Math Nerds, unite!

Chances are if you ask the average person about a math fact, the best they'll be able to come up with is "2+2=4." But not these Redditors. We all know math has to have some cool, complicated concepts, but how could we ever understand them?

Reddit user xxTick asked the masses:

What's the coolest mathematical fact you know of?

Here are some answers that will make you reconsider those days you skipped intro to calc.

e (2.718281828459045...) is the average number of random numbers between 0 and 1 that must be added to sum to at least 1.

there are exactly 10! seconds in six weeks.

10! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 how many seconds in 6 weeks? 6 weeks x 7 days x 24 hours x 60 minutes x 60 seconds = (2 x 3) x 7 x ( 2 x 3 x 4) x (2 x 3 x 10) x (5 x 6 x 2) combine the 3's, combine the extra 2's, stick a 1 in front... = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 seconds.

Cantor's diagonal proof which implies more than one infinity. At least for classical mathematicians.

Gabriel's horn, the volume of the cone is finite, but the surface area is infinite.

https://en.m.wikipedia.org/wiki/Gabriel%27s_Horn

As a PhD student in mathematics, this is not a sexy answer, but one of the reasons I fell in love with math was in my differential equations course when we discussed modeling epidemic using mathematical equations. It was so incredible to me that back in 1927, Kermack and McKendrick came up with a simple formulation of how to model a disease. This idea has been expanded greatly, but their original version of the S-I-R compartmental model is still one of the coolest things. And it can also model rumors as well!

I love Fermat's Last Theorem:

no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.

It just intuitively seems that some n should work, given infinite possible numbers, but it's been proven that nothing but 2 fits.

If you take enough random steps in two dimensions, you'll always eventually get back to your starting point. The same cannot be said of three dimensions.

I just find the idea that you will always get back to where you started by making random moves absolutely mind boggling, and the fact things change just because you can go up and down is even weirder.

ii = 0.20787957635

So an imaginary number to an imaginary power is a real number.

99999999999999999989 is the largest prime number that can also be a Reddit username.

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321

And on it goes

The maximum number of moves needed to solve a Rubik's cube from any configuration is a mere 20.

Expecting Numberphile subscribers to have a strong showing in this thread.

There is a prime number named after one of Satan's devils. It is a 1 followed by 13 0s. 666. 13 more 0s. And a 1.

The Collatz Conjecture: It's an unsolved mathmatical conjecture that can be summarized as follows; Take any positive integer, or "n". If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. For example, start with 21. it's odd so I multiply by 3 and add 1, to get 64. 64 is even so I divide by 2 to get 32, again to get 16, 8, 4, 2, 1. No one has found a number that doesn't follow this rule.

The Birthday Problem.

If you have 23 people in a room, there is a 50% chance that at least two of them have the same birthday. If you put 70 people in, the probability jumps to 99.9%.

It seems f-cking weird to me but I haven't done math since high school so what do I know.

if you fold a piece of paper 103 times, the thickness of it will be larger than the observable universe - 93 billion light-years

Banach-Tarski paradox, in a nutshell what it says is that if you take a (let's make it simpler) 3 dimensional ball, you can partition it in finite number of pieces (which is only true for 3-dim case, otherwise it's countably infinite) and then rotate and translate some of the pieces and you can get two exactly identical balls that we started with. So you might think we doubled the volume, indeed we did.

Astronomer here! Do you remember a few months ago when NASA announced the discovery of seven Earth-sized planets around a star called TRAPPIST-1? Astronomers and mathematicians freaked out a bit about it because it turned out all those planets were in resonance, where objects orbit in a simple multiplicative of another (so, if Earth were to orbit the sun one time every time Venus orbited twice- not really the case). These simple ratios can be good in celestial mechanics for sure- Pluto crosses Neptune's orbit, for example, but they are in a 2:3 resonance so will never crash into each other. But it's also very likely to lead to amplified gravitational forces that then eject planets, and frankly, TRAPPIST-1 should not be stable based on the resonances we see there and is just very luckily in a few million year gap or so where that system can exist according to mathematics and computer simulations.

The cool thing about this though is resonance is a mathematical concept that just describes vibrations, from that in a violin string to stability in a bridge. And acoustic resonance is very important for making music sound good- some resonances work, some make music sound "bad."

The cool thing here though is because mathematics shows up in everything, some Canadian astronomers realized you can "hear" TRAPPIST-1 because it has "good" resonances. (No really, they tried other systems, but apparently they all sounded awful.) They sped up the orbits of the system 212 million times (so you wouldn't have to wait ~18 years to hear the full piece), and frankly the resulting piece is pretty awesome. You should check it out!

Math is everywhere!

One of my favorite is about the number of unique orders for cards in a standard 52 card deck.

I've seen a a really good explanation of how big 52! actually is.

Set a timer to count down 52! seconds (that's 8.0658x1067 seconds)

Stand on the equator, and take a step forward every billion years

When you've circled the earth once, take a drop of water from the Pacific Ocean, and keep going

When the Pacific Ocean is empty, lay a sheet of paper down, refill the ocean and carry on.

When your stack of paper reaches the sun, take a look at the timer.

The 3 left-most digits won't have changed. 8.063x1067 seconds left to go. You have to repeat the whole process 1000 times to get 1/3 of the way through that time. 5.385x1067 seconds left to go.

