aha
June 6, 2011 7:08 AM Subscribe

Longer explanation. Some of these entries don't seem to get it and not many there yet, so contribute some better ones.

posted by DU (54 comments total) 16 users marked this as a favorite

Twitapedia?
posted by jonmc at 7:09 AM on June 6, 2011

I think the color-coded equation is almost worth a post by itself. Sadly, the aha! beta site wouldn't support that great example, so I think the "twitter" part of the equation is probably a fail. Maybe a LaTeX/MathML color mashup would work better.
posted by DU at 7:09 AM on June 6, 2011 [3 favorites]

In the Great Purge of 2012, all ideas and subjects that can't be expressed in 140 characters or less will be deleted from the Noosphere.
posted by Horselover Phattie at 7:20 AM on June 6, 2011 [1 favorite]

Instead of SOHCAHTOA, learn OHSAHCOAT using a mnemonic. Then you get opp = hyp * sin, opp / hyp = sin, etc. with the letters in order. There's usually less manipulation to do from this starting point.

worst aha ever
posted by nathancaswell at 7:20 AM on June 6, 2011 [1 favorite]

But you can't call it Twikipedia or else the Buck Rodgers people will sue our asses off.
posted by inturnaround at 7:23 AM on June 6, 2011 [1 favorite]

"0.1 + 0.2 does not = 0.3, but 0.1 + 0.2 = 0.30000000000000004 instead. This is called machine precision and basically the binary conversion of these values looses the values precision."

What?
posted by oddman at 7:24 AM on June 6, 2011

I think the color-coded equation is almost worth a post by itself. Sadly, the aha! beta site wouldn't support that great example, so I think the "twitter" part of the equation is probably a fail. Maybe a LaTeX/MathML color mashup would work better.

This was the first post I've seen that does this. Skip to the good part.
posted by Jpfed at 7:27 AM on June 6, 2011 [1 favorite]

"We don't actually know how gravity works! We only understand its properties."

Is this a Juggalos project?
posted by oddman at 7:28 AM on June 6, 2011 [5 favorites]

The product of two negative numbers is positive. For example, if you owe 10 dollars to each of 3 friends, and they all die, you've just made 30 dollars.

Um...I don't think "example" means what you think it means....
posted by Philosopher Dirtbike at 7:29 AM on June 6, 2011

Think of the Fourier transform like getting the "source code" for a pattern (instead of seeing the raw output)
...in machine language.
posted by Wolfdog at 7:29 AM on June 6, 2011

Too many tweets make a twat
posted by londonmark at 7:29 AM on June 6, 2011 [1 favorite]

Analogy: Imaginary numbers are a different dimension from real ones. i is a rotation of 90 degrees

I have no idea what application this has, but it creates a nice visualization. Especially when thinking of a number line.
posted by lizjohn at 7:31 AM on June 6, 2011

Sarah Palin reads, like, all of these.
posted by fatllama at 7:32 AM on June 6, 2011

Bah.

1. Think "Colour coded equations? How do they work?"
2. Look at linked example. Think, "fantastic!"
3. Remember that I'm colour blind, and have know idea which part of the equation relates to "Energy at."

Great idea, but sadly one I'm unable to parse.
posted by seanyboy at 7:33 AM on June 6, 2011 [1 favorite]

A fuction is like a vending machine. You enter coins (the Domain), you type the code of a drink (the correspondence rule that a specific function follows) and you get the drink (the Range of Values).

I'm pretty sure I had to use this textbook at one point during Peak Reform Calculus.
posted by Wolfdog at 7:34 AM on June 6, 2011

Oddman, that might be difficult to understand without having working knowledge of binary to decimal conversion. Here's a little more on machine epsilon (machine precision) which goes into detail why round offs are necessary when representing floating points after binary conversion to decimal.
posted by samsara at 7:37 AM on June 6, 2011

I saw the post on induction and recursion, and was intrigued. I typed "recursion" into Google to see what it was. The first thing you get is:

"Did you mean: recursion?"

