thevirtualfer said:
I know that 1, 00001, 0000000000001, are the same quantity.
But how many "places" are on the left of the digit 1 ? Infinity ? Is ...001 a number ?
in a real sense, ...0001 is not a number, nor are 01, nor 1 nor 27 nor 39.4
These are all possible
representations of a number, but the number they represent is a far more abstract thing.
You might try to define the number as an intuitive notion of "oneness" - something that 'one flower', 'one planet', 'one grain of rice' or 'one forum post' all have in common. Because this is intuitive, it is unsatisfactory as a basis for logical reasoning.
You might try to represent the number via a set of axioms... that there is a number which we will call 0, that x+0=x for any number x, that any number has a successor, and that "one" is the successor of 0. This is good for logic, it means we can prove things about numbers, such as the fact that 2+2=4, 3^2+4^2=5^2, every number has a prime factorisation, and that x^n+y^n=z^n has no solutions in integers x,y,z>0, n>2.
One thing we can prove is that any number has a unique representation as a string of digits - so that if we say '384' means 3 x 10^2 + 8 x 10 + 4, that's the only way to write 384 as a string of digits that doesn't begin with 0. We can define rules for adding or multiplying strings of digits, and prove that the strings of digits you get via these methods represent the actual sums and products of the numbers you were trying to add or multiply.
We could then allow strings of digits to start with zeroes... we could even allow infinite strings of digits... then we'd find that ....00000000000000 represents 0, .....00000000001 represents 1, .....999999999999 represents -1 (since .....9999999999 + ....0000000001 = ...0000000000), and ...333333333333 represents -1/3 (since ....00000003 x ...3333333 = ...999999).
Keep in mind that this is a non-standard way to represent numbers, that there may be numbers represented like this that can't be represented as a finite string of digits, and they become something interesting and exciting to think about... Eg, what number is ....66666666666667 ? What string of digits represents 1/7 ?
Have fun!
