After serious thinking conway wrote : >> Zero is the name for no quantity. Its only real use >> is to set bases. >> >> Mitchell Raemsch > > > ANYTHING that is empty...still contains space. You can have "no quantities" > of ANYTHING....that does not mean you have no quantities of space. There is > always space.....you can NOT have NOTHING.....zero is a empty quantity of > space....not an empty quantity of nothing. > > Zero's real use is as a "point of reference" on a number line.......to "set" > bases.....here I agree....
I disagree. Zero's real role is additive identity. One's real role is multiplicative identity.
Going out on a limb...
Two is the first number. There is no need to count one thing, but when you have 'another thing' you have what we now call two and sort of 'back define' one as how many you used to have. Seemingly a mathematician would also have the concept of ratio (I have two times as many as before) and the concept of addition (I have one more than before) follow naturally. The concepts of 'another another thing' and another 'nuther 'nuther thing and so on follows and one can use as many symbols as we want to represent the new inductively discovered (or created) numbers.
With the concept of ratio (I have twice as many as before) and the addition notion, the inverses are desired, so we create (or discover) the integers for addition and subtraction and the rationals for multiplication and division.
Enter the reals:
After seemingly filling the whole number line (or space) they discover (or invent) the 'number' squareroot of two. It's not on the line anywhere, there must be spaces (holes) in our number line. To fill these holes we define them as the spaces strictly between the rationals and call the reals complete and continuous.
Additive exponentiation becomes deprecated due to multiplication, but multiplicative exponentiation still means 'how many' elements are being composited rather than 'how much' of some effect to apply. Geometry and algebra conspire to create (or discover) a polar notation way to describe newly discovered (or invented) complex numbers making use of imaginary numbers and a 'how much' interpretation of exponentials.
The reals get named as such to distinguish them from the imaginary numbers used in the complex plane.
Zero and one are fundamental building blocks, and there is no 'problem' concerning them. There is a perceived problem though when students think that Calculus only approximates answers due to the problem with zero appearing as a denominator in a 'ratio' that need not ever be calculated at all.
Good luck to you in your endeavor, and thanks for dumping that horrible math forum in favor of Google Groups -- even though GG sucks compared to a real NNTP client.