Just looked it up and it turns out this easter egg is well known already. Still a nice surprise for me.
posted by marmaduke_yaverland at 7:42 AM on June 6, 2011 [1 favorite]

seanyboy- to use this technique to reach the broadest possible audience, it's good to put most information in the blue-yellow dimension, followed by luminance, followed by red-green. (While all sighted people can distinguish luminance information, varying luminance too much can impact readability; most sighted people, including most colorblind people, can still distinguish along the blue-yellow dimension).
posted by Jpfed at 7:49 AM on June 6, 2011

Um...I don't think "example" means what you think it means....

I think it does and I also think you might have a little confusion on the topic.
posted by DU at 7:51 AM on June 6, 2011 [1 favorite]

0.99 repeating divided by three = 0.33 repeating; 1 divided by three = 0.33 repeating; 1 = 0.99 repeating. That's why I round 2(pi) to 6.

138 characters! Wait, what are we talking about now?
posted by AzraelBrown at 7:56 AM on June 6, 2011

The 1 = .999... Wikipedia talk page is one of the most depressing ever. I expect intense Internet debates on Monty Hall or evolution or global warming. Not on basic mathematical facts.
posted by kmz at 8:08 AM on June 6, 2011

Me: Um...I don't think "example" means what you think it means....
DU: I think it does and I also think you might have a little confusion on the topic.

The product of two negative numbers is positive. For example, if you owe 10 dollars to each of 3 friends, and they all die, you've just made 30 dollars.

This is not an example of multiplying two negative numbers. This is an example of the use of additive inverses.
posted by Philosopher Dirtbike at 8:11 AM on June 6, 2011

Most people go through school and never realize there are multiple ways to express a single number.
posted by introp at 8:12 AM on June 6, 2011

-3 friends * -10 dollars per friend = 30 dollars.
posted by DU at 8:20 AM on June 6, 2011 [1 favorite]

-3 friends * -10 dollars per friend = 30 dollars.

A good explanation doesn't make up for a bad one.
posted by londonmark at 8:33 AM on June 6, 2011

You can take derivatives of types!
posted by boo_radley at 8:56 AM on June 6, 2011

DU, you're confusing your representation of the problem with the problem posed. They are not the same. That's why it was a bad example; there are perfectly natural of representations of that problem that do not involve multiplying two negative numbers.
posted by Philosopher Dirtbike at 8:58 AM on June 6, 2011

Twikipedia

beedybeedybeedy[citation needed]
posted by Mr. Bad Example at 9:02 AM on June 6, 2011 [1 favorite]

Objects orbiting around the earth are in fact following a parabolic path, only without ever reaching the ground, because the earth isn't flat.

Um, no. Just because is sounds cool and a lot of people voted for it doesn't make it so. This makes me appreciate the overbearing wikipedia editorial process.
posted by TheShadowKnows at 9:15 AM on June 6, 2011 [1 favorite]

There are alternative representations for every multiplication problem, so by your logic it is impossible to give a good example of multiplication.
posted by DU at 9:21 AM on June 6, 2011

So, the articles roll by on an endless scroll based on recent updates, there's effectively no real archive, and the search feature is all but useless on any topic of note?
posted by Eideteker at 9:39 AM on June 6, 2011

There are alternative representations for every multiplication problem, so by your logic it is impossible to give a good example of multiplication.

No, I said "That's why it was a bad example; there are perfectly natural of representations of that problem that do not involve multiplying two negative numbers." I've bolded the critical part.
posted by Philosopher Dirtbike at 9:41 AM on June 6, 2011

Yeah, the forier transform article was really interesting. I had no idea it was that simple. I also have no idea why anyone would want "twitter + wikipedia"
posted by delmoi at 9:47 AM on June 6, 2011

A fuction is like a vending machine. You enter coins (the Domain), you type the code of a drink (the correspondence rule that a specific function follows) and you get the drink (the Range of Values).

