1. It's impossible to comb all the hairs on a tennis ball in the same direction without creating a cowlick. [This is called the hairy ball theorem].
2. There is an entire book explaining how 1+1=2 works and makes sense.
3. The guy that proved the existence of irrational numbers was murdered for his finding.
4. There's exactly 10! seconds in 6 weeks.
n! (pronounced n factorial) means n*(n-1)*(n-2)*(n-3)*....*2*1. So 10! = 10*9*8*7*6*5*4*3*2*1 = 3,628,800.
5. The decimal fractions of seven are the same six recurring digits, in the same order, but starting from a different one each time.
1/7 = 0.142857142857...
2/7 = 0.285714285714...
3/7 = 0.428571428571...
4/7 = 0.571428571428...
5/7 = 0.714285714285...
6/7 = 0.857142857142...
6. The Fibonacci sequence is encoded in the number 1/89:
1/89 = 0.01 + 0.001 + 0.0002 + 0.00003 + 0.000005 + 0.0000008 + 0.00000013 + 0.000000021 + 0.0000000034...
x% of y = y% of x
y(x/100) = x(y/100)
yx/100 = xy/100
So, in order to calculate a percentage in your head it might be easier to turn it around.
What is 2% of 50? It's the same thing as 50% of 2.
8. The creators of Futurama actually had to create a new mathematical formula to solve the brain swapping issue they had. It's called The Futurama Theorem
9. If you put 23 people in a room, there's over 50% chance that at least two of them have the same birthday.
10. The BanachTarski paradox: Given a solid ball, there exists a decomposition of the ball into a finite number of disjoint subsets, which can be put back together in a different way to yield two identical copies of the original ball.
11. There are different kinds of infinity, and some are larger than others.
22/7 is closer to the actual value of pi than 3.14 is.
it relates some of the most important mathematical constants into one equation.
edit: Since there seems to be a lot of confusion or wonder as to why this is so neat, read the explanation of its mathematical beauty.
14. How big is the number of possible permutations when shuffling a 52 cards deck. Specifically the example to give us the faintest perception of how ridiculously big 52 Factorial is.
EDIT II: In this video posted by another redditor the guy uses a different example:
"If every star in our galaxy had a trillion planets, each with a trillion people and each of these people had a trillion pack of cards, and somehow they managed to shuffle all a thousand times at second, and they have been doing this since the Big Bang, they would only just now being starting to repeat shuffles."
15. Tetration is the iteration that follows exponentiation.
n a = aaaa... with an n number of a's.
Also, it is referred to as "a to the superpower of n"
with your friends!