Introduction to light

WEBVTT
Kind: captions
Language: en

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What I want to do in this
video is give ourselves

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a basic introduction to
the phenomenon of light.

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And light is, at least
to me, mysterious.

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Because on one level it
really defines our reality.

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It's maybe the most defining
characteristic of our reality.

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Everything we see, how
we perceive reality,

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is based on light
bouncing off of objects

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or bending around objects or
diffracting around objects,

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and then being
sensed by our eyes,

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and then sending signals
into our brain that

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create models of the
world we see around us.

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So it really is, almost,
the defining characteristic

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of our reality.

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But at the same time, when you
really go down to experiment

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and observe with
light, it starts

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to have a bunch of
mysterious properties.

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And to a large degree it is
not fully understood yet.

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And probably the most
amazing thing about light--

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well, actually there's tons of
amazing things about light--

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but one of the mysterious things
is when you really get down

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to it-- and this is actually
not just true of light,

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this is actually true
of almost anything

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once you get onto a small enough
quantum mechanical level--

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light behaves as both
a wave and a particle.

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And this is probably not
that intuitive to you,

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because it's not
that intuitive to me.

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In my life, I'm used to certain
things behaving as waves,

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like sound waves or
the waves of an ocean.

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And I'm used to certain things
behaving like particles,

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like basketballs or-- I
don't know-- my coffee cup.

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I'm not used to things
behaving as both.

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And it really depends on
what experiment you run

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and how you observe the light.

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So when you observe
it as a particle,

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and this comes out
of Einstein's work

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with the photoelectric effect--
and I won't go into the details

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here, maybe in a
future video when

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we start thinking about
quantum mechanics--

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you can view light as a
train of particles moving

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at the speed of light, which
I'll talk about in a second.

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We call these particles photons.

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If you view light in other
ways-- and you see it

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even when you see light being
refracted by a prism here--

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it looks like it is a wave.

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And it has the
properties of a wave.

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It has a frequency, and
it has a wavelength.

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And like other waves,
the velocity of that wave

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is the frequency
times its wavelength.

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Now even if you ignore this
particle aspect of light,

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if you just look at the
wave aspect of the light,

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it's still fascinating.

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Because most waves require
a medium to travel through.

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So for example, if I think about
how sound travels through air--

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so let me draw a bunch
of air particles.

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I'll draw a sound wave traveling
through the air particles.

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What happens in a sound wave
is you compress some of the air

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particles and those compress
the ones next to them.

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And so you have points in
the air that have higher,

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I guess you could
say, higher pressure

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and points that
have lower pressure,

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and you could plot that.

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So we have high
pressure over here.

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High pressure, low pressure,
high pressure, low pressure.

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And as these things
bump into each other,

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and this wave essentially
travels to the right--

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and if you were to plot
that you would see this wave

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form traveling to the right.

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But this is all
predicated, or this is all

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based on, this energy
traveling through a medium.

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And I'm used to visualizing
waves in that way.

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But light needs no medium.

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Light will actually travel
fastest through nothing,

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through a vacuum.

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And it will travel at an
unimaginably fast speed--

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3 times 10 to the eighth
meters per second.

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And just to give
you a sense of this,

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this is 300 million
meters per second.

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Or another way of
thinking about it is it

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would take light less
than a seventh of a second

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to travel around the earth.

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Or it would travel around the
earth more than seven times

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in one second.

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So unimaginably fast.

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And not only is this
just a super fast speed,

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but once again it
tells us that light

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is something fundamental
to our universe.

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Because it's not just an
unimaginable fast speed.

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It is the fastest speed not just
known to physics, but possible

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in physics.

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So once again something
very unintuitive to us

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in our everyday realm.

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We always imagine
that, OK, if something

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is going at some speed, maybe if
there was an ant riding on top

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of that something and it was
moving in the same direction,

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it would be going even faster.

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But nothing can go faster
than the speed of light.

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It's absolutely impossible based
on our current understanding

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of physics.

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So it's not just
a fast speed, it

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is the fastest speed possible.

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And this right here
is an approximation.

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It's actually 2.99
something something times 10

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to the eighth meters per second.

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But 3 times 10 to the
eighth meters per second

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is a pretty good approximation.

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Now within the visible
light spectrum-- and I'll

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talk about what's beyond
the visible light spectrum

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in a second-- you're probably
familiar with the colors.

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Maybe you imagine them as
the colors of the rainbow.

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And rainbows really
happen because the light

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from the sun, the white
light, is being refracted

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by these little water particles.

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And you can see that
in a clearer way when

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you see light being refracted
by a prism right over here.

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And the different wavelengths
of light-- so white light

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contains all of the
visible wavelengths--

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but the different wavelengths
get refracted differently

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by a prism.

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So in this case the
higher-frequency wavelengths,

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the violet and the blue,
get refracted more.

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Its direction gets bent
more than the low-frequency

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wavelengths, than the reds and
the oranges right over here.

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And if you want to look at the
wavelength of visible light,

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it's between 400 nanometers
and 700 nanometers.

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And the higher the frequency,
the higher the energy of that

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light.

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And that actually
goes into when you

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start talking about the
quantum mechanics of it--

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that the higher frequency means
that each of these photons

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have higher energy.

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They have a better ability
to give kinetic energy

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to knock off electrons or
whatever else they need to do.

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So higher frequency--
let me write

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that down-- higher frequency
means higher energy.

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Now I keep referring to this
idea of the visible light.

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And you might say, what
is beyond visible light?

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And what you'll find
is that light is just

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part of a much
broader phenomenon,

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and it's just the part
that we happen to observe.

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And if we want to broaden
the discussion a little bit,

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visible light is
just really part

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of the electromagnetic spectrum.

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So light is really just
electromagnetic radiation.

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And everything that I told
you about light just now--

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it has a wave property and
it has particle properties--

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this is not just specific
to visible light.

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This is true of all of
electromagnetic radiation.

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So at very low frequencies
or very long wavelengths--

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we're talking about
things like radio waves,

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the things that allow a
radio to reach your car;

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the things that
allow your cellphone

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to communicate with cell towers;
microwaves, the things that

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start vibrating water
molecules in your food

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so that they heat
up; infrared, which

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is what our body
releases, and that's

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why you can detect
people through walls

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with infrared cameras; visible
light; ultraviolet light,

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the UV light coming from the
sun that'll give you sunburn;

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X-rays, the
radiation that allows

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us to see through the soft
material and just visualize

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the bones; gamma rays,

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the super high energy
that comes from quasars

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and other certain types of
physical phenomena-- these

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are all examples of
the exact same thing.

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We just happen to perceive
certain frequencies

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of this as visible light.

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And you might say, hey,
Sal, how come we only

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perceive certain
frequencies of this?

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How can we only see
these frequencies?

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Literally we can see
those frequencies

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with our unaided eye.

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And the reason, or at
least my best guess

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of the reason of that, is
that's the frequency where

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the sun dumps out a lot of
electromagnetic radiation.

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So it's inundating the Earth.

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And if, as a species, you
wanted to observe things

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based on reflected
electromagnetic energy,

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it is most useful to be able to
perceive the things where there

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is the most
electromagnetic radiation.

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So it is possible that in other
realities or other planets

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there are species
that perceive more

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in the ultraviolet range
or the infrared range.

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And even on Earth,
there are some

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that perform better at
either end of the range.

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But we see really well in
the part of the spectrum

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where the sun just happens to
dump a lot of radiation on us.

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Now I'll leave you there.

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I think that's a pretty
good overview of light.

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And if any of this
stuff seems kind

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of unintuitive or
daunting, or really

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on some level confusing--
this wave-particle duality,

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this idea of a transfer of
energy through nothing--

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and it seems
unintuitive, don't worry.

