I realize that we have been taught all of our lives that zero isn't a number, rather it's a placeholder. I have thought about this and come to the conclusion that zero is the only whole number and any other number is an increment or fraction of the base. Counting is a logical process of repeating a cycle of increments contigent upon the base. Zero is either the beginning or end of a base cycle. When the base count is achieved, then the cycle resets to zero and repeats itself. The following is how I have derived my logical conclusion: Base Ten has 10 increments to reach the base. 0=Terminal Start 1 = 10% 2 = 20% 3 = 30% 4 = 40% 5 = 50% 6 = 60% 7 = 70% 8 = 80% 9 = 90% 10 = 100% Terminal Stop Base two has only 2 increments to reach the base. 0 = Terminal Start 1 = 50% 10 = 100% Terminal Stop In either cycle of counting each increment is a fraction of the base, except for zero. Zero percent isn't a fraction. Thus, zero is the only whole number. In a binary table, zero in the one's column will always equate to an even number. One in the one's column will always equate to an odd number. Any other column is a catchall to hold the overflow of zero or one. Example: 111111111110 = Even 000000000001 = Odd Example A light switch is in the off position. Below indicate if the light switch is on/off based upon how many times the switch is flipped. Switch flipped 1234 times = [ ] ON [ ] OFF Switch flipped 4321 times = [ ] ON [ ] OFF The answer is not based upon how many times the switch is flipped but whether it has been flipped an even or odd number of times. A switch flipped an odd number of times is in the opposite state. A switch flipped an even number of times is in the same state. The reason is that zero is an even number and represents the completion of a cycle. One is an odd number and represents that a cycle is incomplete. Any incomplete cycle produces an imbalance. The completion of a cycle produces a balance. If we examine any mathematical equation the problem and solution will always balance with each other. The equal sign symbolizes the state of balance. EXAMPLE 2 + 2 = 4 or 2 - 2 = 0 or 2 X 2 = 4 or 2 / 2 = 0 The problem is assigned to the left side of the equation and the solution is assigned to the right side of the equation. Each side balances with each other. The solution to any problem is to create a balance. Zero = balance One = imbalance Whenever we have problems then it means that a one is hidden in the equation creating an imbalance. Eliminate the one and the problem is solved.
What? Zero is a number. Who told you otherwise?! What?! Do you know what whole numbers are? There is not a logical way to derive whole numbers. They just are. They are definition. Whole numbers = {0, 1, 2, 3, 4, ...}. This is only by custom. Counting does not have to "cycle" (like 1, 2, 3, 4, 5, 6, 7, 8, 9, 10... or one zero). I could make up symbols as I keep going. However, it is much easier to cycle. As I said, you cannot logically conclude zero is a whole number. It's by definition. Thus however your argument works (which I cannot understand) is meaningless.
I've seen a proof for negative zero. Just because -0=0 doesnt mean you cant have a proof where you end up with a negative 0. If I can remember the name of the guy who did it, I will post the proof here and we'll see if the proof holds up under scrutiny.
There isn't a logical way to derive whole numbers they just are. That is an illogical way to explain your answer.
What? What is the highest math course you took? And how long ago what that? I got a lot of books that would disagree with you.
The concept of the integer is fundamental to so much of mathematics. You can't define rational and irrational numbers without it. We wouldn't have Pythagorean triangles, Fermat's Last Theorem, Fibbonaci Series, or the multiple classes of infinities. There's got to be a rigorous definition of "integer" somewhere!
One way of looking at it comes from elementary group theory. The integers form a group under addition. The identity element is 0, which is a member of the group.
So do a lot of subsets of integers... like all numbers divisible by 4. Do my 2-cents contribute to this discussion? No.
let me define the set of positive integers using the notion of an inductive set. An inductive set S is any set containing the number one and also containing k + 1 whenever k is a member of S. The set of positive integers is the smallest inductive set. -S, the set of negatives of S, S itself, and 0 make up the set of integers. 0 is an identity element according to the field axioms. Maybe that clears up something.
Group theory? (G, #) such that 1) a and b in G... a # B is in G 2) There exists an e in G such that e#a = a#e = a 3) For an element a in G, there exists an element we will call a<sup>-1</sup> in G such that a#a<sup>-1</sup> = a<sup>-1</sup>#a = e. 4) (G, #) is associative. That is, for a, b, and c in G, (a#b)#c = a#(b#c).
By inspection, find the solutions to the equation x = x * x Let a, b, c... be those solutions. (I'm not messing around with subscripts in this font. Besides, I don't have subscripts if I don't have integers, do I?) Let O = a + b + c ... (I can't count the number of solutions and say there are exactly two, since I don't have integers.) Define "positive integers" as the set O, O+O, O+O+O... Define "zero" as O - O Define "negative integers" as the set zero - O, zero - O - O, zero - O - O - O... Does that do it?
What is it you are trying to make? Anyway... {a, b, c, d, ...} = {1, 0}. So O = 1. But, what is it you are trying to solve? I am a bit perplexed.
Oh, bad presentation, sorry. What I'm trying to do is define the term "integer" without using concepts that only graduate students can understand. This seems to me to identify 1 as the first-discovered member of the set of integers, and provide a way to both identify and order the others. The question was, how do we define whole numbers, and the answer was that it's really difficult and requires higher math. I don't think so. Am I right?
In that vein, would it be sufficient to define the integers as those number that can be obtained by adding and subtracting ones? Or more formally, how about "the intersection of all additive groups that include one." Pete (not a mathemetician)