Actually, it's a epimorphism between the external direct sumof the power set of coins and the set of buttons you can push and the external direct sum of the set of drinks you can get, and your change. The kernel of which would be not having enough funds to buy a drink.
posted by delmoi at 10:01 AM on June 6, 2011

I really don't get this argument. That is in fact a good example of multiplying by negative numbers. No amount of dirtbike philosophizing makes it not.

An example is a simple version of the problem that illustrates a point. The simplicity could, for instance, derive from being easy to solve via a familiar method. Then when you solve it using the new method, you know you have the right answer. It's really easy to see how you've made $30, which makes this a particularly good example.
posted by DU at 10:04 AM on June 6, 2011

No, I said "That's why it was a bad example; there are perfectly natural of representations of that problem that do not involve multiplying two negative numbers." I've bolded the critical part.

But then is "You have three friends, and they each give you 10 dollars" not a good way to represent 3 x 10 = 30? Because 10 + 10 + 10 is a perfectly natural representation of that expression. The example works as long as the calculation you are trying to show fits and makes sense in the context of the example, the fact that other intuitive representations exist shouldn't affect that.
posted by burnmp3s at 10:05 AM on June 6, 2011

According to delmoi, vending machines can not be generalized to non-Abelian categories. :(
posted by Wolfdog at 10:13 AM on June 6, 2011

"A change in perspective is worth 80 IQ points." - Alan Kay

Yeah, but in which direction?
posted by Afroblanco at 10:19 AM on June 6, 2011 [1 favorite]

Quoting Bertrand Russell from the link:

"The time you enjoy wasting is not wasted."

I guess this justifies masturbation, and Metafilter for that matter.
posted by philip-random at 11:40 AM on June 6, 2011

I also love the illustration of negative*negative = positive. It may not be the best way to attack that problem, but that's the first good explanation of " – * – = +" I've heard that doesn't resort to talking about "additive inverses"

The 'derivatives of types' comment is also cool. I just read a great article in the Wikibooks Haskell book that applies the idea – it's nicely written and not too horribly abstract. Just right for someone trying to learn Haskell and (possibly) ready to move beyond monads.
posted by benito.strauss at 11:47 AM on June 6, 2011

It's also good to remember that mathematical induction is actually a form of deductive reasoning.

Er, is there some sort of mnemonic to keep induction and deduction (and apparently their varietals) straight? I swear I see statements like this and always have the same helpless reaction.

No amount of dirtbike philosophizing makes it not.

Apart from being referenced in a TMBG song, I have no idea what this means. Help?
posted by psoas at 12:18 PM on June 6, 2011

psoas: I'm sorry but I think I'm about to confuse you even more.

Deduction: start from premises/axioms and use certain allowed rules to find out "conclusions": what those axioms/premises imply

Induction 1: get a bunch of data together and try to infer a general rule that explains this data. This isn't deductive reasoning, but it leans on the laws of probability, statistical procedures, and the assumption that the world is "uniform enough".

Induction 2 (the one related to recursion): If you are trying to prove (deductively!) that some statement S(n) is true for all non-negative integers n, then you can break the proof down into two cases:

"Base case": Prove S(0)
"Inductive step": Prove that S(n) is true, assuming that you already know that S(n-1) is true (which can often be a big help!).

Now, the statement doesn't really have to be in terms of non-negative integers (but it often is). It's really just about having a "starting point" (like 0) for whatever is universally quantified, and a method for exhaustively enumerating the universally-quantified variable(s) (like incrementing by 1) that allows you to use previous results.
posted by Jpfed at 12:41 PM on June 6, 2011 [1 favorite]

I tried it out. While clearly very beta, I think the idea is brilliant and look forward to where he takes it going forward.

