Math Nerds Answer Math Questions

What is complex analysis? Just what's up with the different sizes of infinity, and why should I care?

You all know you've asked these sorts of questions before.  ^*

Now's you're chance to get them answered by the resident math nerds with as little jargon as possible.

I'm bad, and that's good. I will never be good, and that's not bad. There's no one I'd rather be than me.

Although I'm a bit of a math nerd myself, I've had a few problems on the education part so there's quite a few things I don't know. Let me start with this:
What is "e" and why is it important?

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I feel like an idiot for even having to ask this, but: How exactly is it that 0.999... is equal to 1?

Heapers’ Hangout

Is this a practical joke?

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CA: If 1/9 = .111... then 9 x 1/9 = .999..., right?

Fractionally, 9 x 1/9 is represented as 9/9 which is equal to 1. That's the simplest way to prove it that I know.

edited 5th Dec '10 8:17:10 PM by Iverum

dysfunctional human artistry

@2: I assume that as a self-professed math nerd, you won't mind a little bit of calculus-level jargon. e is a constant with value a little bit greater than 2.7. The reason why this particular constant got its own letter designation is that it has a few very nice properties. One in particular is that the derivative of e^x is still e^x. For those of you who don't care so much about the abstract properties, e shows up when you consider compound interest. Let's say that you annually compound interest at a rate r on an amount of money (let's call it A). Then the amount of money you have after t years is A(1+r)^t. Now let's say that you used the same yearly rate, but compounded monthly (ie you do the compounding at one twelfth of the rate at the end of each month). The amount of money after t years is A(1+(r/12) )^(12t), which ends up being a nice bit more after a few years. We can keep reducing the time frame (compounded weekly, daily, hourly, etc.) and get an answer, so let's go as far as we can and compound continually. Then the amount of money you have after t years is Ae^(rt)—it can be modeled with the exponential function of base e.

@4: I don't think so—it is category theory, after all.

I'm bad, and that's good. I will never be good, and that's not bad. There's no one I'd rather be than me.

^^ Wow, it's that simple?

That's like a magic trick or something.

Heapers’ Hangout

Meo: This needs a certain amount of complexity to really put across, since e doesn't even pop up until calculus and certainly isn't used much till then, so bear with me.

Imagine a generic exponential curve y = (stuff)^x graphed out. Now as you tweak (stuff), notice how the graph gets horizontally squished or stretched. Tweak (stuff) until the slope of this curve at x=0 is exactly 1. That value of (stuff) is e. [1]

Now why is this important? Well, exponents in base e have some weird calculusy properties that end up popping up behind every corner. The derivative of e^x is equal to itself. Its integral is equal to itself (plus c). It winds up being a logarithmic base that pops up all the time (we call this the "natural log" or ln). Essentially logs and exponents that work in base e have some quirky properties that make them extremely easy to work with in terms of calculus operations, and you'll often see expressions of the form (stuff)^x or log_(stuff)(x) transformed into equivalent A*e^x or A*ln(x) expressions to make things easier in the long run.

ninjad, damn =[

edited 5th Dec '10 8:31:11 PM by Pykrete

Is there a good introductory textbook for this? As if I didn't have enough books to buy >_____<

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Alright, thanks for the help. I started Calculus, but didn't quite finish that class...

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Basically the idea is that if you do everything in base e, you wind up using less pencil in the long run because you don't start hemorrhaging chain-rule coefficients every time you do something to your equation. The interest stuff Ironeye mentioned can be done in any exponential base, but changing bases and writing it as Ae^rt means you can filter it through calculus tools with less fuss.

edited 5th Dec '10 8:34:21 PM by Pykrete

@Tzetze: I don't know. One of my old lecturers wrote a book about category theory for people who don't study category theory, but his publisher wouldn't print it because the category theorists they got to review it thought it was useless because it didn't cover the key proofs in category theory. That was completely missing the point of the book, but it was enough for the publisher.

I'm bad, and that's good. I will never be good, and that's not bad. There's no one I'd rather be than me.

Okay, here I'm going to go into the mathematical definition of real numbers. Basically, a real number is defined by means of an infinite sequence of numbers that get arbitrarily close to each other as you progress  ^{mathematically}

^{for every number ε greater than zero, there exists a number N such that all the numbers past the Nth will be within ε of each other}

. This is called a Cauchy sequence. Two sequences denote the same real number if you still have a Cauchy sequence when you interleave the two sequences, i.e. the first from the first one, then the first from the second one, then the second from the first one, then the second from the second one, etc. Now, for a real number with a given decimal expansion, you can just use the truncated decimal expansions. For example, for pi: 3, 3.1, 3.14, 3.141, 3.14159, 3.141592, etc.

Now to see if 0.9repeating and 1 are the same number, we consider the sequence 0, 1, 0.9, 1, 0.99, 1, 0.999, 1, 0.9999, 1, 0.99999, 1, 0.999999, 1, ... Notice how the 0.9999 and such grow arbitrarily close to 1, and as such, the sequence is Cauchy and corresponds to a real number.

edited 5th Dec '10 8:51:19 PM by Ponicalica

the future we had hoped for

Let me prove that 3 * 0.3333... = 0.9999... = 1.

Let x = 0.3333...

10x = 3.333...
- x = 0.333...
= 9x = 3.000... = 3
9x = 3
3x = 1.

Tadah.

edited 5th Dec '10 8:52:17 PM by sgrunt

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^^ And that's a rigorous explanation compared to what sgrunt and I said.

dysfunctional human artistry

I'm going to get dozens of explanations of this now, aren't I? And some will be more technical than others.

Heapers’ Hangout

I want to hear Dedekind cuts explained next if we're doing that.

dysfunctional human artistry

@CA: Here's a press release and a article for The Other Wiki explaining it, if you want.

Edit: The URL doesn't work... It's here:
http://en.wikipedia.org/wiki/0.999...

edited 5th Dec '10 9:00:24 PM by Meophist

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Yes. The simplest one (that doesn't actually prove the statement in question) is that decimal representations of numbers are not always unique.

edited 5th Dec '10 8:59:55 PM by Ironeye

I'm bad, and that's good. I will never be good, and that's not bad. There's no one I'd rather be than me.

ninjad

edited 5th Dec '10 9:00:36 PM by Pykrete

So, wait, what's a Cauchy sequence exactly?

Ironeye: -_- Thanks anyway... guess I'll trawl through my uni's library.

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@Tzetze: A Cauchy sequence is a sequence where the terms eventually get arbitrarily close to each other.

I'm bad, and that's good. I will never be good, and that's not bad. There's no one I'd rather be than me.

A Cauchy sequence is one whose terms become increasingly close to each other down to arbitrary precision. Basically a convergent sequence, but you use an epsilon-proof to show it.

ninjad again

edited 5th Dec '10 9:03:30 PM by Pykrete

Oh, that reminds me:

Let ε < 0...

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Why is division by zero impossible?