# "there are ten enthusiastic seconds in 6 weeks"

February 14, 2016 3:24 PM Subscribe

I think Benford's Law is the coolest one I can think of, off top.

posted by mordax at 3:49 PM on February 14, 2016 [11 favorites]

posted by mordax at 3:49 PM on February 14, 2016 [11 favorites]

The sum of all positive integers is -1/12.

posted by humanfont at 3:54 PM on February 14, 2016 [6 favorites]

posted by humanfont at 3:54 PM on February 14, 2016 [6 favorites]

π seconds is a nanocentury.

posted by eriko at 4:16 PM on February 14, 2016 [2 favorites]

posted by eriko at 4:16 PM on February 14, 2016 [2 favorites]

Though Euler's Identity is pretty cool as well.

posted by eriko at 4:19 PM on February 14, 2016 [4 favorites]

posted by eriko at 4:19 PM on February 14, 2016 [4 favorites]

My favorite was the one about the Fibonacci's leading to 1/89.

On revisiting the post, I think my favorite comment is one in response to the fact that there's no "formula" for solving the quintic, where the commenter asks "Why can't there [be]?". I think the proper first response to that is a smile and "You're probably not as curious about that as you think you are".

posted by benito.strauss at 4:29 PM on February 14, 2016 [8 favorites]

On revisiting the post, I think my favorite comment is one in response to the fact that there's no "formula" for solving the quintic, where the commenter asks "Why can't there [be]?". I think the proper first response to that is a smile and "You're probably not as curious about that as you think you are".

posted by benito.strauss at 4:29 PM on February 14, 2016 [8 favorites]

"Sometimes two plus two is four. But sometimes it’s five or even three. Sometimes it’s all of those at the same time."

Orwell: 1984

posted by TDavis at 4:31 PM on February 14, 2016 [2 favorites]

Orwell: 1984

posted by TDavis at 4:31 PM on February 14, 2016 [2 favorites]

0.999...=1

Not that 0.999... rounds to 1 but that they are exactly the same number.

posted by LizBoBiz at 4:40 PM on February 14, 2016 [2 favorites]

Not that 0.999... rounds to 1 but that they are exactly the same number.

posted by LizBoBiz at 4:40 PM on February 14, 2016 [2 favorites]

OK, I ordered 5 items from amazon.co.jp and wanted to allocate the $100 shipping cost in a fair, reasonable manner.

The conservative, corporate approach would be that each of the 5 items gets the $100 prorated by weight -- each has to pay its own freight.

Campus Notebooks 46%

Xylitol Gum 26%

MDRV6 Headphones 17%

Seiko Clock 8%

Yuki DVD 2%

But that seemed unfair to the notebooks -- sure, they were heavy but they were also pretty cheap. What if I took the socialist approach, "from each by their ability to pay" and allocated by dollar value?

Campus Notebooks 12%

Xylitol Gum 24%

MDRV6 Headphones 35%

Seiko Clock 18%

Yuki DVD 12%

Then I wondered if I could choose the middle road and take the average of the two allocations:

Campus Notebooks 29%

Xylitol Gum 25%

MDRV6 Headphones 26%

Seiko Clock 13%

Yuki DVD 7%

Indeed I could!

posted by Heywood Mogroot III at 4:43 PM on February 14, 2016 [1 favorite]

The conservative, corporate approach would be that each of the 5 items gets the $100 prorated by weight -- each has to pay its own freight.

Campus Notebooks 46%

Xylitol Gum 26%

MDRV6 Headphones 17%

Seiko Clock 8%

Yuki DVD 2%

But that seemed unfair to the notebooks -- sure, they were heavy but they were also pretty cheap. What if I took the socialist approach, "from each by their ability to pay" and allocated by dollar value?

Campus Notebooks 12%

Xylitol Gum 24%

MDRV6 Headphones 35%

Seiko Clock 18%

Yuki DVD 12%

Then I wondered if I could choose the middle road and take the average of the two allocations:

Campus Notebooks 29%

Xylitol Gum 25%

MDRV6 Headphones 26%

Seiko Clock 13%

Yuki DVD 7%

Indeed I could!

posted by Heywood Mogroot III at 4:43 PM on February 14, 2016 [1 favorite]

I might have to change that to Euler's identity, because:

e^(i * pi) + 1 = 0

rearranges to:

Pi = ln(-1) / i

So that would mean that Pi is undefined or does not exist. I love contradictions!

posted by LizBoBiz at 4:48 PM on February 14, 2016

e^(i * pi) + 1 = 0

rearranges to:

Pi = ln(-1) / i

So that would mean that Pi is undefined or does not exist. I love contradictions!

