%path = "maths/numbers/representation" %kind = chindnum["texts"] %level = 9

The numbers are their own concepts independent from their representations.

The wikipedia article gives a great overview beyond the positional system described here.

Positional System

It is not possible to give every number its own sign. Instead we use signs up to a certain count and then one makes heaps (groups) of that count and starts counting these heaps.

Note

One can compare the numeral system with letters or phonetic systems. In a language one combines phonemes to produce a multitude, i.e. the words. These are associated/mapped to concepts. With Numbers digit signs are combined and then mapped to a count and beyond.

The Decimal System

Position coding:

Instead of writing 3 tens and 4 ones we write 3 at a position for the tens and 4 for the position for the ones: 34. This can be called position coding: Via the position we identify what we mean.

302 means 3 heaps of tens of tens (hundred), 0 (no) tens and 2 ones.

Position value:

The value of the position increases from right to left

… 10³=1000 10²=100 10¹=10 10⁰=1

These are powers of 10. 10 is the base of the decimal system.

Fractions:

As heaps of 10 get a position also fractions in 10th get a position to the right of the dot (.)

,1/10¹=1/10 1/10²=1/100 1/10³=1/1000 …

The Binary System

In the dual system two things make their own heap.

Together with the 0 the binary system has 2 signs, which mean: there or not there

The position values of the positions are these

… 2⁴=16 2³=8 2²=4 2¹=2 2⁰=1 . \(2^{-1}\) \(2^{-2}\)

Example

1011₂ = 11₁₀

The binary system is important, because computers use it and because 2 is the smallest quantity one can still choose from.

The Hexadecimal System

Here we make heaps of 16.

Together with the 0 we have 16 signs:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

The new ones are A=10, B=11, C=12, D=13, E=14, F=15.

The position values are:

… 16⁴=65536 16³=4096 16²=256 16¹=16 16⁰=1 . \(16^{-1}\) \(16^{-2}\)

Because of 2⁴=16, one needs 4 binary digits for one hexadecimal digit. Since the binary system is important, the hexadecimal is important, too, and so are other systems with base power of 2, like. Base 8 (octal), 64 (base64), 128(ASCII) and 256 (ANSI).

Duodecimal System

Twelve has many divisors: 2, 3, 4, 6 This allows an easy representation of fractions with these denominators.

But as with the decimal system (1/3 = 0.333…) the duodecimal system has easy fractions that are periodic (1/5 = 0.2497 2497…).