Tip:
Highlight text to annotate it
X
e^(pi*i) = -1
e^(pi*i) = -1
many people regard this formula as one of the most beautiful in mathematics.
I think it's beautiful because it ties
a lot of different areas in mathematics
together:
trigonometry
elements of the types of limits that you learn in calculus
and the beginnings of
complex numbers and
complex analysis
but the question is: what dose it mean
to raise a number like e
to an imaginary power? Crazy, right?
1 = sqrt(-1)
That means when we square i,
we get negative one.
If you think about that for a little while you'll see why it's unusual
when you square
*real* numbers you always get positive answer
if I square a negative number (like negative two squared)
I get positive 4
and if I square a positive number
like four, I get positive 16. Always positive! Either way!
so in order to square something and get a negative number
it has to be very unusual
indeed!
and, in fact that's what
we call i, it is the number is designed to be squared
to make it negative
so, question still remains what does it mean
to take a number like e
too the pi i power?
the pi part isn't too bad
because pi
is a real number
3.1415 ... roughly, butt i
isn't even real, come on...
so what does it do? Well,
what you'll see from the proof
that I'm going to offer today
is that this ends up being
exactly negative one. So let's embark on our journey!
e is equal to the limit as n goes to infinity of one plus one over n to the n power
this is the definition of e, that means that e
to the x
is equal to the limit as n goes to infinity of one plus one over n to the n-x power
now for the purposes of our proof
it will be more convenient
if we write this limit in a slightly different form. so let's let y be equal to n times x
that way I can replace n-x with y.
and if y is equal to n-x then we also know
that x over y will be equal to one over n
(I just manipulated that equation in to the second one.)
so now, making the replacements we have
e to the x is equal to the limit
so i can replace the limit as n goes to infinity witha limit as y
goes to infinity of 1 plus
(replace 1 over n with x over y)
and replace n-x with y. so this is
a definition of e that will be better for us to work with
the next formula that I would like to use
is called the binomial theorem
it says what happens when we raise (a+b) to soe power.
that's in part
elements
city and power
diplomacy
this formula allows us to
c_n_n_'s
i don't know it's like
so now i'm excited about coming up
teeth here
weaknesses
has lied
wilson
of the songs
zero
supplier i'm going to be placed flowers again
shoes changed
no matter who's in
and multiple this place actually without it
easy credit
she is zero
zero
zeros
and somewhere
thursday correct
as follows
talent
complex freshen up what i can see about
the sandwich it's your own sounds fun
indiscretions power so that that's most well salient points
about what she was mine
that's going to be
what actuarial
lamar and i just want to
testimony
you
along with this work here this is an automatically
now the next time
respect them
next ceremony as well
all right thanks for your home
times lives there
what we'll see what happened is that
we cancel out
not getting enough supplies
and wimax one
times my mind
who made for that part
timestamps
and something happens to the next time
head of state for the next step
for me
now i want to see it
blasted nothing here
and i see one of his one person in science and in return
com so i'm going to fight limited by a quiet
and his attorney
products in the united states
well i don't know what it's like should
finally signed the letter by monday morning
science it's sad
mudslides
prefect
so now the
every session heading to happen
hasn't happened says one of those things
as quite close to finishing one of the weapons zero cynicism planks fashioned
after clinic is a key to the act
easy answers
this is the trustee recession
for he said yes
and once you have each of the accident twelve groceries
extensively
steven nation ***
its limits
ensue if your
next