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It seems unintuitive even
for the best of physicists.

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So you're already at the leading
edge of physics thinking.


i as the principal root of -1 (a little technical)

WEBVTT
Kind: captions
Language: en

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In your mathematical
careers you might

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encounter people who
say it is wrong to say

00:00:05.890 --> 00:00:10.510
that i is equal to the principal
square root of negative 1.

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And if you ask them
why is this wrong,

00:00:12.950 --> 00:00:15.550
they'll show up with this
kind of line of logic

00:00:15.550 --> 00:00:17.900
that actually seems
pretty reasonable.

00:00:17.900 --> 00:00:19.800
They will tell
you that, OK, well

00:00:19.800 --> 00:00:21.320
let's just start
with negative 1.

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We know from definition
that negative 1

00:00:23.630 --> 00:00:26.050
is equal to i times i.

00:00:26.050 --> 00:00:28.520
Everything seems pretty
straightforward right now.

00:00:28.520 --> 00:00:30.270
And then they'll say,
well look, if you

00:00:30.270 --> 00:00:33.604
take this, if you assume
this part right here,

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then we can replace each of
these i's with the square root

00:00:36.020 --> 00:00:36.860
of negative 1.

00:00:36.860 --> 00:00:37.770
And they'd be right.

00:00:37.770 --> 00:00:39.270
So then this would
be the same thing

00:00:39.270 --> 00:00:42.530
as the square root of negative
1 times the square root

00:00:42.530 --> 00:00:43.940
of negative 1.

00:00:43.940 --> 00:00:45.450
And then they would
tell you that,

00:00:45.450 --> 00:00:48.030
hey, look just from
straight up properties

00:00:48.030 --> 00:00:50.190
of the principal
square root function,

00:00:50.190 --> 00:00:53.700
they'll tell you that the
square root of a times b

00:00:53.700 --> 00:00:55.840
is the same thing as the
principal square root

00:00:55.840 --> 00:00:58.969
of a times the principal
square root of b.

00:00:58.969 --> 00:01:01.260
And so if you have the
principal square root of a times

00:01:01.260 --> 00:01:02.470
the principal square
root of b, that's

00:01:02.470 --> 00:01:04.470
the same thing as the
square root of a times b.

00:01:04.470 --> 00:01:09.520
So based on this property of the
radical of the principal root,

00:01:09.520 --> 00:01:11.150
they'll say that
this over here is

00:01:11.150 --> 00:01:13.600
the same thing as
the square root

00:01:13.600 --> 00:01:17.130
of negative 1 times negative 1.

00:01:20.290 --> 00:01:23.481
If I have the principal root
of the product of two things,

00:01:23.481 --> 00:01:24.980
that's the same
thing as the product

00:01:24.980 --> 00:01:26.600
of each of their
principal roots.

00:01:26.600 --> 00:01:28.380
I'm doing this in
the other order here.

00:01:28.380 --> 00:01:30.730
Here I have the principal
root of the products,

00:01:30.730 --> 00:01:32.730
over here I have
this on the right.

00:01:32.730 --> 00:01:35.370
And then from that we all
know that negative 1 times

00:01:35.370 --> 00:01:36.470
negative 1 is 1.

00:01:36.470 --> 00:01:41.450
So this should be equal to the
principal square root of 1.

00:01:41.450 --> 00:01:44.050
And then the principal
square root of 1-- Remember,

00:01:44.050 --> 00:01:45.830
this radical means
principal square root,

00:01:45.830 --> 00:01:51.410
positive square root, that is
just going to be positive 1.

00:01:51.410 --> 00:01:52.860
And they'll say, this is wrong.

00:01:52.860 --> 00:01:59.160
Clearly, negative 1 and positive
1 are not the same thing.

00:01:59.160 --> 00:02:02.040
And they'll argue therefore,
you can't make this substitution

00:02:02.040 --> 00:02:04.510
that we did in this step.

00:02:04.510 --> 00:02:06.080
And what you should
then point out

00:02:06.080 --> 00:02:09.199
is that this was not
the incorrect step.

00:02:09.199 --> 00:02:11.340
That it is true negative
1 is not equal to 1,

00:02:11.340 --> 00:02:13.610
but the faulty line
of reasoning here

00:02:13.610 --> 00:02:21.220
was in using this property
when both a and b are negative.

00:02:21.220 --> 00:02:26.660
If both a and b are negative,
this will never be true.

00:02:26.660 --> 00:02:36.317
So a and b both
cannot be negative.

00:02:36.317 --> 00:02:38.650
In fact normally when this
property is given-- sometimes

00:02:38.650 --> 00:02:40.120
it's given a little
bit in the footnotes

00:02:40.120 --> 00:02:42.110
or you might not even notice
it because it's not relevant

00:02:42.110 --> 00:02:43.680
when you're learning
it the first time--

00:02:43.680 --> 00:02:45.470
but they'll usually give a
little bit of a constraint

00:02:45.470 --> 00:02:45.970
there.

00:02:45.970 --> 00:02:52.300
They'll usually say for a and
b greater than or equal to 0.

00:02:52.300 --> 00:02:53.990
So that's where they
list this property.

00:02:53.990 --> 00:02:56.070
This is true for a and b
greater than or equal 0.

00:02:56.070 --> 00:03:00.930
And in particular, it's false
if both a and b are both,

00:03:00.930 --> 00:03:03.460
if they are both negative.

00:03:03.460 --> 00:03:07.310
Now, I've just spent
the last three minutes

00:03:07.310 --> 00:03:11.260
saying that people who tell you
that this is wrong are wrong.

00:03:11.260 --> 00:03:12.940
But with that said,
I will say that you

00:03:12.940 --> 00:03:16.780
have to be a little
bit careful about it.

00:03:16.780 --> 00:03:19.390
When we take traditional
principal square roots.

00:03:19.390 --> 00:03:22.070
So when you take the
principle square root of 4.

00:03:22.070 --> 00:03:24.490
We know that this is
positive 2, that 4 actually

00:03:24.490 --> 00:03:26.100
has two square roots.

00:03:26.100 --> 00:03:34.860
There's negative 2 also
is a square root of 4.

00:03:34.860 --> 00:03:36.500
If you have negative
2 times negative 2

00:03:36.500 --> 00:03:38.000
it's also equal to 4.

00:03:38.000 --> 00:03:42.470
This radical symbol here
means principal square root.

00:03:42.470 --> 00:03:44.770
Or when we're just dealing
with real numbers, non

00:03:44.770 --> 00:03:46.459
imaginary, non
complex numbers, you

00:03:46.459 --> 00:03:48.500
could really view it as
the positive square root.

00:03:48.500 --> 00:03:50.820
This has two square roots,
positive and negative 2.

00:03:50.820 --> 00:03:52.570
If you have this radical
symbol right here

00:03:52.570 --> 00:03:57.750
principal square roots, it means
the positive square root of 2.

00:03:57.750 --> 00:03:59.604
So when you start
thinking about taking

00:03:59.604 --> 00:04:01.020
square roots of
negative numbers--

00:04:01.020 --> 00:04:03.510
or even in the future
we'll do imaginary numbers

00:04:03.510 --> 00:04:05.500
and complex numbers
and all the rest--

00:04:05.500 --> 00:04:07.120
you have to expand
the definition

00:04:07.120 --> 00:04:08.930
of what this radical means.

00:04:08.930 --> 00:04:12.670
So when you are taking
the square root of really

00:04:12.670 --> 00:04:16.240
of any negative
number, you're really

00:04:16.240 --> 00:04:19.170
saying that this is no longer
the traditional principal

00:04:19.170 --> 00:04:20.420
square root function.

