In praise of the trinity

Date: Sun, 31 Oct 1999 12:46:15 -0800

Theory: Three is the most magnificent of all numbers.

The Evidence:
Three is the only natural number that is the sum of all the preceding numbers. It is the only number that is the sum of all the factorials of the preceding numbers: 3= 1! + 2!. In Babylonia there were three main gods: the Sun, Moon, and Venus. In Egypt there were three main Gods: Horus, Osiris, and Isis. In Rome there were three main Gods: Jupiter, Mars, and Quirinus. In classical literature, there were Three Fates, Three Graces, Three Furies.

In languages, there are three genders (masculine, feminine, and neutral) and three degrees of comparison (positive, comparative, and superlative). Note that German Chancellor Otto von Bismarck signed three peace treaties, served under three emperors, waged three wars, owned three estats, and had three children. He also organised the union of three countries. His family crest bore the motto 'in trinitate fortitudo' ("In trinity, strength"). There is also a German saying, 'alle gute Dinge sind drei ("All good things come in Threes").

Also, it is written that an Arahat once presented this number to a student and said: "What do you find significant about 69,696?".

The student entered the Void for a few seconds of eternity, and replied: "That is too easy, Master. It is the largest undulating square known to humanity."

(An undulating number is a number of the form abababab... For example, 232 232 and 56 565 are both undulating numbers.)

69,696 is a palindrome of 696, which in turn is a palindrome of 69. 69 is 3 multiplied by 23, in other words it represents the most chaotic and unholy form of the Trinity.

Also, the largest narcissistic number in base 4 (note that 4=3+3/3, a trinity of 3s) is 3303 (note the trinity of 3s). For the unenlightened, narcissistic numbers are numbers 'in love with themselves'. Variously called Armstrong numbers, or perfect digital variants, they are numbers that are the sums of powers of their digits, e.g. 153 is a narcissistic number because 153 = 1^3 + 5^3 + 3^3.

In 1911, the brilliant mathematician Srinivasa Ramanujan presented an equation, containing an infinite set of nested roots, to the Journal of the Indian Mathematical Society. Not one of the journals readers could determine a solution. The answer was, of course, 3.

In Number Theory, "Super-3" numbers are integers i such that, when raised to the power of 3 and then multiplied by 3 (i.e. numbers of the form 3i^3), contain three consecutive 3s (333!). (The smallest super-3 number is 261, because 3x261^3=53,338,743.) Anybody born in 1923, 1926 or 1928, is therfore under the special auspices of the Trinity because these are all super-3s.

Now for something simply mind blowing.....

Q: How many numbers contain the digit 3?
A: All of them.

The proof:

In the first ten numbers, there is only one number that contains the digit 3. This means that 1/10, or 10%, of the numbers have the number 3, when considering the first ten numbers. In the first 100 numbers the occurence of numbers with at least one 3 seems to be growing. In fact, there are 19 such numbers: 3,13,30,31,32,33,34,35,36,37,38,39.

From these observations we can derive a formula to describe the proportion of threes. This formula is 1-(9/10)^n. This formula can be best understood as follows. The probability of having a 3 as a digit in a one-digit number is 1/10 and of not having a 3 is 9/10. For a two-digit number, the probability of not having a 3 as the first digit or the second digit (that is, there are no 3s in the wo-digit number) is simply the product of not having a 3 for the first digit multiplied by the probability of not having a 3 for the second digit: (9/10)x(9/10)=0.81. The probability of having a digit 3 is 1-0.81. for a three-digit number we have (9/10)x(9/10)x(9/10)=0.729, and so on. For an n-digit number we therefore have the probability of not having any 3s:(9/10)^n.

We can use this formula to construct the following table:

The first n numbers      number of numbers that contain the digit 3
  10                              1
        100                             19
        1000                            271
Note that the 'number of numbers that contain the digit 3' is rapidly increasing, indicating that almost all numbers have a 3 in them. If we were to continue this table to encompass the infinite number of numbers, it would indicate that all numbers *do* have a digit 3 in them.

Try me.

The Purple Priest.

"Aleph zero bottles of beer on the wall,
Aleph zero bottles of beer.
Take one down,
pass it around,
Aleph zero bottles of beer on the wall."