Imagine these running on the side of Wikipedia as a forum/living dialogue adjacent to the more permanent encyclopedia. I don't know how much time he has invested but already he's got 8 out of 10 start-ups that spent millions in the early 00s beat. My bet is that this is one idea that could go far with a little critical mass and curation. Awesome post (my favorite this month). Thanks DU!!!
posted by astrobiophysican at 3:22 PM on June 6, 2011

No amount of dirtbike philosophizing makes it not.

Apart from being referenced in a TMBG song, I have no idea what this means. Help?

Check the nick of the user to whom I'm replying.
posted by DU at 6:49 PM on June 6, 2011

...and there goes my aha! moment.
posted by psoas at 8:46 PM on June 6, 2011

delmoi, I don't think it's epic. I've never seen a machine where you can purchase more than one drink at a time. (Of course I always translate 'epic' into 'surjective'. Maybe it actually is right-cancellable.) But I love your characterization of the kernel.

Well, I was thinking that the output of the function would be the same for {{nickle, nickle,dime,quarter},"press Pepsi"} as it would be for {{dime,dime,quarter},"press Pepsi"} making it surjective.

Also, the Wikipedia article isn't defining it in the regular way, but by saying that if you have three functions g₁(y), g₂(y) and f(x) then if g₁(f(x)) == g₂(f(x)) then the function g₁ = g₂ if (and only if) f is an epimorphism. Then they go on to say most authors use it to mean surjective homomorphism but apparently that's the technical definition and in most cases they're the same. The diagram with the two outputs represents g₁ and g₂ to some third set

_{So, in terms of the vending machine you could cash in the pop using either function g1 or g2 you just got for some z in Z, and if you always get the same z for the same coins + selection, then you know, in fact, that g1 and g2 are the same.

_{But you couldn't say that if you considered pops that you couldn't get out of the vending machine. Like if you can't get zima out of the vending machine, then you have no way of knowing if g₁(zima) = g₂(zima) and thus whether or not g₁ = g₂ in the space of all beverages)}}
posted by delmoi at 12:27 AM on June 7, 2011 [1 favorite]

If you consider a stochastic vending machine which dispenses soda i with probability p_i, then for sufficiently large numbers of trials N, the number of times that soda i is dispensed should be close to Np_i, and the larger N is, the closer we should expect it to be.

That's the law of beverages.
posted by Wolfdog at 4:19 AM on June 7, 2011 [1 favorite]

I have discovered a truly marvelous proof that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two, sed CXL litterarum exiguitas non caperet.
posted by No-sword at 4:53 AM on June 7, 2011

Curse you, <super> tag-disrespecting MeFi style!
posted by No-sword at 4:54 AM on June 7, 2011

delmoi, my project for tonight is to understand your epic post and also understand a case where 'epic' is not the same as 'surjective'.

But when I was thinking about the fact you expressed as " ...the output of the function would be the same for ...", I thought the category folk would express that as: 'The vending machine map can be factored through the value map', i.e. we can write

vend = vend' ○ value

where value() maps a set of of coins to a non-negative integer.

(I indulge in category theory sporadically. Sometimes it seems to give great insight. Other times it seems like the mathematical equivalent of producing a Monty Python quote for absolutely every incident.)
posted by benito.strauss at 8:38 AM on June 7, 2011

No-sword, I use <sup>, not <super>. Not sure if it's standard, but it ^{seems to} work.
posted by benito.strauss at 8:40 AM on June 7, 2011

delmoi: I think your example is one where epimorphic doesn't necessarily imply surjective.

Most of the examples of this in Wikipedia are categories where to be epimorphic your morphism doesn't t have to hit everything in the range, it just has to hit enough (where the structure of the category determines what 'enough' means).

So for the vending machine, you may not be able to buy a Dr. Pepper and a bag of Cheez-Its in one purchase (or application of the function), but you can buy any one them, one at a time. I'm not sure that's what you were getting at in your 'footnotes', but it works for me.
posted by benito.strauss at 4:18 PM on June 7, 2011

Thanks 10⁶, benito!
posted by No-sword at 8:04 PM on June 7, 2011

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