00:04:20.420 --> 00:04:24.080
You're now talking that this
is the principal complex square

00:04:24.080 --> 00:04:25.720
root function, or
this is now defined

00:04:25.720 --> 00:04:28.120
for complex inputs
or the domain,

00:04:28.120 --> 00:04:31.590
it can also generate
imaginary or complex outputs,

00:04:31.590 --> 00:04:34.005
or I guess you could
call that the range.

00:04:34.005 --> 00:04:36.770
And if you assume that,
then really straight

00:04:36.770 --> 00:04:42.120
from this you get that negative,
the square root of negative x

00:04:42.120 --> 00:04:47.020
is going to be equal to i
times the square root of x.

00:04:47.020 --> 00:04:49.202
And this is only-- and I'm
going to make this clear

00:04:49.202 --> 00:04:51.160
because I just told you
that this will be false

00:04:51.160 --> 00:04:52.980
if both a and b are negative.

00:04:52.980 --> 00:04:55.610
So this is only
true-- So we could

00:04:55.610 --> 00:05:06.540
apply this when x is
greater than or equal to 0.

00:05:06.540 --> 00:05:08.210
So if x is greater
than or equal to 0,

00:05:08.210 --> 00:05:10.720
then negative x is
clearly a negative number,

00:05:10.720 --> 00:05:12.370
or I guess it could also be 0.

00:05:12.370 --> 00:05:13.620
It's a negative number.

00:05:13.620 --> 00:05:16.300
And then we can apply
this right over here.

00:05:16.300 --> 00:05:18.230
If x was less than
0, then we would

00:05:18.230 --> 00:05:20.250
be doing all of this
nonsense up here.

00:05:20.250 --> 00:05:23.052
And we start to get
nonsensical answers.

00:05:23.052 --> 00:05:25.010
And if you look at it
this way and you say hey,

00:05:25.010 --> 00:05:27.740
look, i can be the square
root of negative one,

00:05:27.740 --> 00:05:30.220
if it's the principal branch
of the complex square root

00:05:30.220 --> 00:05:31.180
function.

00:05:31.180 --> 00:05:33.240
Then you could rewrite
this right over here

00:05:33.240 --> 00:05:41.040
as the square root of negative
1 times the square root of x.

00:05:41.040 --> 00:05:45.790
And so really the real fault
in this logic when people

00:05:45.790 --> 00:05:48.130
say, hey, negative 1
can't be equal to 1,

00:05:48.130 --> 00:05:52.860
the real fault is
using this property

00:05:52.860 --> 00:05:57.230
when both a and b, where both
of these are negative numbers.

00:05:57.230 --> 00:06:00.460
That will come up with something
that is unambiguously false.

00:06:00.460 --> 00:06:03.320
If you expand your
definition of the complex

00:06:03.320 --> 00:06:05.810
or expand your definition
of the principal root

00:06:05.810 --> 00:06:09.790
to include negative
numbers in the domain

00:06:09.790 --> 00:06:13.260
and to include imaginary
numbers, then you can do this.

00:06:13.260 --> 00:06:16.280
You can say the square
root of negative x

00:06:16.280 --> 00:06:19.010
is the square root
of negative 1 times--

00:06:19.010 --> 00:06:22.370
Or you should say the principal
square root of negative x--

00:06:22.370 --> 00:06:23.900
I should be
particular my words--

00:06:23.900 --> 00:06:26.010
is the same thing as the
principal square root

00:06:26.010 --> 00:06:30.120
of negative 1 times the
principal square root of x when

00:06:30.120 --> 00:06:32.110
x is greater than or equal to 0.

00:06:32.110 --> 00:06:34.240
And I don't want to
confuse you, if x

00:06:34.240 --> 00:06:39.020
is greater than or equal
to 0, this negative x, that

00:06:39.020 --> 00:06:41.080
is clearly a
negative, or I guess

00:06:41.080 --> 00:06:44.370
you should say a
non positive number.


Calculating i raised to arbitrary exponents

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.810
Now that we've seen
that as we take i

00:00:02.810 --> 00:00:06.740
to higher and higher powers,
it cycles between 1, i,

00:00:06.740 --> 00:00:11.260
negative 1, negative i, then
back to 1, i, negative 1,

00:00:11.260 --> 00:00:12.290
and negative i.

00:00:12.290 --> 00:00:14.155
I want to see if we can
tackle some, I guess

00:00:14.155 --> 00:00:15.780
you could call them,
trickier problems.

00:00:15.780 --> 00:00:17.110
And you might see these surface.

00:00:17.110 --> 00:00:18.526
And they're also
kind of fun to do

00:00:18.526 --> 00:00:22.180
to realize that you can use
the fact that the powers of i

00:00:22.180 --> 00:00:23.320
cycle through these values.

00:00:23.320 --> 00:00:25.990
You can use this to really,
on a back of an envelope,

00:00:25.990 --> 00:00:29.100
take arbitrarily
high powers of i.

00:00:29.100 --> 00:00:31.650
So let's try, just
for fun, let's

00:00:31.650 --> 00:00:35.310
see what i to the
100th power is.

00:00:35.310 --> 00:00:39.280
And the realization here is
that 100 is a multiple of 4.

00:00:39.280 --> 00:00:43.800
So you could say that this
is the same thing as i

00:00:43.800 --> 00:00:47.467
to the 4 times 25th power.

00:00:47.467 --> 00:00:50.050
And this is the same thing, just
from our exponent properties,

00:00:50.050 --> 00:00:55.167
as i to the fourth power
raised to the 25th power.

00:00:55.167 --> 00:00:57.000
If you have something
raised to an exponent,

00:00:57.000 --> 00:00:59.090
and then that is
raised to an exponent,

00:00:59.090 --> 00:01:02.300
that's the same thing as
multiplying the two exponents.

00:01:02.300 --> 00:01:04.170
And we know that
i to the fourth,

00:01:04.170 --> 00:01:05.420
that's pretty straightforward.

00:01:05.420 --> 00:01:07.390
i to the fourth is just 1.

00:01:07.390 --> 00:01:09.590
i to the fourth is
1, so this is 1.

00:01:09.590 --> 00:01:12.300
So this is equal to
1 to the 25th power,

00:01:12.300 --> 00:01:15.910
which is just equal to 1.

00:01:15.910 --> 00:01:18.867
So once again, we use this
kind of cycling ability of i

00:01:18.867 --> 00:01:20.450
when you take its
powers to figure out

00:01:20.450 --> 00:01:22.672
a very high exponent of i.

00:01:22.672 --> 00:01:24.880
Now let's say we try something
a little bit stranger.

00:01:27.730 --> 00:01:31.200
Let's try i to the 501st power.

00:01:31.200 --> 00:01:34.620
Now in this situation, 501,
it's not a multiple of 4.

00:01:34.620 --> 00:01:36.310
So you can't just
do that that simply.

00:01:36.310 --> 00:01:38.226
But what you could do,
is you could write this

00:01:38.226 --> 00:01:41.500
as a product of two
numbers, one that

00:01:41.500 --> 00:01:44.140
is i to a multiple
of fourth power.

00:01:44.140 --> 00:01:45.580
And then one that isn't.

00:01:45.580 --> 00:01:47.050
And so you could rewrite this.

00:01:47.050 --> 00:01:50.390
500 is a multiple of 4.

00:01:50.390 --> 00:01:56.000
So you could write this as
i to the 500th power times i

00:01:56.000 --> 00:01:56.960
to the first power.

00:01:56.960 --> 00:01:57.230
Right?

00:01:57.230 --> 00:01:58.070
You have the same base.

00:01:58.070 --> 00:01:59.840
When you multiply,
you can add exponents.

00:01:59.840 --> 00:02:02.960
So this would be i
to the 501st power.

00:02:02.960 --> 00:02:05.170
And we know that this
is the same thing

00:02:05.170 --> 00:02:07.920
as-- i to the 500th power
is the same thing as i

00:02:07.920 --> 00:02:10.050
to the fourth power.

