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Current time:0:00Total duration:10:17

[SINGING] On the first
day of Christmas, my true love gave to me,
the multiplicative identity. I always hated the song
the "12 days of Christmas" when I was younger. Not that the tune
or the words are any worse than any
other Christmas song, but it's just so
long and repetitive. Singing it sucks too,
because like math class, it seems no matter how hard
you try to pay attention, you lose focus out of the
sheer tedium of it all and forget where you are. Unless you keep vigorous records
by drawing complicated graphs. [SINGING] On the
second day of Christmas my true love gave to
me, the only even prime, and the absolute value
of e to the i pi. See? It doesn't have
to be repetitive, but the "12 days of Christmas"
is more fun to think about, then to actually
sing or listen to. I do like these kinds
of visualizations. And you can think of
other ways to visualize the process of
singing the "12 Days." Which might come
in useful next time you're at a family
Christmas party, and someone insists on singing
through the whole thing. [SINGING] On the
third day of Christmas my true love gave
to me, the number of spatial dimensions, at
least macroscopically-- don't yell at me string
theorists-- the number of points that define
a line, and the limit of sine x over x
as x goes to zero. Unlike a normal, reasonable
songs, adding 12 more verses wouldn't just make
it twice as long, because the stupid thing
just grows and grows. Even 99 bottles of
beer on the wall has the promise of
getting to zero, unless you believe in anti-beer. But "12 Days" is
just disheartening, because the closer you think
you're getting to the end, the longer the verses get. Dragging it out
to the bitter end. [SINGING] On the
fourth day of Christmas my true love gave to me,
the smallest possible number of sides on a
polyhedron, the number of points that define a plane,
the divisor of even numbers, and any other number
to the power of 0. If I had a time machine, and was
not bitterly anti-time travel-- and yes I've actually
protested with a sign-- one of the things
on my list would be to go back to the year 0,
pick your era, and be like, hey three kings of orient,
hurry it up, will ya? Also what's myrrh? Because it sounds like
the noise a camel makes. Really though, five
days of Christmas would be more reasonable. Even eight I could live with. [SINGING] On the fifth
day of Christmas, my true love gave to
me, five golden ratio producing pentagons,
the number of sides on a square, the number of
sides on that rigid, functional, and beautiful creature
called the triangle, and I guess a two,
and the number of sides on a Mobius strip. Graphing the numbers like
this may seem trivial, but I think it's
nice to be reminded that these numbers aren't
just squiggles on a page, symbols to be manipulated, but
actually represent something. And I think it's
nice that it results in such a lovely triangle. You know how I feel
about triangles. [SINGING] On the sixth
day of Christmas, my true love gave to me,
my name in Roman numerals, number of feet in
iambic pentameter, my name in Roman
numerals backwards, the first Mersenne
prime, the number of syllables in a foot of
iambic pentameter, and sine x squared plus cosine x squared. Right. So if you want to know
how many times you're going to have to sing
a line about whatever stuff your true
love got you, it's like counting up all the things
in one of these triangles. That has however many
layers, in this case 12. The answer will be what's
called a triangular number, for obvious reasons. You can also shift
the things around to get an equilateral
triangle, which is how triangular numbers
are usually explained. So the first triangular
number is 1, the second is 3, the third is 6,
the fourth is 10. [SINGING] On the seventh
day of Christmas, my true love gave to me, the
most common lucky number, the first perfect number, the
only prime ending in five. The number of colors
sufficient for coloring in a map, the only
prime triangular number, the highest number that is its
own factorial, and 1/2 plus 1/4 plus 1/8 plus 1/16 and so on. Almost. Here's a famous
story that I've heard told a few different ways, and
the actual facts are fuzzy, but basically, here's
the gist of it. Carl Gauss was bored
in his math class. I imagine why, because not only
was Gauss a pretty smart guy, but math classes
sucked 300 years ago, just as much as they suck today. And Gauss would get himself
in trouble when he was bored. Maybe he also liked to
escape via the window. So his teacher got
fed up and was like, Gauss-- I mean, he
probably called him Carl, but that's not
the point-- Guass, if you're so freaking
bored with my class, how about you go
sit in the corner and add up all the
numbers between 1 and 100. That ought to keep
you busy for a while. So Gauss goes to the corner,
but he's just sitting there. And the teacher gets all mad
and he's like, hey Gauss. I guess that means
you've already added up all those
numbers, right? And Gauss is like,
sure, it's 5,050. [SINGING] On the eighth
day of Christmas, my true love gave to me,
the only perfect cube that's a Fibonacci
number besides one, the number of Frieze patterns,
the number of sides on a cube, the number of platonic solids,
the first composite number, the number of regular polytropes
in all dimensions greater than four, the
Euler characteristic of polyhedra
homeomorphic to a sphere, and the number where if
it's the base of a logarithm it's undefined. His teacher of course,
didn't believe him. I think the teacher spent
the next 10 minutes adding up the numbers by hand in an
effort to catch Gauss in a lie, and when he saw that
Gauss was right, he probably gave him
detention any way. Or more likely whacked
him with a ruler a few times for having the
gall to be smarter than him. Or it could be that the
story's mostly made up. I don't know. Nevertheless, here's
how he did it. Instead of adding up the
numbers individually, like his teacher did, which
would have been super boring, he realized this fact. The numbers 1 through 100 come
in pairs that add up to 101. 1 plus 100. 2 plus 99. 3 plus 98. 4 plus 97. And so on. There's 50 such pairs of
numbers, and 50 times 101 is really easy to
do in your head, because it's 50 times
100 plus 50, or 5050. [SINGING] On the
ninth day of Christmas my true love gave to me, an
upside down six, infinity sideways, flipped
over L, an upside down nine, a funny looking
S, a sail for a boat, backwards E, half
a heart on a plate, and a simple, boring,
short and straight line. So yeah. If you want to add up all the
numbers between 1 and anything, this trick works. For example, all the
numbers between 1 and 12. 1 plus 12 is 13. 2 plus 11 is 13. All the way in to 6 plus 7. That's 6 times 13, which I do
my head as 60 plus 18 equals 78. So 78 is the 12th
triangular number, while 5,050 is the
100th triangular number. At least it's not "100
Days of Christmas." I'll stick with
"Bottles of Beer." [SINGING] On the tenth
day of Christmas, my true love gave
to me, the base of our Arabic numeral system,
the base of a nonary numeral system, the base used in octal,
and the base of septinary, and the base of senary,
and a number five, quaternary's base, ternary's
base also, and binary two, and the base of unary. Here's my favorite way of
visualizing the triangular numbers. Say you've got them in this
configuration where they make a nice right equilateral
triangle, or half a square. Finding the area of
a square is easy, because you just square
the length of it. In this case, 12 times 12. And the triangle
is half of that. Only not really, because
half the square means you only get half
of this diagonal, so you've got to add
back in the other half. But that's easy because there's
12 things in the diagonal, and 12/2 is 6. So to get the n-th
triangular number, just take n squared
over 2 plus n over 2, or n squared plus n over 2. [SINGING] On the eleventh
day of Christmas, my true love gave
to me the number my amp goes up to, the number
of fingers on my hands, the German word for no,
what I did after I eat, the number of heads on a
hydra, at least until you start cutting them off,
the number of strings on a guitar, the number
I like to do high, the amount of horsemen
of the Apocalypse, the number of notes
in a triad, the number of pears in a pair of
pears, and the number of partridges in a pear tree. [SINGING] On the twelfth
day of Christmas, my true love gave to me-- Actually, enough is enough. Merry Christmas.