## Notes on Representation – Prime Bags

In our normal decimal number system, we represent quantity by writing down pieces that are supposed to be added together: 23 has a twenty piece and a three piece, and together that makes twenty-three.  The basic building blocks in this system are one, ten, hundred, thousand, etc.  With these building blocks, “23” is short hand for {ten, ten, one, one, one}, or {2 tens, 3 ones}.  Roman numerals are another short hand for this: XXIII is also short hand for {ten, ten, one, one, one}. The decimal number 103 is short hand for {1 hundred, 0 tens, 3 ones} or {hundred, one, one, one}, in Roman numerals CIII.  The decimal number 1002 is short hand for {1 thousand, 0 hundreds, 0 tens, 2 ones} or {thousand, one, one}, in Roman numerals MII.

Eachof the above notations has an underlying assumption that the pieces are all to be added together.  Could we construct another system of representation of quantity that is based on multiplication rather than addition?  I’ll show one way we can do that in this post, a way that is based on prime numbers.  Let’s be clear that I’m not suggesting you drop the decimal number system in favor of the this one.  I’m offering an alternative representation of quantity, to suggest that our elementary school experience of addition being easy and multiplication hard may be an artifact of the representations of quantity we are used to.

In the prime bag system, the building blocks are prime numbers: two, three, five, seven, eleven, etc.  (To these, we would later have to add the number zero, half, negative one, and others, to extend the system beyond the counting numbers.)  We can put these prime building blocks in a bag, e.g. [two, two, five].  This bag now represents a quantity.  What quantity does [two, two, five] represent?  Well, in this system the pieces are supposed to be multiplied.  Two times two times five is the quantity twenty, so [two, two, five] represents the quantity twenty.  The prime bag [three, seven] represents the quantity twenty-one.  Note that, for us who are used to the decimal system, it seems difficult to have to multiply all these pieces two, two and five.  But in this system, you don’t need to do the multiplication any more than you have to do the addition in the decimal representation {thousand, one, one} for the quantity thousand-two.

In the prime bag system, multiplication is easy.  To multiply [three, five] with [two, three] I merely toss the contents of the two bags together in a new bag: [two, three, three, five].  I’m done!  I’ve got the answer, at least in this system of representation.  The quantity represented by [three, five] is more commonly known as fifteen, the quantity represented by [two, three] is more commonly known as six, and the result of multiplication, [two, three, three, five] is more commonly known as ninety.  Multiplication works, and it is easy.

One way to look at the various representations is that they allow us to defer certain computations.  In the normal decimal representation 23, we’re not actually having to add twenty and three.  We don’t need to!  It is left implied in the representation.  Similarly, with the prime bags, the representation serves to defer the multiplication of primes.  As in any representation, certain aspects are highlighted at the expense of other aspects.  In the prime bag representation, the make up of numbers into prime factors is highlighted.  In this representation, not only multiplication is easy, but also factoring.  Simplifying fractions is really easy when numerator and denominator are both represented as prime bags.

Representations suppress certain aspects, as they highlight others.  In the normal decimal representation, prime factorization is not easy to do – in fact, modern systems of cryptography are based precisely on the difficulty of factoring numbers presented in the decimal system (or variants like the binary system).  Conversely, the downfall of the prime bag system is that comparison is fairly hard to do: is [two, two, five] more or less than [three, seven]?  Which one represents “more” quantity?  The question can be answered, but not straightforwardly.

$\begin{matrix} \text{decimal} & \text{prime bag} \\ 1 & [ ] \\ 2 & \text{[two]} \\ 3 & \text{[three]} \\ 4 & \text{[two two]} \\ 5 & \text{[five]} \\ 6 & \text{[two three]} \\ 7 & \text{[seven]} \\ 8 & \text{[two two two]} \\ 9 & \text {[three three]} \\ 10 & \text{[two five]} \\ 11 & \text{[eleven]} \\ 12 & \text{[two two three]} \end{matrix}$

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