Piflik said:
Coldie said:
Piflik said:
Actually, there really is no such thing as infinity. It is a theoretical concept, but it doesn't exist. Every number has an end. Always.
Infinity is very prominent a mathematical concept. It's quite real, I assure you. And, as redundant as it is redundant, infinitely long numbers are, in fact, infinitely long. They never have an end. Here, let me show you, an infinitely long number that has an infinite number of digits after the decimal point and no end ever:
Quite a famous transcendent number, especially prominent in trigonometry and geometry. Also it's infinitely long and has a finite value.
Some infinitely long numbers have a repeating pattern, for instance 1/3 is just infinitely repeating threes. The notation for such numbers is 0.(periodic pattern), so 1/3 = 0.(3) or 1/7 = 0.(142857).
So if you take 0.(9): 1 - 0.(9) = 1 x 10[sup]-infinity[/sup] = 0.
You can find the detailed calculations in the first couple pages of the thread.
Don't try to understand it, just accept it as the universal and absolute truth. Because that's what it is. Elementary Arithmetics.
So you agree with my prof that 0 = 1? Because if you want to do traditional maths (or Elementary Arithmetics as you call it...) with infinity, you would have to...
A very good argument
Let me debunk it *g*
At first, i will backtrack from some earlier statements and partially agree with what jaime wolf said earlier - though, probably in a different way than would be expected of me.
I'll abolish numbers. Or rather, i'll define that "number" is just the dividend of the base, in a base-unitsystem. In more simple words: 1/3 -> 1 is the dividend/number, and 3 the base.
With VALUE, i will mean the following: 1/3 and 1/4 - the "number/dividend" in both cases is identical, but the value is not.
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Okay, lets rock:
1. Let's start easy. We'll enter 1/3 into our calculator. Notice that this is actually an operation in the style number/base. This is important. Our "value" is actually represented as a formula/process/function/howeverYouWannaCallIt.
2. Next, we want to store this in decimal. So, a num/1 system with digits ranging from 0-9. Problem is that this system cannot accurately represent 1/3. No matter how many fractions we add, we'll never reach the accuracy of 1/3.
Therefore, the value changes. If we have to discard something, we lose something. A difference is a difference. 0.333... therefore cannot be equal to 1/3, for the plain simple reason that this isn't a matter of digits... decimal simply cannot represent this value accurately, no matter what you do to the digits.
3. For the same reason, 0.999... can never become 1. Why? Quite simply because it is the wrong operation to get what we want. Before i explain this, remember that what we're defining is a formula. The "0.999"-part actually means "0.999 / 1". Okay?
Good. If we add infinity, the formula is this: "zero point infiniteTimes9 / 1" (this isnt absolutely correct, but it doesnt affect what i'm gonna show). So, we just added a command to append "9" digits forever. But just as with 1/3, no matter how many digits you add, the value is not identical to 1, for the simply reason that there again is a loss in accuracy, and therefore there is a loss, which in turn means, there is a difference. The accuracy-loss may become ever smaller, but it never vanishes, as long as all you're doing is appending 9's.
So, it is simply the wrong operation to achieve a result of 1. What we'd need to do, is more like - as so many "naive" posters proposed - ROUND UP.... but then we'd have to acknowledge, that we have to change the value to get it to 1, and that therefore it cannot have been 1 before the rounding.
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4. Ready for pi? Okay. So far, the formula-style we used to represent values has been of the scheme num/base. But since we've thrown the traditional idea of "numbers" overboard, we can say that it really doesn't need to be that way. We can represent a value with any formula we like (which is why relations work at all). Some values can be represented with multiple formulas ( 2/4 = 1/2 ). There are however also formulas for which currently there is no representation in the num/base format possible. Which is why they run "forever" in those. The supposed "infinity" in pi, is nothing else, then the infinity in 1/3 when converted to decimal. In other words: The only reason why pi is running forever in decimal, is that the decimal system cannot accurately represent it.
Notice a pattern? In the above example, values always could "forever not resolved" in a num/base system, precisely BECAUSE of a difference! Only because the system cannot accurately represent the desired value, we never reach total accuracy. Or to put it super bluntly: 1/3 converted to decimal results in endless 3's BECAUSE there is forever a difference!
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To finish this wall of text, a little addition about infinities. The above "infinite inability to be equal" has very little to do with what i'd call singularities. You know, stuff like "infinitely small" or "infinitely high" - totally different thing. So, that unit-conversions results in such weird things, proves in no way at all that singularities exist in reality - that is a different kind of "infinity". Bodies in space may follow curves according to some formula called pi, that by coincidence doesn't fit into our dumb num/base system.... but that does not mean that "there is some kind of infinity in those bodies flying around" - it's just an anomaly in our number system, nothing else - call it a maths-bug if you want to.
Infinity is very prominent a mathematical concept. It's quite real, I assure you.
Yes, indeed - very "real".
