Number Systems

This page assumes:

Mathematics is a superset of logic operating on numbers (symbols). I will also use non-mathematical operators, but those belong to set theory

Numbers

  1. Numbers (in mathematics) is a symbol for an amount.
  2. Amounts may range from no amount to a infinite amount.
  3. Unique number symbols represent unique amounts.
  4. One number represents a greater or smaller amount than another number.

In other fields, number symbols might be used simply as unique identifiers. A example of this is the enumeration of my examples in formal language so they can be identified and explained in the natural language English afterwards.

Number systems

, we would need infinitely many symbols to represent all possible amounts. To make it not impossible to represent all numbers, we use number systems. Number systems can take all sorts of forms, but I will only describe the most widely used ones here.

The first thing we want to do is to reduce the number of required symbols from a infinite amount to a rational finite amount. We do this by introducing the concept of ciphers and base amounts. In a base ten number system (the most common one), one cipher can represent the whole amounts from none to nine. If we now define the symbol 1 as our fundemental amount, we can symbolically represent everything from one fundamental amount to nine fundemental amounts. Mathematics does not bother thinking about what 1 really is. It is just used as a fundamental amount used to compose new amounts (whatever it is). It might (in Physics) correspond to 1 meter or 1 second

$1 = 1$

$2 = 1 + 1$

$3 = 1 + 1 + 1$

$4 = 1 + 1 + 1 + 1$

$5 = 1 + 1 + 1 + 1 + 1$

$6 = 1 + 1 + 1 + 1 + 1 + 1$

$7 = 1 + 1 + 1 + 1 + 1 + 1 + 1$

$8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1$

$9 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1$

Now we have limited the amount we can represent with one cipher from infinity to the finite amount represented by the symbol 9. I will refer to these symbols as numbers for convenience from here on. To represent amounts greater than 9, we now use additional ciphers instead of inventing more symbols. The amount we in English refer to as twelve we can be expressed the following ways:

3 + 9

4 + 8

5 + 7

6 + 6 (one half of twelve)

7 + 5

8 + 4

9 + 3

But because it is impractical to represent simple amounts as expressions, we need to add some additional rules. the first thing we have to stop with is using operators and only number symbols. The next thing we need is a consistent way to use addition so that we are not limited to the amounts 1 to 9. The way we do this is by introducing the artificial amount of nothing. We use the symbol 0 to represent this amount. The next thing we do is the following: