<- this has sparked lively discussion elsewhere, and I thought some heads on here would enjoy this ->
This started in the Math SubReddit where I refused to give ground with much much better math minds than myself. I am known to be stubborn and I will not grant the Tall order that calculating the infinite series, 1+1/2+1/4+1/8+1/16… comes out to 2.
This is bullshit.
darkon points out some valid holes in my conjecture
http://www.reddit.com/r/math/c…
It doesn’t progress. It’s already there.Consider an arrow hitting a target. It goes halfway there, then half of the remaining distance, then half of the still remaining distance, then half again, and so on.
The arrow still gets there even though it has to go through an infinite number of “halfways” to do it.
In the same way, 1/1 + 1/2 + 1/4 + 1/8 +1/16 + 1/32 + … adds up to 2 even though there are an infinite number of terms in the sum. Actually it sums to 2 because there are an infinite number of terms. Any finite number of terms and it’s not equal to 2, although it can get very close with a finite number of terms.
This is to ignore the value of the some of the parts. It is the boundries of the sum of all these units that are in question. The sum of the whole fellas.
Of course if the sum of units without disregard for their individual value will equal two in an infinite set of numbers. It’s really not that hard to do.
1+1=2
The proof for that is a little long, so I won’t post it here.
I refuse to believe you cannot add more value to the sum of the whole as 1.999… without getting to the value 2.
If it already equals 2, it is 2, the is the bound of the value of that unit.
The distance between 2 and 1.999… is infinitely small as you can always add half of the unit before it, one more unit in the sum of the whole which gets infinitely close to the bounds of the value 2.
To deny that such an infinitely small distance cannot exist is to deny that infinity can not go the other direction. It’s the same concept.
That would be a major flaw in the entire system.
This guy explains the problem.
lydianrain
http://www.reddit.com/r/math/c…
Since you’re being downmodded without real explanation, (not by me) I’ll try to give this a shot.The trouble here is the nature of the “…” in the title post. It’s a little vague, and many people who try to talk about what an infinite series is miss the point.
The statement “the sum from n=0 to infinity of 1/2^n EQUALS 2” is a formal statement that means what you are saying: the partial sums can come as close to 2 as we wish. (more rigorously: for every epsilon there is an N blah blah blah) What it is not is a statement of equality of two numbers.
In mathematics, it is often standard to abuse this notation a little bit, and treat the sum as a number, and say that the above statement is a statement of equality of two numbers. This is usually only problematic in cases where existence of the limit is under question.
In any case, the people downmodding you presumably haven’t thought much about this distinction, and are criticizing the idea that an infinite series is not truly equal to something. Hence the snarky “we do math” as though this notational sleight-of-hand should be plainly obvious to any fool.
In any case, Soupstorm is basically right that we can “allow” a series to have a value, and I think a lot of people are interpreting you as contradicting him. However, the whole thing is taking on cultish tones with the invocation of the name of “mathematics” to justify a very vague, mostly implicit position–what does it mean to “allow” anyway?–so I’m inclined to take your side here.
For some reason the mere questioning of Tall brought about the wraths of some acolytes from some D&D scenario, all chanting “Mathematics, Mathematics”.
A two word comment would have sufficed:
It’s shorthand.
The notion that the notion behind that shorthand could be challenged will be challenged.
It must always be challenged to give validity to the value for it to be a fact, otherwise it is an act of faith.
Excuse the muddy boots on the ivory floors, I did not know “we do math” was going to be a sleight of hand magic show performed by high priests.
Do not look down upon 1.999…, denying it equality of a value independent of two, just watch the acolytes saw lady rules in half in half, and magically put her back together again.
It’s quite a show.
1.999… does not equal 2, ever, except when needed as shorthand in computation.
Add that to the magic cult’s bylaws.
2 comments
1.99999999999999999999999999999999999999999999999999 isn’t 2. It’s pretty f*cking close, but it’s not there. There’s always a gap. It’s a f*cking small gap, but it’s a gap if you’re writing down the 9’s. When there’s an infinite number of 9’s after the decimal, you get so close that the gap is infinitely small. Some folks are gonna say that makes the gap nonexistent, that it makes it 2, it just makes the gap so small that most folks won’t acknowledge it. That might even be the convention, but that’s not quite 100% (make that 99.999999999999999999999%) right.
I ♥ numbers.
I’m certain that you’ve heard the old story about the accountant who got the job after answering his employer’s question “How much is 2+2?” with the answer -“However much you want it to be.”