Is Zero a Whole Number?
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Pete said:
Z is called an infinite cyclic group. The informal definition of this is a group (a set and an operation as defined by a previous post by me) that can be defined by the products of one element in the group. These products are g<sup>0</sup>, g<sup>1</sup>, g<sup>2</sup>, ... and g<sup>-1</sup>, g<sup>-2</sup>, g<sup>-3</sup>, ... Product just refers to the opperation. In this case, addition. So, g<sup>-2</sup>, = g<sup>-1</sup>g<sup>-1</sup>, (-1) + (-1) = -2. The element in g is 1.
I think I did a bad job of explaing that. Just look for it in wiki or mathworld.
Zephyr said:
As far as I know... one and zero "just are." It's an assumption that 0 != 1. But nothing is said about 2, -2, 2.435... ect.
However, a neat proof I did one time was prove that 1+1 != 0 or 1. Since that is the case.. what do you do? You make up a number. In this case, we call it "2." Then you do the same thing for the next numbers.
Show 1 + 2 != 0, 1, 2. So call it 3.
Show 1 + 3 != 0, 1, 2, 3. So call it 4.
Then you just show a general rule for defining new numbers.
It seems like a coward way to go.. .define a new number. However, don't you remember imaginary numbers? Sqrt(-1). WTF? Well, let's just call it i.
Well, one 'is' in rings with unity ... 2Z with the usual operators, on the other hand, has no multiplicative unitAbsane said:
And I think giving names to stuff you don't understand is a better approach than ignoring them ... wasn't there a group of Greeks who insisted there could be no irrational numbers? That must have put a bit of a dent in their research ability...
"Hey Obstinopolis, you've spent five years looking for the fractional form of the squareroot of two. Don't you think you should take a break?"
"You're right, Tumuchondaplato ... I'll start on the squareroot of three tomorrow."
I think it's quite alright to look at x<sup>2</sup> - 2 = 0 and think... wow. What is x? Well, it seems I can get closer and closer to a number but cannot quite find it. Why not just call whatever this number is Sqrt(2)? There. Problem solved.
It's funny but hey.. what else can you do? WHen you work with faulty assumptions, you don't get anywhere. Assuming sqrt(2) is a fraction of two integers leads to a contradition.
It's funny but hey.. what else can you do? WHen you work with faulty assumptions, you don't get anywhere. Assuming sqrt(2) is a fraction of two integers leads to a contradition.
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Fraggle Rocker
Staff member
No. 0!=1 follows from extending the recursive form of the definition of factorials downward. It's not exactly an assumption, but more properly a convention.Absane said:
(n+1)! = n!*(n+1), therefore
(n-1)! = n!/n
There is no definition for negative numbers because of the division by zero problem. I suppose by this same convention (-1)! = infinity.
As for non-integers... Has anyone tried to find a function that provides the right solutions for integers, which could then be interpolated for fractions?
Fraggle Rocker said:
For negative numbers.. there is a definition for cominations and such. Like:
C(-4,2) = (-4)*(-4 - 1)/2!.
And for non-interegers.. the same thing applies. I got a formal definition somewhere but I don't think I really need to dig it up. It's used for things like Newton's Binomial Theorem.
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I have never met anybody who would agree with this.
Counting is not a process of repeating a cycle of increments contigent upon the base. Only for convenience do we use a base to count. What about match stick counting, i.e. 1, 11, 111, 1111, 11111...? How is this repeating a cycle of increments dependent upon a base? How would your idea work if you were counting in Roman numerals? And even in base counting the cycle doesn't reset to zero, otherwise we'd be trapped counting in modular arithmetic where 10=0 (in base 10).
DJ Erock said:
Do you see what he was doing? He made a faulty analogy.
2 + 2 = 4 : 2 - 2 = 0 :: 2 X 2 = 4 : 2 / 2 = 0.
Division is defined in terms of addition and multiplication. Subtraction is defined in terms of addition. He assumes that multiplication is repeated addition (in some ways, it is.. but not in general). He also assumes that division is repeated subtraction (in some ways, yes.. but not in general). Again, division is defined in terms of multiplication and addition, not subtraction.
Much of this thread is nonsense which seems to have been pointed out as such by others.
As far as I know, mathematicians do not view zero as having a sign, although there are functions which approach zero from below. I have never heard of any claim that the limit of such a function is different from the limit of a function which approaches zero from above.
The only context in which I have encoutered negative zero is in the arithmetic units of (circa 1957-1963) computers, using signed magnitude values. For at least one such computer, the machine language branch (Goto) if zero command did not branch (go) for negative zero. This caused a bit of extra work for competent programmers.
Due to an incompetent programming staff at some department stores, an inactive account could be dunned for a negative zero balance. Almost any activity would get rid of the anomaly. It occurred if you had a credit balance of $X and charged an item costing exactly $X.
Several of us amused ourselves by deliberately causing a negative zero balance and collecting the dunninbg letters, climaxed by one threatening legal action for failure to pay the zero balance, which appeared on statements as zero without a sign.
As far as I know, mathematicians do not view zero as having a sign, although there are functions which approach zero from below. I have never heard of any claim that the limit of such a function is different from the limit of a function which approaches zero from above.
The only context in which I have encoutered negative zero is in the arithmetic units of (circa 1957-1963) computers, using signed magnitude values. For at least one such computer, the machine language branch (Goto) if zero command did not branch (go) for negative zero. This caused a bit of extra work for competent programmers.
Due to an incompetent programming staff at some department stores, an inactive account could be dunned for a negative zero balance. Almost any activity would get rid of the anomaly. It occurred if you had a credit balance of $X and charged an item costing exactly $X.
Several of us amused ourselves by deliberately causing a negative zero balance and collecting the dunninbg letters, climaxed by one threatening legal action for failure to pay the zero balance, which appeared on statements as zero without a sign.