posted by LizBoBiz at 4:48 PM on February 14, 2016

I would love it if a new Ramanujan popped up in threads like these.

posted by rhizome at 4:54 PM on February 14, 2016 [1 favorite]

posted by rhizome at 4:54 PM on February 14, 2016 [1 favorite]

LizBoBiz, did you know that

posted by benito.strauss at 5:12 PM on February 14, 2016 [9 favorites]

*i*^{i}=*e*^{-π/2}, so an imaginary number raised to an imaginary power is a real number? I've always thought of that as the extra spicy version of Euler's identity.posted by benito.strauss at 5:12 PM on February 14, 2016 [9 favorites]

This all makes perfect sense, however, because with complex multiplication, you multiply the magnitudes, but you add the angles. Multiplying a number by

posted by I-Write-Essays at 5:31 PM on February 14, 2016 [4 favorites]

*i*is a counterclockwise rotation by pi/2.posted by I-Write-Essays at 5:31 PM on February 14, 2016 [4 favorites]

Benford's Law is very useful when one is attempting to generate plausible-looking numbers.

posted by Mr.Encyclopedia at 5:33 PM on February 14, 2016 [4 favorites]

posted by Mr.Encyclopedia at 5:33 PM on February 14, 2016 [4 favorites]

Probably too technical for a general audience, but the MathOverflow thread "Jokes in the sense of Littlewood" is one of my favourites.

Here are a couple of lay-accessible excerpts:

Here are a couple of lay-accessible excerpts:

You are standing in a room; at every tick of the clock, someone throws in a pair of numbered ping-pong balls: 1 & 2, then 3 & 4, etc... and you only have enough time to throw outandoneof them before the next tick. If you throw out the one with thelargestnumber, then after [infinitely many] ticks of the clock, you are in the room with all the odd-numbered balls, whereas if you always threw out the ball with thesmallestnumber, you would be rid of them all!

And what if the balls arenotnumbered? A good way to get non-mathematicians thinking about infinity.

The fundamental axioms of mathematics are inconsistent if and only if we can prove that they are consistent.posted by a car full of lions at 5:45 PM on February 14, 2016 [3 favorites]

(Because, you know, it follows from "logic." See Second Incompleteness theorem)

A million seconds is 12 days.

A billion seconds is 31 years.

A trillion seconds is 31,688 years.

posted by Ian A.T. at 5:49 PM on February 14, 2016 [3 favorites]

A billion seconds is 31 years.

A trillion seconds is 31,688 years.

posted by Ian A.T. at 5:49 PM on February 14, 2016 [3 favorites]

*Pi = ln(-1) / i*

So that would mean that Pi is undefined or does not exist. I love contradictions!

So that would mean that Pi is undefined or does not exist. I love contradictions!

Actually...

posted by jpdoane at 5:55 PM on February 14, 2016 [1 favorite]

Loving that thread and

posted by benito.strauss at 5:56 PM on February 14, 2016

**F**_{un}, car full of lions.posted by benito.strauss at 5:56 PM on February 14, 2016

*The sum of all positive integers is -1/12.*

Only if you play games with what "sum" means.

posted by Mrs. Davros at 5:59 PM on February 14, 2016 [4 favorites]

They're all games, Mrs. Davros. Some are closer to MLB and some are closer to calvinball.

posted by benito.strauss at 6:03 PM on February 14, 2016 [2 favorites]

posted by benito.strauss at 6:03 PM on February 14, 2016 [2 favorites]

To expand on that, the natural logarithm of the complex number z is defined* as:

ln(z) = ln |z| + i*arg(z)

where |z| is the magnitude of z and arg(z) is the angle in radians between z and the positive real axis. So for negative real numbers, arg(z) is 180deg or pi radians.

Thus: ln(-a) = ln(a) + i*pi, for a>0.

*technically, this is just the principal value, since ln(z) is multivalued. If that sounds interesting, look up Reimann surfaces - its really cool...

posted by jpdoane at 6:06 PM on February 14, 2016 [1 favorite]

ln(z) = ln |z| + i*arg(z)

where |z| is the magnitude of z and arg(z) is the angle in radians between z and the positive real axis. So for negative real numbers, arg(z) is 180deg or pi radians.

Thus: ln(-a) = ln(a) + i*pi, for a>0.

*technically, this is just the principal value, since ln(z) is multivalued. If that sounds interesting, look up Reimann surfaces - its really cool...

posted by jpdoane at 6:06 PM on February 14, 2016 [1 favorite]

In the 1+1 =2 thread:

"What's the name of this book?"

