%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.
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.
For a count below ten there are signs: 1, 2, 3, 4, 5, 6, 7, 8, 9.
For “none” there is 0. Together these are 10 signs.
For a count ten and above one makes heaps of ten and counts these separately.
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.
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 …
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.
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).
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…).