator, can be solved easily. The formula for x_④y is:

x_④y = x^xy-1

so we're really solving:

x^xy-1 = x^y

If x>1 we have:

x^y-1 = y

and thus:

x = y^1/(y-1)

(If you change y to l you can see the similarity to the solution of x^y=y^x. The same values of l and x that solve the x^y=y^x equation also solve x_④l = x^l.)

We can also express x^y in terms of y:

x^y =(y^1/(y-1))^y =y^y/(y-1)

and now notice that this is equal to the product xy, because:

xy =yx =y y^1/(y-1) =y^{(y-1+1)/(y-1)} =y^y/(y-1)

Therefore, the solutions to x_④y=x^y also simultaneously solve x^y=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/a^a

The smallest non-trivial solution is when a=2, we get x=9/4 and y=3/2, and the value of x_④y =x^y =xy is 27/8=3.375.

3.5

Implicitly mentioned in the Bible in a few laces, see 1260.

4

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 x^y=xy is for x=y=2, and x^y=xy=4.

4.6692016091...

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(e^π) (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 e^x = gamma(x+1), and corresponds to the point where the Gamma function starts to exceed the e^x function.

6

A perfect number. See here for more.

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 A006883).

Tests for divisibility by 7 are difficult. About the best one I've found is as follows (demonstrated by testing 4156726):

• Remove the last digit and subtract twice this digit from the remaining number: 415672 - 2×6 = 415660 (You are subtracting 21×d and dividing by 10, where d is that last digit.)
• Repeat this process: 41566 - 0×2 = 41566, 4156 - 6×2 = 4144, 414 - 4×2 = 406, 40 - 6×2 = 28
• The result is a multiple of 7 if and only if the original number is.

There is another divisibility test that works as follows:

• Alternately add and subtract groups of 3 digits starting from the right: 726-156+004=574. (If the result is negative, drop the sign) (This is done because 7 is a factor of 1001.)
• If greater than 999, repeat the previous step.
• Add the 1's digit, 3 times the 10's digit and twice the 100's digit: 4+3×7+2×5=35. (The reason for the multipliers: 1 = 1 mod 7, 10 = 3 mod 7, and 100 = 2 mod 7)
• If desired, repeat previous step until result is small enough to tell at a glance if it's a multiple of 7: 5+3×3=14.
• The result is a multiple of 7 if and only if the original number is.

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.

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.

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

Number of fingers on the average 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

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".

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 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:

Some interesting things to note:

• The sequence sometimes jumps a lot (like from 2520 to 5040) and sometimes a little (like from 50400 to 55440). This behavior continues, but the gaps get generally smaller; the last time it jumps by a factor of 2 appears to be the jump from 2520 to 5040.
• If a number is a record-setter, that number times 10 is also sometimes a record-setter, but not the number times 100.
• All the factorials up to 7!=5040 are record-setters, but 8!=40320 and all higher factorials are not.
• The record-setters gradually incorporate bigger and bigger prime factors, but sometimes a prime factor is added, goes away, and then comes back. For example, the first multiple of 11 in the sequence is 27720; the next two (45360 and 50400) are not multiples of 11; after that 11 comes back to stay. Most higher prime factors come and go multiple times.
• The record-setters with repeating digits, starting with 720720, continue for about 20 terms. The digit-pattern results from the factors of 1001.

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.

13

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

• Remove the last digit, then add 4 times that digit to the remaining number: 74624595267+4×3 = 74624595279. (Each time you are adding 39d=(40-1)d and dividing by 10, where d is that removed digit.)
• Repeat until it gets down to 40 or less: 7462459527+4×9 = 7462459563; 746245956+4×3 = 746245968; 74624596+4×8=74624628; 7462462+4×8 = 7462494; 746249+4×4 = 746265; 74626+4×5 = 74646; 7464+4×6 = 7488; 748+4×8 = 780; 78+4×0 = 78; 7+4×8 = 39.
• If this process results in 0, 13, 26 or 39 the original number is a multiple of 13; otherwise it isn't.

Another method goes like this:

• Alternately add and subtract groups of 3 digits starting from the right: 673-952+245-746=-780. (If the result is negative, drop the sign.) (This is done because 13 is a factor of 1001.)
• If greater than 999, repeat the previous step.
• Add 4 times the 100's digit plus 3 times the 10's digit, and subtract the 1's digit: 4×7+3×8-0=52. (The reason for the multipliers: 1 = 1 mod 13, 10 = -3 mod 13, and 100 = -4 mod 13; the signs are all reversed to make the result be usually positive.)
• If necessary, repeat the previous step until the result is less than 40, or small enough to tell at a glance if it's a multiple of 13: 3×5-2=13.
• The result is a multiple of 13 if and only if the original number is.

14

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):

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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