00:02:10.050 --> 00:02:11.700
4 times what?

00:02:11.700 --> 00:02:14.760
4 times 125 is 500.

00:02:14.760 --> 00:02:17.280
So that's this part right
over here. i to the 500th

00:02:17.280 --> 00:02:21.510
is the same thing as i to the
fourth to the 125th power.

00:02:21.510 --> 00:02:26.150
And then that times
i to the first power.

00:02:26.150 --> 00:02:27.800
Well, i to the fourth is 1.

00:02:27.800 --> 00:02:31.690
1 to the 125th power
is just going to be 1.

00:02:31.690 --> 00:02:33.130
This whole thing is 1.

00:02:33.130 --> 00:02:37.140
And so we are just left
with i to the first.

00:02:37.140 --> 00:02:39.222
So this is going
to be equal to i.

00:02:39.222 --> 00:02:41.430
So it seems like a really
daunting problem, something

00:02:41.430 --> 00:02:43.180
that you would have
to sit and do all day,

00:02:43.180 --> 00:02:46.090
but you can use this cycling
to realize look, i to the 500th

00:02:46.090 --> 00:02:47.620
is just going to be 1.

00:02:47.620 --> 00:02:51.690
And so i to the 501th is just
going to be i times that.

00:02:51.690 --> 00:02:55.060
So i to any multiple of 4--
let me write this generally.

00:02:55.060 --> 00:03:00.450
So if you have i to any multiple
of 4, so this right over here

00:03:00.450 --> 00:03:04.030
is-- well, we'll just restrict k
to be non-negative right now. k

00:03:04.030 --> 00:03:06.380
is greater than or equal to 0.

00:03:06.380 --> 00:03:10.250
So if we have i to any
multiple of 4, right over here,

00:03:10.250 --> 00:03:16.130
we are going to get 1, because
this is the same thing as i

00:03:16.130 --> 00:03:19.280
to the fourth power
to the k-th power.

00:03:19.280 --> 00:03:22.180
And that is the same thing
as 1 to the k-th power,

00:03:22.180 --> 00:03:23.960
which is clearly equal to 1.

00:03:23.960 --> 00:03:25.510
And if we have
anything else-- if we

00:03:25.510 --> 00:03:29.340
have i to the 4k plus 1 power,
i to the 4k plus 2 power,

00:03:29.340 --> 00:03:31.640
we can then just do this
technique right over here.

00:03:31.640 --> 00:03:33.640
So let's try that with a
few more problems, just

00:03:33.640 --> 00:03:35.920
to make it clear that
you can do really,

00:03:35.920 --> 00:03:38.200
really arbitrarily crazy things.

00:03:38.200 --> 00:03:45.020
So let's take i to
the 7,321st power.

00:03:45.020 --> 00:03:47.540
Now, we just have
to figure out this

00:03:47.540 --> 00:03:52.940
is going to be some multiple
of 4 plus something else.

00:03:52.940 --> 00:03:55.870
So to do that, well, you could
just look at it by sight,

00:03:55.870 --> 00:03:58.870
that 7,320 is divisible by 4.

00:03:58.870 --> 00:04:00.270
You can verify that by hand.

00:04:00.270 --> 00:04:02.160
And then you have
that 1 left over.

00:04:02.160 --> 00:04:08.020
And so this is going to
be i to the 7,320 times

00:04:08.020 --> 00:04:09.770
i to the first power.

00:04:09.770 --> 00:04:12.905
This is a multiple of 4-- this
right here is a multiple of 4--

00:04:12.905 --> 00:04:17.240
and I know that because
any 1,000 is multiple of 4,

00:04:17.240 --> 00:04:21.209
any 100 is a multiple of 4,
and then 20 is a multiple of 4.

00:04:21.209 --> 00:04:24.497
And so this right over
here will simplify to 1.

00:04:24.497 --> 00:04:26.080
Sorry, that's not i
to the i-th power.

00:04:26.080 --> 00:04:28.960
This is i to the first power.

00:04:28.960 --> 00:04:33.240
7,321 is 7,320 plus 1.

00:04:33.240 --> 00:04:37.287
And so this part right over
here is going to simplify to 1,

00:04:37.287 --> 00:04:38.870
and we're just going
to be left with i

00:04:38.870 --> 00:04:41.100
to the first power, or just i.

00:04:41.100 --> 00:04:42.600
Let's do another one.

00:04:42.600 --> 00:04:50.860
i to the 90-- let me try
something interesting.

00:04:54.030 --> 00:04:56.230
i to the 99th power.

00:04:56.230 --> 00:04:58.860
So once again, what's
the highest multiple

00:04:58.860 --> 00:05:01.490
of 4 that is less than 99?

00:05:01.490 --> 00:05:02.590
It is 96.

00:05:05.230 --> 00:05:08.930
So this is the same thing
as i to the 96th power times

00:05:08.930 --> 00:05:11.400
i to the third power, right?

00:05:11.400 --> 00:05:14.320
If you multiply these, same
base, add the exponent,

00:05:14.320 --> 00:05:16.840
you would get i
to the 99th power.

00:05:16.840 --> 00:05:20.410
i to the 96th power, since
this is a multiple of 4,

00:05:20.410 --> 00:05:23.740
this is i to the fourth, and
then that to the 16th power.

00:05:23.740 --> 00:05:26.850
So that's just 1 to the
16th, so this is just 1.

00:05:26.850 --> 00:05:29.670
And then you're just left
with i to the third power.

00:05:29.670 --> 00:05:32.940
And you could either remember
that i to the third power

00:05:32.940 --> 00:05:35.630
is equal to-- you
can just remember

00:05:35.630 --> 00:05:36.880
that it's equal to negative i.

00:05:36.880 --> 00:05:39.270
Or if you forget that,
you could just say, look,

00:05:39.270 --> 00:05:42.480
this is the same thing
as i squared times i.

00:05:42.480 --> 00:05:45.360
This is equal to
i squared times i.

00:05:45.360 --> 00:05:48.800
i squared, by definition,
is equal to negative 1.

00:05:48.800 --> 00:05:55.340
So you have negative 1 times
i is equal to negative i.

00:05:55.340 --> 00:05:58.890
Let me do one more
just for the fun of it.

00:05:58.890 --> 00:06:01.840
Let's take i to the 38th power.

00:06:01.840 --> 00:06:03.450
Well, once again,
this is equal to i

00:06:03.450 --> 00:06:07.230
to the 36th times i squared.

00:06:07.230 --> 00:06:09.040
I'm doing i to the 36th
power, since that's

00:06:09.040 --> 00:06:11.920
the largest multiple
of 4 that goes into 38.

00:06:11.920 --> 00:06:13.730
What's left over is this 2.

00:06:13.730 --> 00:06:15.870
This simplifies
to 1, and I'm just

00:06:15.870 --> 00:06:20.530
left with i squared, which
is equal to negative 1.


Introduction to i and imaginary numbers

WEBVTT
Kind: captions
Language: en

00:00:00.906 --> 00:00:04.667
In this video, I want to introduce you to the number i,

00:00:04.667 --> 00:00:10.333
which is sometimes called the imaginary, imaginary unit

00:00:10.333 --> 00:00:12.562
What you're gonna see here, and it might be a little bit difficult,

00:00:12.562 --> 00:00:17.333
to fully appreciate, is that its a more bizzare number than

00:00:17.333 --> 00:00:20.178
some of the other wacky numbers we learn in mathematics,

00:00:20.178 --> 00:00:25.983
like pi, or e. And its more bizzare because it doesnt have a tangible value

00:00:25.983 --> 00:00:29.002
in the sense that we normally, or are used to defining numbers.