"Why Did We Fucking Bother, Fuck This Shit, by Whitehead and Russell."

posted by marienbad at 6:34 PM on February 14, 2016 [3 favorites]

"What's the name of this book?"

"Why Did We Fucking Bother, Fuck This Shit, by Whitehead and Russell."

posted by marienbad at 6:34 PM on February 14, 2016 [3 favorites]

*So that would mean that Pi is undefined or does not exist. I love contradictions!*

No, it just means that log is a multivalued function, like square root, and that one of the infinitely many values taken by log(-1) is pi i.

posted by escabeche at 6:41 PM on February 14, 2016 [1 favorite]

Remember. e to the i times pi plus one equals fucks given.

posted by eriko at 7:02 PM on February 14, 2016 [2 favorites]

posted by eriko at 7:02 PM on February 14, 2016 [2 favorites]

8675309 is prime. A twin prime with 8675311.

posted by Mitheral at 7:29 PM on February 14, 2016 [16 favorites]

posted by Mitheral at 7:29 PM on February 14, 2016 [16 favorites]

I think the best thing I learned in that thread is that Ft. Wayne, IN had a long term mayor named Harry Baals.

posted by waitingtoderail at 7:37 PM on February 14, 2016 [2 favorites]

posted by waitingtoderail at 7:37 PM on February 14, 2016 [2 favorites]

Perform 8 perfect out-shuffles on a new deck of cards and the deck will be back in suits order.

posted by Mitheral at 7:59 PM on February 14, 2016

posted by Mitheral at 7:59 PM on February 14, 2016

**humanfont**

*The sum of all positive integers is -1/12.*

**Mrs. Davros**

*Only if you play games with what "sum" means.*

and "="

posted by yeolcoatl at 8:22 PM on February 14, 2016 [3 favorites]

I'm somewhat ashamed to admit that threads like this and the one linked by a car full of lions above are my favorite things on MathOverflow (or Math StackExchange, or /r/math). Anyway, here's my favorite one to tell (and the source of my mefi handle).

Everyone knows that you can keep a slice of pizza from flopping over as you try to eat it by curling it along a line from the crust to the point of the slice. The reason why this works is a famous theorem in differential geometry dealing with the curvature of surfaces.

Starting at any point on a surface you can calculate the curvature in a particular direction--imagine traveling in a straight (that is, "straight" from the perspective of someone standing on the surface) path in that direction and measuring how much the surface curves up or down as you go. The minimum and maximum values you can get, among all the directions you could measure in, are called the principal curvatures at that point, and they always come from two perpendicular directions (unless the two are equal).

Now here's the theorem: you can bend a surface to change its principal curvatures without stretching or compressing it, but you

posted by egregious theorem at 9:33 PM on February 14, 2016 [13 favorites]

Everyone knows that you can keep a slice of pizza from flopping over as you try to eat it by curling it along a line from the crust to the point of the slice. The reason why this works is a famous theorem in differential geometry dealing with the curvature of surfaces.

Starting at any point on a surface you can calculate the curvature in a particular direction--imagine traveling in a straight (that is, "straight" from the perspective of someone standing on the surface) path in that direction and measuring how much the surface curves up or down as you go. The minimum and maximum values you can get, among all the directions you could measure in, are called the principal curvatures at that point, and they always come from two perpendicular directions (unless the two are equal).

Now here's the theorem: you can bend a surface to change its principal curvatures without stretching or compressing it, but you

*can't change their product.*Your pizza started out flat, with both principal curvatures zero, but once you fold it, you've introduced a nonzero principal curvature. Since the product must remain zero, the curvature along the path from crust to tip is now forced to be zero!posted by egregious theorem at 9:33 PM on February 14, 2016 [13 favorites]

This is a pretty active thread so I assume that for most of you, your skulls are intact and your brains are safely ensconced in them. If you're interested in changing that fact, read on.

We don't usually think about it, but it makes sense that if you pick a number uniformly at random (from any interval you care to choose) that almost certainly, the number you pick will be irrational. That's fine, not too surprising.

One way you can represent an irrational number is with a continued fraction. Here's an example:

You could imagine picking a number uniformly at random (from any interval you care to choose) and finding the continued fraction for that number. By all appearances the continued fraction that results should be quite unpredictable. But as you take the geometric mean of more and more of the coefficients (the numbers to the left of the plus signs) of that apparently randomly-derived continued fraction, you almost certainly approach a single known constant, 2.685452...

Pretty damn spooky. But get this.

