Arithmetic of Infinity
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Strange but True: Infinity Comes in Different Sizes
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Teaching the Arithmetic of Infinity - Conferences - UEA
Rizza D. However, with the subtraction and division cases listed above, it does matter as we will see. Here is one way to think of this idea that some infinities are larger than others. This is a fairly dry and technical way to think of this and your calculus problems will probably never use this stuff, but it is a nice way of looking at this.
For a much better and definitely more precise discussion see,. So, pick any two integers completely at random.
Start at the smaller of the two and list, in increasing order, all the integers that come after that. Eventually we will reach the larger of the two integers that you picked. Because we could list all these integers between two randomly chosen integers we say that the integers are countably infinite. Again, there is no real reason to actually do this, it is simply something that can be done if we should choose to do so.
In general, a set of numbers is called countably infinite if we can find a way to list them all out. In a more precise mathematical setting this is generally done with a special kind of function called a bijection that associates each number in the set with exactly one of the positive integers. To see some more details of this see the pdf given above. It can also be shown that the set of all fractions are also countably infinite, although this is a little harder to show and is not really the purpose of this discussion.
To see a proof of this see the pdf given above. It has a very nice proof of this fact. The following is similar to the proof given in the pdf above but was nice enough and easy enough I hope that I wanted to include it here. This means that there should be a way to list all of them out.
We could have something like the following,.
In the case of our example this would yield the new number. The reason for going over this is the following. An infinity that is uncountably infinite is significantly larger than an infinity that is only countably infinite. So, if we take the difference of two infinities we have a couple of possibilities. Depending upon the context there might still have some ambiguity about just what the answer would be in this case, but that is a whole different topic. Again, we avoided a quotient of two infinities of the same type since, again depending upon the context, there might still be ambiguities about its value.
Be careful when dealing with infinity.