00:00:29.002 --> 00:00:35.667
"i" is defined as the number whose square is equal to negative 1.

00:00:35.667 --> 00:00:44.002
This is the definition of "i", and it leads to all sorts of interesting things.

00:00:44.002 --> 00:00:46.324
Now some places you will see "i" defined this way;

00:00:46.324 --> 00:00:50.968
"i" as being equal to the principle square root of negative one.

00:00:50.968 --> 00:00:55.240
I want to just point out to you that this is not wrong, it might make sense to you,

00:00:55.240 --> 00:00:57.800
you know something squared is negative one, then maybe

00:00:57.800 --> 00:01:00.674
its the principle square root of negative one.

00:01:00.674 --> 00:01:02.671
And so these seem to be almost the same statement,

00:01:02.671 --> 00:01:04.900
but I just want to make you a little bit careful, when you do this

00:01:04.900 --> 00:01:06.733
some people will even go so far as to say this is wrong,

00:01:06.733 --> 00:01:09.467
and it actually turns out that they are wrong to say that this is wrong.

00:01:09.467 --> 00:01:12.841
But, when you do this you have to be a little bit careful about what it means

00:01:12.841 --> 00:01:16.649
to take a principle square root of a negative number, and it being defined

00:01:16.649 --> 00:01:19.900
for imaginary, and we'll learn in the future, complex numbers.

00:01:19.900 --> 00:01:22.918
But for your understanding right now, you dont have to differentiate them,

00:01:22.918 --> 00:01:26.587
you don't have to split hairs between any of these definitions.

00:01:26.587 --> 00:01:31.333
Now with this definition, let us think about what these different powers of "i" are.

00:01:31.333 --> 00:01:33.321
because you can imagine, if something squared is negative one,

00:01:33.321 --> 00:01:37.667
if I take it to all sorts of powers, maybe that will give us weird things.

00:01:37.667 --> 00:01:40.566
And what we'll see is that the powers of "i" are kind of neat,

00:01:40.566 --> 00:01:45.302
because they kind of cycle, where they do cycle, through a whole set of values.

00:01:45.302 --> 00:01:50.225
So I could start with, lets start with "i" to the zeroth power.

00:01:50.225 --> 00:01:53.569
And so you might say, anything to the zeroth power is one,

00:01:53.569 --> 00:01:57.005
so "i" to the zeroth power is one, and that is true.

00:01:57.005 --> 00:01:59.978
And you could actually derive that even from this definition,

00:01:59.978 --> 00:02:03.646
but this is pretty straight forward; anything to the zeroth power, including "i" is one.

00:02:03.646 --> 00:02:07.000
Then you say, ok, what is "i" to the first power,

00:02:07.000 --> 00:02:11.667
well anything to the first power is just that number times itself once.

00:02:11.667 --> 00:02:14.002
So that's justgoing to be "i".

00:02:14.002 --> 00:02:16.324
Really by the definition of what it means to take an exponent,

00:02:16.324 --> 00:02:18.000
so that completely makes sense.

00:02:18.000 --> 00:02:19.993
And then you have "i" to the second power.

00:02:19.993 --> 00:02:22.933
"i" to the second power, well by definition,

00:02:22.933 --> 00:02:29.002
"i" to the second power is equal to negative one.

00:02:29.002 --> 00:02:33.000
Lets try "i" to the third power ill do this in a color i haven't used.

00:02:33.000 --> 00:02:41.634
"i" to the third power, well that's going to be "i" to the second power times "i"

00:02:41.634 --> 00:02:44.838
And we know that "i" to the second power is negative one,

00:02:44.838 --> 00:02:48.067
so its negative one times "i" let me make that clear.

00:02:48.067 --> 00:02:51.067
This is the same thing as this, which is the same thing as that,

00:02:51.067 --> 00:02:52.965
"i" squared is negative one.

00:02:52.965 --> 00:02:58.492
So you multiply it out, negative one times "i" equals negative "i".

00:02:58.492 --> 00:03:01.325
Now what happens when you take "i" to the fourth power,

00:03:01.325 --> 00:03:06.897
I'll do it up here. "i" to the fourth power.

00:03:06.897 --> 00:03:10.659
Well once again this is going to be "i" times "i" to the third power.

00:03:10.659 --> 00:03:14.235
So that's "i" times "i" to the third power. "i" times "i" to the third power

00:03:14.235 --> 00:03:21.990
Well what was "i" to the third power? "i" to the third power was negative "i"

00:03:21.990 --> 00:03:28.306
This over here is negative "i". And so "i" times "i" would get negative one,

00:03:28.306 --> 00:03:31.650
but you have a negative out here, so its "i" times "i" is negative one,

00:03:31.650 --> 00:03:35.000
and you have a negative, that gives you positive one.

00:03:35.000 --> 00:03:37.966
Let me write it out. This is the same thing

00:03:37.966 --> 00:03:43.167
as, so this is "i" times negative "i", which is the same thing as

00:03:43.167 --> 00:03:46.975
negative one times, remember multiplication is commutative,

00:03:46.975 --> 00:03:49.251
if you're multiplying a bunch of numbers you can just switch the order.

00:03:49.251 --> 00:03:52.316
This is the same thing as negative one times "i" times "i".

00:03:52.316 --> 00:03:55.984
"i" times "i", by definition, is negative one.

00:03:55.984 --> 00:03:59.839
Negative one times negative one is equal to positive one.

00:03:59.839 --> 00:04:02.672
So "i" to the fourth is the same thing as "i" to the zeroth power.

00:04:02.672 --> 00:04:05.319
Now lets try "i" to the fifth.

00:04:05.319 --> 00:04:09.266
"i" to the fifth power. Well that's just going to be "i" to to the fourth

00:04:09.266 --> 00:04:14.653
times "i". And we know what "i" to the fourth is. It is one.

00:04:14.653 --> 00:04:19.669
So its one times "i", or it is one times "i",

00:04:19.669 --> 00:04:21.341
or it is just "i" again. So once again it is exactly the same thing

00:04:21.341 --> 00:04:23.012
as "i" to the first power.

00:04:23.012 --> 00:04:25.067
Lets try again just to see the pattern keep going.

00:04:25.067 --> 00:04:27.006
Lets try "i" to the seventh power.

00:04:27.006 --> 00:04:28.400
Sorry, "i" to the sixth power.

00:04:28.400 --> 00:04:35.319
Well that's "i" times "i" to the fifth power, that's "i" times "i" to the fifth,

00:04:35.319 --> 00:04:38.570
"i" to the fifth we already established as just "i",

00:04:38.570 --> 00:04:44.235
so its "i" times "i", it is equal to, by definition,"i" times "i" is negative one.

00:04:44.235 --> 00:04:47.997
And then lets finish off, well we could keep going on this way

00:04:47.997 --> 00:04:50.598
We can keep putting high and higher powers of "i" here.

00:04:50.598 --> 00:04:52.502
An we'll see that it keeps cycling back.

00:04:52.502 --> 00:04:56.263
In the next video I'll teach you how taking an arbitrarily high power of "i",

00:04:56.263 --> 00:04:58.333
how you can figure out what that's going to be.

00:04:58.333 --> 00:05:00.489
But lets just verify that this cycle keeps going.

00:05:00.489 --> 00:05:06.667
"i" to the seventh power is equal to "i" times "i" to the sixth power.

00:05:06.667 --> 00:05:11.914
"i" to the sixth power is negative one. "i" times negative one is negative "i".

00:05:11.914 --> 00:05:14.839
And if you take "i" to the eighth, once again it'll be one,

00:05:14.839 --> 99:59:59.999
"i" to the ninth will be "i" again, and so on and so forth.


Stellar parallax

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:02.229
Let's say I'm walking
along some trail

00:00:02.229 --> 00:00:04.270
and there are some trees
on the side of the road.