We know this property is true of almost all numbers. But the only numbers we've found that satisfy this property are the ones we've specifically constructed to satisfy it. It's not just lurking somewhere out of view... it's lurking everywhere out of view.

posted by Jpfed at 10:02 PM on February 14, 2016 [5 favorites]

We don't usually think about it, but it makes sense that if you pick a number uniformly at random (from any interval you care to choose) that almost certainly, the number you pick will be irrational. That's fine, not too surprising.

One way you can represent an irrational number is with a continued fraction. Here's an example:

1 1 + ------------------------ 1 1 + ----------------- 1 1 + ---------- 1 1 + ---- ...which is the continued fraction for everybody's second-favorite irrational, phi.

You could imagine picking a number uniformly at random (from any interval you care to choose) and finding the continued fraction for that number. By all appearances the continued fraction that results should be quite unpredictable. But as you take the geometric mean of more and more of the coefficients (the numbers to the left of the plus signs) of that apparently randomly-derived continued fraction, you almost certainly approach a single known constant, 2.685452...

Pretty damn spooky. But get this.

We know this property is true of almost all numbers. But the only numbers we've found that satisfy this property are the ones we've specifically constructed to satisfy it. It's not just lurking somewhere out of view... it's lurking everywhere out of view.

posted by Jpfed at 10:02 PM on February 14, 2016 [5 favorites]

Consider my mind blown. Although your last paragraph makes sense to me, or is at least consistent with, the fact that almost all numbers have never been seen, used, or mentioned in all of mathematics.

posted by benito.strauss at 10:10 PM on February 14, 2016

posted by benito.strauss at 10:10 PM on February 14, 2016

*We know this property is true of almost all numbers. But the only numbers we've found that satisfy this property are the ones we've specifically constructed to satisfy it. It's not just lurking somewhere out of view... it's lurking everywhere out of view.*

I guess I don't find this too surprising? My sense is that this is true of a lot of statements that hold for almost all reals, because we have access to so few of them. Normality is another example where the only known instances that satisfy the property are constructed.

posted by invitapriore at 10:45 PM on February 14, 2016

That in

posted by drnick at 12:48 AM on February 15, 2016 [2 favorites]

*Principia Mathematica*it took 86 pages before Whitehead & Russell could prove that 1+1=2 from first principles. Here is a scan of the page that I've been carrying around in my wallet for some 15 years. I think about that a lot.posted by drnick at 12:48 AM on February 15, 2016 [2 favorites]

That there are more reals than natural numbers.

And Gödel's proofs that Russell/Whitehead-style logicism fails.

Both are mind blowing.

posted by persona au gratin at 2:11 AM on February 15, 2016 [2 favorites]

And Gödel's proofs that Russell/Whitehead-style logicism fails.

Both are mind blowing.

posted by persona au gratin at 2:11 AM on February 15, 2016 [2 favorites]

*We know this property is true of almost all numbers. But the only numbers we've found that satisfy this property are the ones we've specifically constructed to satisfy it*

Um hello, poindexter? Why are you mathematismists always trying to say six unpossible things before breakfast?

Don't you realise that you can easily find lots of numbers that are satisfied with their property - all you need to do is to do like you said: "imagine picking a number uniformly at random (from any interval you care to choose) and finding the continued fraction for that number". Just pick that number, and BAM! The number you are looking for is the one you picked.

If it's too difficult, just roll 1dC. DUUUH. Try being smart instead of being a mathlete!!!

posted by the quidnunc kid at 6:03 AM on February 15, 2016 [5 favorites]

This, above, is how we know the quidnunc kid's number is #1.

posted by AugustWest at 7:27 AM on February 15, 2016 [2 favorites]

posted by AugustWest at 7:27 AM on February 15, 2016 [2 favorites]

Actually I'm 0

posted by the quidnunc kid at 7:59 AM on February 15, 2016 [1 favorite]

^{#}! My existence is unprovable in ZFC! But who the hell eats at ZFC anyway??? Zambia Fried Chicken??? I eat at KFC, like Jesus did - and when I've had my zinger burger meal, chips and coke, I can prove ANYTHING. Memo to everybody: you are all doing math WRONG.posted by the quidnunc kid at 7:59 AM on February 15, 2016 [1 favorite]

Oh, here's another one that still weirds me out a little. The harmonic series--the sum of all numbers 1/n for integer n--is infinite: the sum for n running from 1 to k is about log(k), so it grows without bound as you include more terms. Euler proved that the same is true if you only include reciprocals of primes, although the divergence is even slower: it grows like log(log(k)).

On the other hand, if you include only reciprocals of numbers whose decimal representation does not contain the digit '9', the sum is finite.* The same holds true if you replace '9' with any finite-length string of digits, or 'decimal' with any other base.

