Our number system?

Question

As I take more and more statistics classes, I'm amazed at what can be determined with numbers. Therefore I often wonder--is our number system considered perfect? If not, what weaknesses does it have?

Answer 1

I presume you're meaning base 10 notation... The concept of a number system being perfect doesn't work for me; my point is that any number system is an arbitrary system for representing quantity, and as long as it is convenient and comprehensible to the one using the system, it's fine.

Several systems are actually in use, although polynomial (or positional) notation in base 10 is by far the dominant one. These systems include:

positional notation in base 2, 8 or 16

Factorial notation

Odd Primorial

Primorial

P-Adic

and I'm sure others I have no clue about.

No one system is perfect. Base 10's greatest weakness is that it is somewhat cumbersome to convert to and from base 8 or 16, which have been heavily used in computer work. Ohter than that, there's no really major trouble with it, and it is so familiar...

Answer 2

We use the base-10 number system for historical & cultural reasons -- mostly because we have ten fingers and ten toes. Fingers are good for counting.

But the ancient Babylonians had a better system ... they used base 60!

Somewhere in-between is base-12.

The reason these others could be better is because of how the composite numbers factor. 10 = 5 x 2, and that's it. 12 works better: 12 = 4 x 3 = 6 x 2 = 3 x 2^2. And 60 is awesome:

60 = 12 x 5 = 15 x 4 = 20 x 3 = 30 x 2 = 5 x 3 x 2^2.

Think about the system of money as an example ... using pennies, nickels, dimes, and quarters is not very efficient. Who likes to have four pennies in their pocket? Who likes to have a pocketful of quarters when they go to the laundromat to wash clothes?

Actually, a good number system might be based on the product of primes 2 x 3 x 5 = 30. Then suppose you wanted to represent some large number, say, 72,398,043 (decimal). In base 30, this is 2(30^5) + 29(30^4) + 11(30^3) + 12(30^2) + 8(30^1) + 3(30^0) = 48,600,000 + 23,490,000 + 297,000 + 10,800 + 240 + 3 = 72,398,043, and it would be written as a 6-digit number, something like this:

72,398,043 (decimal) = 2wbc83 (base-30)

where w, b, and c respectively represent the base-30 digits for 29, 11, and 12.

(I used this for the base-30 digits:
0 1 2 3 4 5 6 7 8 9 a b c d e
f g h j k m n p q r s t u v x)

It looks confusing, but that's just because we're not used to it.

You might say there are too many different digit characters in a base-30 system, but that's not so. The alphabet has 26 characters, and you didn't have any trouble learning that, did you?

Answer 3

This depends on what you mean by "perfect". A number system serves a particular purpose. For example, complex numbers is a superset of our number system. Complex numbers however, are not suitable for our usage but very appropriate for RCL (resistor, capacitor, inductor) calculations.

We mostly use a base 10 system. Computer uses base 2 or 8 or 16. So what is perfect depends on how suitable the number system is for that particular field.