So to kill that time you try something else.

Shuffle a deck of cards, deal yourself 5 cards every billion years

Each time you get a royal flush, buy a lottery ticket

Each time that ticket wins the jackpot, throw a grain of sand in the grand canyon

When the grand canyon's full, take 1oz of rock off Mount Everest, empty the canyon and carry on.

When Everest has been levelled, check the timer.

There's barely any change. 5.364x1067 seconds left. You'd have to repeat this process 256 times to have run out the timer.

Bayes' theorem.

Suppose a drug test is 99% sensitive and 99% specific. That is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users. Suppose that 0.5% of people are users of the drug. If a randomly selected individual tests positive, what is the probability that he is a user?

The answer is around 33.2%

69! (69 factorial; approximately 1.711224524×1098 ) is the largest factorial number that most hand-held calculators can handle. This is because it also happens to be the last factorial number that is less than a googol (10100 ), and these calculators can't handle numbers larger than a googol.

1729 is the smallest number that is the sum of two positive cubes in two different ways:

1729 = 1^3 + 12^3 = 9^3 + 10^3

Graham's number! Short version: it's really big. I'll try to explain how big, but you won't understand it. You literally can't. I'll explain that bit, too.

First, we need to understand iterative operations. We'll start with easy stuff, but we'll get to the fun stuff soon. First, a so-called "zero order" operation called the "sequence function." If you give it a number, it gives the next one. So if you give it a four, it gives a five. If you give it 283, it returns 284.

Now, the main first order operation is used as shorthand for how many times you want to do the sequence function. You can take a six, and say "start here, and do the sequence function four times." You'll end up with ten. You might recognize this as addition. 6+4 just means 6 -> 7 ->8 -> 9 -> 10.

Now, the second-order function is a way to compress a lot of addition. If you want to take six and add it until you have four sixes together, you write 6 x 4, which means 6 + 6 + 6 + 6. Multiplication, of course.

Exponentiation is just iterated multiplication: 64 just means four sixes, multipled: 6 x 6 x 6 x 6.

That's as far as most people need to know, but you can keep going. Tetration is iterated exponentiation. 6 tetrated by four means four sixes raised to each other: 6666. And 7 pentated by three means seven tetrated by seven tetrated by seven.

Now we're ready to begin. We're going to start with three sexated by three. That is, three pentated by three pentated by three, where three pentated by three equals three tetrated by three tetrated by three, and that tetration means 333 = 7.6 billion. So if you take 3333333... until you have 7.6 billion threes, you'll have three pentated by three. This number is incomprehensibly large. Trust me. Then if you pentate three by that number, you'll have three hexated by three. And this number is truly beyond the realm of human comprehension. But this number is not Graham's number. This number is called G(1).

Notice how each level of operations creates huge numbers far, far faster than even one level down. Sequentation is just counting. Addition gets bigger numbers a little faster. Multiplication with small numbers can get you into the hundreds quickly. Exponentiation very swiftly takes us into pretty big numbers, and tetration accelerates much faster than most real-world things ever call for. Remember how even just with two threes, tetration creates 7 billion.

Now, remember G(1)? What we're going to do now is take two threes, and the operation we're going to perform on them is a G(1)-order operation. Even one step up the operation orders makes a tremendous difference. Now we're taking a number of steps that is an unbelievable number. And when we're done, we have a number we'll call G(2).

Now keep going. Don't even begin to think of how big G(2) is. It's actually impossible. Just do a G(2)-order operation on two threes, and call it G(3). And then keep going. I'll skip to the end now: Graham's number is G(64).

I want to explain why I said you literally can't imagine it. I was not exaggerating. It's been proven, because numbers are information, and information has a fundamental relationship with entropy, and entropy with energy, and energy with mass. All that means that there is no way, even with quantum physics, to compute this number, in any fashion, without something that cannot exist.

Do you know the Planck length? The smallest measurable space that exists, the resolution size of reality. There are about 100000000000000 of them to cross the approximate diameter of a quark. Now imagine that every cubic space on Planck3 could be used to store one binary digit. One quark would have 10 with about 3000 zeroes of them, enough to store information about every atom in the solar system. But we don't need one quark. If we stored a bit on every cubic Planck length in the known universe we would still not have enough space to store Graham's number. You wouldn't even fit G(1). A complete computation of G(1) would literally destroy the universe.

That's what I love about Graham's number. We begin with numbers that without exaggeration are too big to fit in our reality, and then raise them to powers beyond comprehension. It's not nuclear overkill. It's cosmic scales of nuclear overkill repeated in terms no one can imagine, all before we've even really begun, and the power of words is exhausted. And yet... we can write it, in a recursive formula, on a sticky note of the palm of your hand in about thirty seconds.

Of course, it's not the biggest number. You could have Graham's number plus one. Graham's number times 2. G(65). G(Graham's number). But at that point, what difference does it make? If math is the language of the universe, what's the point of numbers the universe itself can never represent? Human language is the greatest limiting factor in human thought and communication, but human thought cannot keep pace with its own vision into the language of math.

Graham's number: for those times when someone's just learned Googolplex and you need to top them. Just make sure that guy's not in the room who knows about TREE functions.

Monty Hall Problem

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

The answer is yes.