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MariaDroujkova

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**Answer** by Jack
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Feb 03, 2013 at 12:56 PM

As Maria mentioned, one interpretation of infinity is with limits. For instance, let f(x) = 3 + 1/x. Then as x gets bigger, f(x) tends to 3. By that I mean that the difference between 3 and f(x) gets as small as you like as x gets bigger. Suppose f(x) tends to A and g(x) tends to B. Then f(x) + g(x) tends to A+B and f(x)g(x) tends to AB, and f(x)/g(x) tends to A/B if B is non-zero. Some other functions grow without bound as their parameter increases. For instance, f(x) = x is one of them. As x grows, the function grows, and it grows larger (in modulus) than any particular number if the parameter x is large enough. If f(x) grows without bounds, and g(x) tends to A, then f(x)g(x) grows without bound, as does f(x) + g(x) and f(x)/g(x) if A is non-zero. So if decides that if f(x) grows without bound you consider it tending to the limit Infinity, and you decide the following rules (on top of the usual rules of arithmetic): 1. Infinity + A = Infinity (for all A, including A = Infinity) 2. Infinity * A = Infinity (for all non-zero, including A = Infinity) Then you can say "if f(x) tends to A and g(x) tends to B, then f(x)g(x) tends to AB", where A and B are allowed to be infinity as well. You've sorta added a number in and extended operations to modal limits. The problem is though that if A = Infinity and B = 0, the rules above don't tell us what Infinity * 0 tends to. So it doesn't answer this question. But the thing is, it cannot. For instance, if f(x) = x and g(x) = k/x, then f(x) tends to Infinity and g(x) tends to 0, but f(x)g(x) = k, which obviously(?) tends to k. So given we can choose k to be what we want (except infinity), this means that 0*Infinity might equal anything (except this example does not show it could be infinity. But it could be that too, take f(x) = x^2, g(x) = 1/x, then f(x)g(x) = x) So 0*Infinity might be anything, and as Maria's example showed, Infinity/Infinity has the same problem. So f(x) = kx, g(x) = x has f(x)/g(x) = k, which can be anything except Infinity, and the Infinity case is possible too by letting f(x) = x^2. So in the system of limits, it definitely cannot be decided. Floating point numbers on computers use an infinity which models limits. But limits aren't the only concept of infinity! Sometimes infinity is just introduced to make an equation more simple. For instance, if you have two resistors in parallel in a circuit, the overall resistance is 1/(1/a + 1/b), where a and b are the resistances of the resistors. If you try and use this equation when the resistance of one of the resistors is zero, the equation breaks down, as 1/0 doesn't exist. But if you say "using the convention that 1/0 = Infinity, Infinity + a = Infinity, 1/Infinity = 0" then it works out - if one or both of the parameters are infinity then the overall things goes to 0, which is physically the correct answer. So quite often when you want an equation to work you say what you want these operations to equal. In this case, the rules didn't contradict the limit case - but it can. For instance, sometimes you want 0^0 = 1, or sometimes 0^0 = 0, for an equation to work in general (or maybe for some terms of a sum to be correct with a general formula). So you just say that is what it should equal. You're not actually asserting anything about the mathematical world - you're just saying for now, that's what you want it to be. The idea of right or wrong doesn't enter into it. Note that 0^0 is indeterminate as a limit - for instance, f(x) = Exp[-x] tends to zero, and g(x) = k/x tends to zero, but f(x)^g(x) = Exp[-k], which can be anything non-zero for choice of k. It can also be zero if f(x) = Exp[-x^2], or f(x) = 0. It can't go to infinity though. Another example is Möbius Transformations. So these are equations of the form f(x) = (ax + b)/(cx + d). They are pretty interesting things with lots of nice properties. Now it turns out their properties are much, much nicer if you consider it mapping points in the Riemann Sphere. Riemann Sphere = Complex numbers + Infinity. Since this post isn't about complex numbers, I'll ignore the complex number bit, but it does mean that we have to interpret f(Infinity). We need to decide it in a way which gives the transformations nice properties, and it turns out the f(Infinity) = a/c does it nicely. Actually, this is the limits version of it too. What point is mapped to infinity? Well, when cx + d = 0, so x = -d/c works. But again, it is just defining the rules of infinity so that nice properties happen. It's not actually an assertion about infinity. There are plenty of other infinities, and I'll only mention them in passing. But Conway's Surreal Numbers or Robinson's Non-standard Analysis both introduce infinitely big and infinitely small numbers. With both of them there isn't just one infinity, so Infinity/Infinity depends on which infinities they are. However, definitely in Robinson's and I think also in Conway's, x/x = 1 even for infinite x, so if they are the same one, it is 1. Then there are the Cardinals and Ordinals of Cantor. In this, again, there are lots of infinities. However, you don't really have division so you'll have to rephrase it as what k as there such that Infinity = k * Infinity. Well, it depends on what Infinity. But if you take Infinity to be omega, the smallest infinite cardinal/ordinal, then any non-zero finite k works. For cardinals, k = omega works too, but it doesn't work with ordinals. Note that with ordinals, although omega = k * omega for each finite non-zero k, omega != omega * k. Ordinal arithmetic is not commutative. Cardinal arithmetic is. *So there are lots of different types of infinity and the answer depends on what makes sense. You are free to make your own idea of infinity too. There aren't rules of maths you must obey - you are free to be creative!*

**Answer** by Sue VanHattum
·
Feb 02, 2013 at 03:48 PM

It might be anything. You need to know more about the infinity you're looking at to answer that question.

**Answer** by dendari
·
Feb 02, 2013 at 05:21 PM

Just one thought.
There are an infinite number of counting numbers and in between each counting number is an infinite number of fractions. So depending on which infinity was the numerator we could have an infinity or zero or there abouts.

**Answer** by MariaDroujkova , Make math your own, to make your own math
·
Feb 02, 2013 at 04:14 PM

Infinity is something you approach, but you don't "get" there. Now imagine two things approaching infinity at the same speed. For example, imagine two space probes flying forever, each at 2 kilometers per second. The distance each starship covers will get larger and larger - it will be infinite in the fullness of time! At t seconds, the distance is 2t. Now let us ask the question - what is the RATIO of the distance the first starship covers to the distance the second starship covers? Why, the distances are exactly the same, so their ratio is always 1 - now, a year later, and in the fullness of time too. What would happen if the first starship went 10 kilometers per second and the second 2 kilometers per second? If you tracked them forever, both would still go to infinity. But the ratio of the distance they covered would stay at 5. You can see if for yourself if you graph the function 10t/2t as the number t goes really large. Try graphing different functions, like (t^2)/2t or (10t+3)/(5t-4) and so on and see what happens as t grows large - or goes to infinity. You can see for yourself that these different-sized infinities can give you different-sized ratios. I used a spreadsheet to calculate and graph three different ratios of things that go toward infinity. Two of the ratios also go toward infinity, but the third stays around 5 at all times! ![Different Infinities][1] [1]:
http://i281.photobucket.com/albums/kk220/12_drakon/DifferentInfinities_zps4d862e06.png

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