00:00:04.270 --> 00:00:06.760
And let's just say
these are some plants.

00:00:06.760 --> 00:00:08.622
And these are the
barks of the trees--

00:00:08.622 --> 00:00:10.830
maybe I should do it in
brown-- but you get the idea.

00:00:10.830 --> 00:00:13.980
These are some plants that are
along the side of the road,

00:00:13.980 --> 00:00:16.905
or at least the stem of the
plant or the bark of the tree.

00:00:16.905 --> 00:00:18.780
And in the background,
I have some mountains.

00:00:18.780 --> 00:00:20.900
Maybe those mountains
are several miles away.

00:00:23.510 --> 00:00:25.770
We know just from
experience that if I'm

00:00:25.770 --> 00:00:29.270
walking-- let me draw
myself over here--

00:00:29.270 --> 00:00:32.800
we know that if I'm
walking this way,

00:00:32.800 --> 00:00:35.790
the trees look like
they're going past me much

00:00:35.790 --> 00:00:38.780
faster than the mountains.

00:00:38.780 --> 00:00:41.500
Like I'll just be going
past one tree after another.

00:00:41.500 --> 00:00:43.880
And they'll just whizz past
me-- maybe if I'm running.

00:00:43.880 --> 00:00:47.410
But the mountains don't seem
to be moving that quickly.

00:00:47.410 --> 00:00:50.670
And this idea, that as
you change your position

00:00:50.670 --> 00:00:52.280
the things that
are closer to you

00:00:52.280 --> 00:00:55.370
seem to move more than the
things that are further

00:00:55.370 --> 00:01:02.810
from you, this property
is called parallax.

00:01:02.810 --> 00:01:04.560
And what we're going
to do in this video--

00:01:04.560 --> 00:01:06.290
and maybe it's
especially obvious

00:01:06.290 --> 00:01:08.623
if you're driving in a car,
then the things close to you

00:01:08.623 --> 00:01:11.026
are whizzing by you, maybe
the curve of the street

00:01:11.026 --> 00:01:13.150
or whatever, while the
things that are further away

00:01:13.150 --> 00:01:15.544
don't seem to be
whizzing by you as fast.

00:01:15.544 --> 00:01:16.960
What I want to do
in this video is

00:01:16.960 --> 00:01:19.130
think about how we
can use parallax

00:01:19.130 --> 00:01:21.221
to figure out how far
certain stars are.

00:01:21.221 --> 00:01:23.220
And what I want to emphasize
is that this method

00:01:23.220 --> 00:01:25.870
is only good for
relatively close stars.

00:01:25.870 --> 00:01:28.020
We don't have
instruments sensitive

00:01:28.020 --> 00:01:31.320
enough yet to use parallax to
measure stars that are really,

00:01:31.320 --> 00:01:32.790
really, really far away.

00:01:32.790 --> 00:01:34.290
But to think about
how this is done,

00:01:34.290 --> 00:01:37.460
how we use stellar parallax--
just let me write "stellar"

00:01:37.460 --> 00:01:40.560
up here-- how we use stellar
parallax, the parallax

00:01:40.560 --> 00:01:43.250
of stars, to figure out
how far away they are,

00:01:43.250 --> 00:01:45.820
let's think a little bit
about our solar system.

00:01:45.820 --> 00:01:50.460
So here is our sun
in the solar system.

00:01:50.460 --> 00:01:54.530
And here is Earth at
one point in the year.

00:01:54.530 --> 00:01:56.220
And what I want to
do is-- and let's

00:01:56.220 --> 00:01:58.540
just say this is the North
Pole, kind of popping out

00:01:58.540 --> 00:01:59.440
of the screen here.

00:01:59.440 --> 00:02:03.910
And so the Earth is
rotating in that direction.

00:02:03.910 --> 00:02:07.160
And I also want to think
about a star that is obviously

00:02:07.160 --> 00:02:08.680
outside of our solar system.

00:02:08.680 --> 00:02:12.190
And I'm really underestimating
the distance to this star.

00:02:12.190 --> 00:02:14.970
As we'll see, or as
you might already know,

00:02:14.970 --> 00:02:18.460
the distance to the nearest
star from our solar system

00:02:18.460 --> 00:02:25.127
is 250,000 times the distance
between the Earth and the sun.

00:02:25.127 --> 00:02:26.960
So if I wanted to draw
this to scale-- well,

00:02:26.960 --> 00:02:29.980
first of all, the Earth would
be this unnoticeable dot here--

00:02:29.980 --> 00:02:32.210
but you would also, whatever
distance this is, you

00:02:32.210 --> 00:02:35.580
would have to multiply
that by 250,000

00:02:35.580 --> 00:02:38.280
to get the distance
to this nearest star.

00:02:38.280 --> 00:02:41.010
Anyway, with that
said, let's think

00:02:41.010 --> 00:02:43.740
about what that star would
look like from the surface

00:02:43.740 --> 00:02:44.600
of the Earth.

00:02:44.600 --> 00:02:46.767
So let me pick a
point on the surface.

00:02:46.767 --> 00:02:48.600
Maybe if we're thinking
about North America,

00:02:48.600 --> 00:02:51.160
we're right there in
the northern hemisphere.

00:02:51.160 --> 00:02:52.880
So let's take that
little patch of land

00:02:52.880 --> 00:02:55.770
and think about how the position
of that star would look.

00:02:55.770 --> 00:02:57.310
So that's the patch of land.

00:02:57.310 --> 00:02:59.250
Maybe this is my
house right over here,

00:02:59.250 --> 00:03:01.440
jutting out the
side of the Earth.

00:03:01.440 --> 00:03:02.650
Maybe this is me standing.

00:03:02.650 --> 00:03:04.720
I'm drawing everything
sideways because I'm

00:03:04.720 --> 00:03:07.260
trying to hold this perspective.

00:03:07.260 --> 00:03:09.400
So this is me looking up.

00:03:09.400 --> 00:03:11.920
And let's say at
this point in time,

00:03:11.920 --> 00:03:13.960
the way I've drawn
this patch, the sun

00:03:13.960 --> 00:03:16.860
will just be coming
over the horizon.

00:03:16.860 --> 00:03:19.740
So the sun is
essentially at sunrise.

00:03:19.740 --> 00:03:24.870
So let me do my best at drawing
the sun from my point of view.

00:03:24.870 --> 00:03:30.230
Remember, the Earth is
rotating in this way-- the way

00:03:30.230 --> 00:03:32.720
I've drawn it, it's
rotating counterclockwise.

00:03:32.720 --> 00:03:34.620
But from the
surface of the Earth

00:03:34.620 --> 00:03:36.750
it would look like the
sun is coming up here.

00:03:36.750 --> 00:03:39.090
It's rising in the east.

00:03:39.090 --> 00:03:42.380
But right at that
dawn, on this day when

00:03:42.380 --> 00:03:47.630
the Earth is right over here,
what would that star look like?

00:03:47.630 --> 00:03:51.190
So if you look at this
version of the Earth,

00:03:51.190 --> 00:03:55.500
the star is kind of skewed
a little bit-- not straight

00:03:55.500 --> 00:03:57.200
up, straight up would
be this direction,

00:03:57.200 --> 00:03:59.440
from the point of view
of my house-- it is now

00:03:59.440 --> 00:04:02.041
skewed a little bit
closer to the sun.

00:04:02.041 --> 00:04:03.790
So if you go in this
is zoomed-in version,

00:04:03.790 --> 00:04:06.140
straight up would look
something like that.

00:04:06.140 --> 00:04:07.710
And maybe based
on my measurement,

00:04:07.710 --> 00:04:15.330
it would look like the
star is right over there.

