Beauty of Parts

All numbers have parts, save One alone. Two has parts both of which are 1 and three has three parts all 1, which makes 3 a Prime number. Indeed 2 is the only Even Prime number since a prime number is one which is divisible only by itself and the Monad, or unit, 1.

But of numbers some are Prime, divisible only by themselves and the number of units which compose them, a number equal to their wholeness. Thus the parts of 5 are both 5 and 1 of which there are 5. In a sense these numbers are indivisible.

Of those which are composed of parts we have three varieties. Among the rarest of species is the Perfect, which is a number whose parts add up to the number itself. Such a number is 6, composed of 1, 2 and 3, for 6 has 6 units, three doubles and two triples to make it up.

The next perfect number is 28 composed of 14, 7, 4, 2, 1. All of these numbers have a certain aloofness and delicacy about them. As if they were somehow fragile. There is 1 perfect number between 1 and 10, one between 10 and 100, one between 100 and 1000 which is 496. These are rare indeed.

Of the other numbers some are deficient in which parts total less than the number itself and some are super-abundant in which the parts total more than the number itself.

Before exploring these it is worthwhile mentioning amicable numbers. These are another rare collection of numbers in which the parts of one total the other number, whose parts reciprocate by adding to the total of the first number. There is only one such pair below 10000, which the Greeks took as an indication of true friendship and what a rare thing this is. For the Greeks 10000 was the ideal size of a polis, since they reasoned we could recognise around 10000 people.

Lets consider deficient numbers first. These are those which often have only one other division besides the unit. Such a number as 4 as it is composed of 1 and 2, yielding 3, a number less in quantity than 4 itself. The next number to have parts is 6 which we have already considered and may discard, which brings us to 8. 8 is again in the duple series 1, 2, 4, 8, 16. It is the characteristic of these numbers that they are deficient by 1 in the total of their parts. We have seen 4 has parts totalling 3. 8 has parts totalling 7, 1, 2, and 4.
16, since we mention it, has parts 1, 2, 4, 8 total 15.

Above 8 we find 9 which has parts 1 and 3, yielding 4. This is interesting as both 4 and 9 are squares so it is significant to me that the second square in energy should have the value of its parts equalling the first square in energy. This is part of the elegance of numbers that make them such a joy and symbol of Beauty to me.

10 has parts of 1, 2 and 5, yielding 8. Again an elegant relationship. 8 is the first ‘established’ expression of numbers as the first cube. 10 is the completion of the series of numbers (in the decimal system which has been most common throughout the world prior to the advent of the computer), an establishment in its own right. Yet still deficient.

The first two deficient numbers, separated by two Primes and the first perfect number, are both in the duple series. In addition to this the next deficient number, 10, reflects back to 8 the second of the deficient series and a duple number.

It is only when we get to 12 that we find the superabundant species of numbers begins.

The parts of 12 are 1, 2,3, 4, and 6. Making a total of 16 – the next duple number.
12 is followed by 13 a prime,
14, a deficient number with parts 1,2 and 7 totalling 10 (!).

15 follows with parts 1,3, 5 adding to 9. The second square in expression.
16 a duple whose parts total 15.
17 a prime and it is not until we reach

18 that we encounter another superabundant number. The parts of 18 are 1, 2, 3, 6, 9 giving a total of 21.

19 is a prime again and
20 is the third superabundant number. 1, 2, 4, 5, 10 totalling 22. 22 is an interesting number we will consider elsewhere.

21 is again deficient, with parts 1, 3 and 7 totalling 11, a prime.

22 has for part 1, 2, 11 which add to 14.
23 is a prime and
24 is the next superabundant number with parts 1, 2, 3, 4, 6, 8, 12 adding to 36. the next multiple of 12.

25 returns us to deficient numbers with parts 1 and 5 giving a total of 6, the first perfect number.

In this way numbers can be analogous to, or imply, other values than those that immediately they represent themselves.

26 gives 16 with parts 13, 2 and 1.
27, the second cube in extension yields 13, itself an interesting number we shall consider again.

28 is the second perfect number,
29 is a Prime

30 is superabundant again with parts 15, 10, 6, 5, 3, 2, 1 totalling 42

31 is again Prime
32 a duple and so deficient by one or 31

33 is again a deficient number and gives parts of 11, 3, and 1 a total of 15!
34 is deficient with parts 17, 2, and 1, a total of 20. 17 is an important number as we shall see when we consider squares.

35 has a total of 13 in its parts 7, 5, 1 and
36, where I shall end this reflection a total of 55 with parts 18, 12, 9, 6, 4, 3, 2, and 1.

55 is the total of the numbers from 1 to 10 added together, a triangular number. (With parts 11, 5, and 1 yielding 17!)

In this collection of 36, 55 and 17 we have an interesting rounding off, for the article and more.