It seems, intuitively, you are removing much "more" from the series by excluding composite numbers than by excluding a digit or digit string, but it's the latter that causes convergence of the series. Yet another example to file under "series summation isn't

*Proof by counting: if n has k decimal digits, 1/n is bounded by 10^-(k-1). After deleting those with the digit '9' there are 8*9^(k-1) numbers with k digits. As a result, you can bound the sum of all terms with exactly k digits in the denominator by 8*0.9^-(k-1); therefore, there's a subsequence of partial sums that is bounded by the partial sums of a convergent geometric series. Since the partial sums increase monotonically, we're done.

posted by egregious theorem at 8:21 AM on February 15, 2016

On the other hand, if you include only reciprocals of numbers whose decimal representation does not contain the digit '9', the sum is finite.* The same holds true if you replace '9' with any finite-length string of digits, or 'decimal' with any other base.

It seems, intuitively, you are removing much "more" from the series by excluding composite numbers than by excluding a digit or digit string, but it's the latter that causes convergence of the series. Yet another example to file under "series summation isn't

*really*addition," I guess.*Proof by counting: if n has k decimal digits, 1/n is bounded by 10^-(k-1). After deleting those with the digit '9' there are 8*9^(k-1) numbers with k digits. As a result, you can bound the sum of all terms with exactly k digits in the denominator by 8*0.9^-(k-1); therefore, there's a subsequence of partial sums that is bounded by the partial sums of a convergent geometric series. Since the partial sums increase monotonically, we're done.

posted by egregious theorem at 8:21 AM on February 15, 2016

e.g.: to put it another way, the density of primes is ~ 1/log(x), and the density of numbers without a 9 is ~ (9/10)^log(x) since there are about log(x) digits in x. So in units of k = log(x), removing the composites leaves you with 1/k, vs. (9/10)^k from removing the 9s.

That doesn't prove the sum of the latter is finite, but it's an easy way to see that primes are far more common than numbers without 9.

posted by mubba at 10:12 AM on February 15, 2016 [3 favorites]

That doesn't prove the sum of the latter is finite, but it's an easy way to see that primes are far more common than numbers without 9.

posted by mubba at 10:12 AM on February 15, 2016 [3 favorites]

Ah, of course. I always overlook things like "there are log(x) digits in x" when I'm thinking about stuff like this. Thanks--now that no longer has to bug me.

posted by egregious theorem at 11:28 AM on February 15, 2016

posted by egregious theorem at 11:28 AM on February 15, 2016

Looks like Brouwer's fixed-point theorem hasn't been mentioned yet...

posted by klausness at 12:05 PM on February 15, 2016 [1 favorite]

posted by klausness at 12:05 PM on February 15, 2016 [1 favorite]

*That there are more reals than natural numbers.*

That doesn't sound all that surprising by itself. What makes it surprising is that that there are the same number of rational numbers as natural numbers, but there are more real numbers. In fact, we can throw in all the roots (square roots, cube roots, etc.) of rational numbers (giving us the algebraic numbers), and we still wouldn't have more than the natural numbers. Almost all real numbers are transcendental.

posted by klausness at 1:48 PM on February 15, 2016 [1 favorite]

Nobody's mentioned the Hairy Ball Theorem yet? ☹

posted by un petit cadeau at 4:30 PM on February 15, 2016 [1 favorite]

posted by un petit cadeau at 4:30 PM on February 15, 2016 [1 favorite]

Godel gets my vote, not just because of what it says about the basic nature of logic, truth and formal systems, but that in breaking mathematics quite so thoroughly it led to Turing's insights and the development of computers - which are simply superb mathematical machines that have made raw live applied mathematics the underlying practical genius of the entire modern world.

Practical paradox is as close as I'll ever get to genuine magic.

posted by Devonian at 6:46 PM on February 15, 2016 [1 favorite]

Practical paradox is as close as I'll ever get to genuine magic.

posted by Devonian at 6:46 PM on February 15, 2016 [1 favorite]

*Normality is another example where the only known instances that satisfy the property are constructed.*

Your statement could become false, though, if someone were to prove the (widely-suspected, I think?) normality of pi. There's isn't any such pre-existing candidate for Kinchin's constant (I have to say that I'm a little disappointed that the property doesn't hold for Kinchin's constant itself).

posted by Jpfed at 6:50 PM on February 15, 2016

Gregory Chaitin's work on undecideability never fails to boggle my mind a bit. It builds on Turing's and Church's work and on Berry's paradox (which is related to the Busy Beaver problem above) then takes a left turn into magic. Fascinating stuff

posted by Death and Gravity at 9:39 AM on February 16, 2016

posted by Death and Gravity at 9:39 AM on February 16, 2016

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