00:04:15.330 --> 00:04:17.640
So it's a little bit
skewed towards where

00:04:17.640 --> 00:04:21.690
the sun is rising, towards the
east relative to straight up.

00:04:21.690 --> 00:04:24.670
Now let's fast-forward
six months so

00:04:24.670 --> 00:04:28.250
that the Earth is on the other
side of its orbit from the sun.

00:04:28.250 --> 00:04:31.290
So let's fast-forward
six months.

00:04:31.290 --> 00:04:32.510
We're over here.

00:04:32.510 --> 00:04:37.040
And let's wait for a time of
day where we are-- essentially

00:04:37.040 --> 00:04:39.160
that little patch of
the Earth is pointed

00:04:39.160 --> 00:04:43.150
in the same direction, at
least in our galaxy, maybe.

00:04:46.660 --> 00:04:49.490
And if you think about it, if we
go back to this patch of Earth,

00:04:49.490 --> 00:04:53.540
now the Earth is still
rotating in that direction.

00:04:53.540 --> 00:04:56.150
But now the sun is on the west.

00:04:56.150 --> 00:04:58.660
The sun is going to
be right over here.

00:04:58.660 --> 00:05:01.000
Maybe I'll do it like this
just to make it clear.

00:05:01.000 --> 00:05:03.471
I'll draw this side of the
sun with this greenish color.

00:05:03.471 --> 00:05:05.720
Obviously the sun is not
green, but it will make clear

00:05:05.720 --> 00:05:08.580
that now we're about the sun
is going to be over here.

00:05:11.340 --> 00:05:14.440
The patch is going to be
turning away from the sun.

00:05:14.440 --> 00:05:16.330
So it'll look to that
observer on Earth

00:05:16.330 --> 00:05:17.460
like the sun is setting.

00:05:17.460 --> 00:05:21.380
So it'll look like the sun is
going down over the horizon.

00:05:21.380 --> 00:05:22.980
But the important
thing is, once we're

00:05:22.980 --> 00:05:26.820
at this point in the year,
what will that star look like?

00:05:26.820 --> 00:05:28.840
Well, if we have
this large diagram,

00:05:28.840 --> 00:05:34.670
we see that the star,
relative to straight up,

00:05:34.670 --> 00:05:39.160
it is now a little bit to the
west now, a little bit more

00:05:39.160 --> 00:05:40.840
on the side of that setting sun.

00:05:40.840 --> 00:05:46.290
So the star would now look
like it is right there.

00:05:46.290 --> 00:05:49.620
And if we have good
enough instruments,

00:05:49.620 --> 00:05:54.050
we can measure the angle
between where that star was

00:05:54.050 --> 00:05:56.770
six months ago and
where it is now.

00:05:56.770 --> 00:06:01.260
And let's call that
angle 2 times theta.

00:06:01.260 --> 00:06:02.760
And the reason why
I call it 2 times

00:06:02.760 --> 00:06:10.100
theta-- we could call theta
the angle between the star

00:06:10.100 --> 00:06:11.570
and being straight up.

00:06:11.570 --> 00:06:15.030
So this would be theta,
and that would be theta.

00:06:15.030 --> 00:06:17.850
And I care about that
because if I know theta,

00:06:17.850 --> 00:06:20.930
and if I know the distance
from the Earth to the sun,

00:06:20.930 --> 00:06:23.210
I can then use a little
bit of trigonometry

00:06:23.210 --> 00:06:25.630
to figure out the
distance to that star.

00:06:25.630 --> 00:06:28.400
Because if you think about
it, this theta right over here

00:06:28.400 --> 00:06:29.990
is the same as this angle.

00:06:29.990 --> 00:06:31.610
So if this is straight up.

00:06:31.610 --> 00:06:33.940
That is looking straight
up into the night sky.

00:06:33.940 --> 00:06:37.030
This would be the angle theta.

00:06:37.030 --> 00:06:40.460
If you know that angle from
basic trigonometry, or actually

00:06:40.460 --> 00:06:42.620
even basic geometry,
if you say this

00:06:42.620 --> 00:06:49.840
is a right angle over here,
this would be 90 minus theta.

00:06:49.840 --> 00:06:51.900
And then you could use
some basic trigonometry.

00:06:51.900 --> 00:06:55.940
If you know this
distance right here,

00:06:55.940 --> 00:06:58.770
and you're trying to figure
out this distance, the distance

00:06:58.770 --> 00:07:03.570
to that nearest star,
we could say, look,

00:07:03.570 --> 00:07:05.180
we need a trigonometric
function that

00:07:05.180 --> 00:07:09.430
deals with the opposite
angle of what we know.

00:07:09.430 --> 00:07:11.910
We know this thing
right over here.

00:07:11.910 --> 00:07:14.160
And the adjacent
angle-- we already

00:07:14.160 --> 00:07:15.750
know this thing right over here.

00:07:15.750 --> 00:07:19.160
So let me call this
the Earth-sun distance,

00:07:19.160 --> 00:07:22.350
or let me just call this d.

00:07:22.350 --> 00:07:24.240
And we want to figure out x.

00:07:24.240 --> 00:07:26.502
So some basic
trigonometry-- and you

00:07:26.502 --> 00:07:27.960
might want to do
this if you forget

00:07:27.960 --> 00:07:30.112
the basic trigonometric
functions-- SOHCAHTOA.

00:07:33.230 --> 00:07:35.650
Sine is opposite/hypotenuse.

00:07:35.650 --> 00:07:37.970
Cosine is adjacent/hypotenuse.

00:07:37.970 --> 00:07:41.530
Tangent is opposite/adjacent.

00:07:41.530 --> 00:07:43.890
So the tangent function
deals with the two sides

00:07:43.890 --> 00:07:47.630
of this right triangle
that we can now deal with.

00:07:47.630 --> 00:07:54.890
So we could say that the tangent
of 90 minus theta, this angle

00:07:54.890 --> 00:08:07.060
right over here--
let me write it--

00:08:07.060 --> 00:08:09.550
is equal to the
opposite side-- is

00:08:09.550 --> 00:08:15.420
equal to x over the
adjacent side, over d.

00:08:15.420 --> 00:08:18.680
Or another way, if you assume
that we know the distance

00:08:18.680 --> 00:08:21.660
to the sun, you multiply both
sides times that distance.

00:08:21.660 --> 00:08:28.610
You get d times the tangent of
90 minus theta is equal to x.

00:08:28.610 --> 00:08:33.200
And you can figure out the
distance from our solar system

00:08:33.200 --> 00:08:35.230
to that star.

00:08:35.230 --> 00:08:37.909
Now I want to make it
very, very, very clear--

00:08:37.909 --> 00:08:39.549
these are huge distances.

00:08:39.549 --> 00:08:42.020
I did not draw this to scale.

00:08:42.020 --> 00:08:44.390
The distance to the
nearest star is actually

00:08:44.390 --> 00:08:47.860
250,000 times the
distance to our sun.

00:08:47.860 --> 00:08:51.541
So this angle is going to be
super, super, super, super

00:08:51.541 --> 00:08:52.040
small.

00:08:52.040 --> 00:08:54.206
So you would need to have
very good instruments even

00:08:54.206 --> 00:08:57.070
to measure-- even to
observe-- the stellar parallax

00:08:57.070 --> 00:08:58.860
to the nearest stars.

00:08:58.860 --> 00:09:01.880
And we're constantly
having better instruments--

00:09:01.880 --> 00:09:03.680
and actually the
Europeans right now

00:09:03.680 --> 00:09:07.730
are in the process of a mission
called Gaia to measure these

00:09:07.730 --> 00:09:11.160
to enough accuracy that we can
start to measure the accurate

00:09:11.160 --> 00:09:15.071
distance to stars several tens
of thousands of light years

00:09:15.071 --> 00:09:15.570
away.

