one way to think about the function e to
the T is to ask what properties does it
have probably the most important one and
from some points of view the defining
property is that it is its own
derivative together with the added
condition that inputting zero returns
one it's actually the only function with
this property and you can illustrate
what this means with a physical model if
e to the T describes your position on a
number line as a function of time then
you start at the number one and what
this equation is saying is that your
velocity the derivative of position is
always equal to that position the
farther away from zero you are the
faster you move so even before knowing
how to compute e to the T exactly going
from a specific time to a specific
position this ability to associate each
position with a velocity paints a very
strong intuitive picture of how the
function must grow you know that you'll
be accelerating and at an accelerating
rate with an all-around feeling of
things getting out of hand quickly and
if you add a constant to that exponent
like e to the two times T the chain rule
tells us that the derivative is now two
times itself so at every point on the
number line rather than attaching a
vector corresponding to the number
itself first double the magnitude of the
position then attach it moving so that
your position is always e to the two T
is the same thing as moving in such a
way that your velocity is always twice
your position the implication of that
too is that our runaway growth feels all
the more out of control if that constant
was negative say negative zero point
five then your velocity vector is always
negative zero point five times your
position vector meaning you flip it
around 180 degrees and scale its length
by a half
moving in such a way that your velocity
always matches this flipped and squished
copy of your position vector you'd go
the other direction slowing down in an
exponential decay toward zero but what
about if that constant was I the square
root of negative one if your precision
was always e to the I T how would you
move as the time T takes forward
well now the derivative of your position
will always be I times itself and
multiplying by I has the effect of
rotating numbers 90 degrees so as you
might expect things only make sense here
if we start thinking beyond the number
line and in the complex plane so even
before you know how to compute e to the
I times T you know that for any position
this might give for some value of time
the velocity at that time will be a 90
degree rotation of that position drawing
this for all possible positions you
might come across you get a vector field
where as usual with vector fields you
shrink things down to avoid clutter at
time T equals zero e to the I T will be
one that's our initial condition and
there's only one trajectory starting
from that position where your velocity
is always matching the vector that it's
passing through a 90 degree rotation of
the position it's when you go around a
circle of radius one at a speed of one
unit per second
so after PI seconds you've traced a
distance of pi around so e to the I
times pi should be negative one after
tau seconds you've gone full circle e to
the I times tau equals one and more
generally e to the I times T equals a
number that's T radians around this unit
circle in the complex plane
nevertheless something might still feel
immoral about putting an imaginary
number up in that exponent and you would
be right to question that what we write
as e to the T is a bit of a notational
disaster giving the number e and the
idea of repeated multiplication we more
emphasis than they deserve but my time
is up so I'll spare you the full rant
until the next video
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