The method used in the video is being basically the same method used in the false proof that .999 repeating = 1.
I believe that both of those proofs are wrong. The problem is that they are engaging in mathematical legerdemain and using an infinite series as a real number. But I don't believe it is mathematically possible to use infinite series as a real number. You can't multiply it by anything. It is infinite. You end up with terms that cannot be mixed, like a blue mile or a red 65 miles per hour. It just doesn't make sense or describe anything real.
This is here is some fucking crazy calculus, but I think it is basically saying you cannot treat an infinite series as a sum, which is what the .999.. = 1 and the infinity video are trying to do. See the part beginning with:
Before we move on to a different topic let’s discuss multiplication of series briefly.It says that you cannot treat multiplying a series as just multiplying the constant terms, you have to distribute each term into each other and then combine them, which is literally impossible when dealing with an infinite series. The only way to do it is to represent it with notation (which is another infinite series), but you cannot derive a real number out of it. And that is the key. An infinite series is not a sequence of numbers. See the concluding two paragraphs:
We’ll leave this section with an important warning about terminology. Don’t get sequences and series confused! A sequence is a list of numbers written in a specific order while an infinite series is a limit of a sequence of finite series and hence, if it exists will be a single value.In that the infinity video, the speaker is giving us an infinite series and then claiming that is identical to a sequence of digits that can be cancelled out using another sequence. But you can't do that. You can't compare an infinite series to a sequence in that way. It's an abuse of terms.
So, once again, a sequence is a list of numbers while a series is a single number, provided it makes sense to even compute the series. Students will often confuse the two and try to use facts pertaining to one on the other. However, since they are different beasts this just won’t work. There will be problems where we are using both sequences and series so we’ll always have to remember that they are different.
An infinite series is a single number, provided it makes sense to even compute the series. And in that case the computation of the series is a number, but that is not the same thing as saying the series itself is the number. That's the same as how a fully grown apple seed is a tree, but an apple seed itself is not a tree.
If you want to perform mathematics using series you must use only series. In that case you are using notation so you would use .999.. and never resolve it to a real number. If you have an infinite series and you want to use it with real numbers you must take the limit of it, which is more crazy calculus but as I understand it it is a way to find the real number to which the series gets the closest (if any) with infinitely diminishing margin of error. In the case of .999.. that is in fact 1, because there is no real number closer to .999.. than 1.
BUT, in this case we aren't saying that .999.. equals 1, we are saying that the limit of .999.. is equivalent to one. Those are not the same statements.
LONG STORY SHORT:
You can say that 1 = 1 or that .999 to infinity = .999 to infinity, but you cannot say that .999 to infinity equals 1. It is exactly like this:
An apple equals a fruit.But you can't compare apples to oranges!
An orange equals a fruit.
If you cancel out the fruit term, you get an apple equals an orange.
The infinity video I believe both has the method wrong and the conclusion wrong. Infinity is not equal to -1, because the limit of infinity is not -1.
The .999 repeating equals 1 has the method wrong, because you can't perform sequential number calculations on an infinite series, and the conclusion is syntactically wrong. It's the limit of .999.. that is equal to 1.
I know some posters on Dark Taco are much better trained math wizards than I. What say you?
Posts
I've said it, in the context of another argument:
ReplyDeletehttp://www.darktaco.com/2009/09/god-is-not-infinite.html
Your claim amounts to an outright rejection of calculus. The whole point is that you can use an infinite series to represent a theoretically derived constant. Rejecting that will put a serious dent in... well, almost everything since the mid 1600's.
From there, you dive into what accounts to eloquent numerology, culminating in a syllogism that requires two false premises to make your point. Apple is a subset of Fruit, not an equivalent.
In the end, I'll refer to a quote in that other blog from a favorite logic professor of mine:
" if you had a decimal and an infinite string of nines, then if this number really was less than one, what number would you add to it to get one? The answer, of course, is a decimal followed by an infinite string of zeros with a one at the end....which, of course, is incoherent. No infinite string of zeros can have a one at the end!"