00:09:15.570 --> 00:09:18.540
So that'll start to give
us a very accurate map

00:09:18.540 --> 00:09:21.080
of a significant chunk
of our galaxy, which

00:09:21.080 --> 00:09:25.470
is about 100,000 light
years in diameter.


Simpler Insertion Sort Function

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:02.940
It was pointed out in the
comments for the last video,

00:00:02.940 --> 00:00:05.660
where I defined this
insertion_sort, that I didn't

00:00:05.660 --> 00:00:08.440
necessarily have
to do this break.

00:00:08.440 --> 00:00:09.820
And this is one
of those examples

00:00:09.820 --> 00:00:12.278
that sometimes when you just
program something-- in the way

00:00:12.278 --> 00:00:15.000
that, at least, your brain
is thinking about it--

00:00:15.000 --> 00:00:17.612
you don't always do it in
maybe the most elegant way.

00:00:17.612 --> 00:00:18.820
And that commenter was right.

00:00:18.820 --> 00:00:22.580
There was actually maybe
a lot more elegant way

00:00:22.580 --> 00:00:25.310
to actually do this
insertion_sort.

00:00:25.310 --> 00:00:27.570
So over here, if we go
into the while loop,

00:00:27.570 --> 00:00:30.300
I have this while
loop happening while i

00:00:30.300 --> 00:00:32.220
is greater than or equal to 0.

00:00:32.220 --> 00:00:34.810
But then I want it to
essentially break out

00:00:34.810 --> 00:00:42.480
of that while loop, if the
variable value is not less

00:00:42.480 --> 00:00:46.620
than the i-th element in list.

00:00:46.620 --> 00:00:49.290
So one way to do
this-- because really,

00:00:49.290 --> 00:00:52.380
I'm just defining the parameters
on when to do this while loop.

00:00:52.380 --> 00:00:55.800
Instead of doing
this if the value is

00:00:55.800 --> 00:00:58.172
less than the i-th element,
do this and otherwise break.

00:00:58.172 --> 00:01:00.380
What I could have done--
and this would have actually

00:01:00.380 --> 00:01:01.920
been a more elegant
way to do this--

00:01:01.920 --> 00:01:09.730
while i is greater than
or equal to 0 and value

00:01:09.730 --> 00:01:16.860
is less than the
i-th element in list.

00:01:16.860 --> 00:01:20.210
So now this is still
the same as before.

00:01:20.210 --> 00:01:21.844
But now I know
that value is less

00:01:21.844 --> 00:01:23.260
than the i-th
element in the list.

00:01:23.260 --> 00:01:24.930
So I will execute
this code over here.

00:01:24.930 --> 00:01:26.650
So I know that this
is going to be true.

00:01:32.190 --> 00:01:36.110
Let me take all of
these one level back.

00:01:36.110 --> 00:01:37.610
And then I don't
have to do an else.

00:01:37.610 --> 00:01:40.050
And I don't have to break
out of the loop anymore.

00:01:40.050 --> 00:01:43.320
And so that should simplify
the program a good bit.

00:01:43.320 --> 00:01:47.500
But let's verify for ourselves
that this actually works.

00:01:47.500 --> 00:01:49.120
So let me save it.

00:01:49.120 --> 00:01:51.490
And then let me run it.

00:01:56.310 --> 00:01:58.080
Let me define a list.

00:01:58.080 --> 00:02:15.460
So I'll just call it c is equal
to 1, 5, 6, 7, 2, 4, 14, and 2.

00:02:15.460 --> 00:02:16.690
So let's just define that.

00:02:16.690 --> 00:02:20.090
And let's try our
insertion_sort.

00:02:20.090 --> 00:02:22.749
Insertion_sort on c.

00:02:22.749 --> 00:02:23.790
So let's see what we get.

00:02:23.790 --> 00:02:25.930
So let's see.

00:02:25.930 --> 00:02:27.410
Let's print c now.

00:02:27.410 --> 00:02:28.340
And there you go.

00:02:28.340 --> 00:02:29.130
It's sorted it.

00:02:29.130 --> 00:02:31.190
So thank you for the comment.

00:02:31.190 --> 00:02:34.790
I think this does
simplify it a little bit.

00:02:34.790 --> 00:02:36.700
Sometimes you always
have to question,

00:02:36.700 --> 00:02:38.610
especially if you have
a while statement,

00:02:38.610 --> 00:02:40.834
is there a better way
instead of using the break.

00:02:40.834 --> 00:02:43.000
Because that's essentially--
I'm putting a condition

00:02:43.000 --> 00:02:44.880
on a reason to break
out of the while loop.

00:02:44.880 --> 00:02:48.450
But why don't I implicitly
just put it up here?


Tau versus pi

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.030
What I want to do in this
video is revisit a little bit

00:00:03.030 --> 00:00:05.210
of what we know
about pi, and really

00:00:05.210 --> 00:00:07.960
how we measure
angles in radians.

00:00:07.960 --> 00:00:11.550
And then think about
whether pi is necessarily

00:00:11.550 --> 00:00:14.510
the best number to be
paying attention to.

00:00:14.510 --> 00:00:16.900
So let's think a little
bit about what I just said.

00:00:16.900 --> 00:00:19.610
So pi, we know, is
defined-- and I'll

00:00:19.610 --> 00:00:22.900
write defined as a
triple equal sign,

00:00:22.900 --> 00:00:25.560
I guess you could call it
that way-- pi is defined

00:00:25.560 --> 00:00:28.720
as the ratio of the
circumference of a circle

00:00:28.720 --> 00:00:32.009
to its diameter, which is
the same thing as the ratio

00:00:32.009 --> 00:00:36.390
of the circumference of the
circle to two times the radius.

00:00:36.390 --> 00:00:40.060
And from that, we get all
these interesting formulas

00:00:40.060 --> 00:00:42.270
that you get in geometry
class that, hey, if you

00:00:42.270 --> 00:00:45.130
have the radius and you want
to calculate the circumference,

00:00:45.130 --> 00:00:48.400
multiply both sides of this
definition, or this equation,

00:00:48.400 --> 00:00:50.100
by two times the radius.

00:00:50.100 --> 00:00:52.660
And you get two times
the radius times

00:00:52.660 --> 00:00:55.090
pi is equal to
the circumference,

00:00:55.090 --> 00:01:01.210
or more familiarly, it
would be circumference

00:01:01.210 --> 00:01:04.985
is equal to 2 pi r.

00:01:04.985 --> 00:01:06.610
This is one of those
fundamental things

00:01:06.610 --> 00:01:08.790
that you learn early
on in your career

00:01:08.790 --> 00:01:11.060
and you use it to find
circumferences, usually,

00:01:11.060 --> 00:01:13.750
or figure out radiuses if
you know circumference.

00:01:13.750 --> 00:01:18.490
And from that comes how we
measure our angles in radians

00:01:18.490 --> 00:01:20.100
once we get to
trigonometry class.

00:01:20.100 --> 00:01:26.310
And just as a review here,
let me draw myself a circle.

00:01:26.310 --> 00:01:29.280
Let me draw myself
a better circle.

00:01:29.280 --> 00:01:33.000
So there is my--
it'll do the job.

00:01:33.000 --> 00:01:35.910
And here is the
positive x-axis, and let

00:01:35.910 --> 00:01:38.220
me make some angle here.

00:01:38.220 --> 00:01:39.760
I'll make the angle
kind of obvious

00:01:39.760 --> 00:01:44.140
just so that it-- so
let me make this angle.

00:01:44.140 --> 00:01:48.176
And the way that
we measure angles,

00:01:48.176 --> 00:01:49.550
when we talk about
radians, we're

00:01:49.550 --> 00:01:51.600
really talking about
the angle subtended

00:01:51.600 --> 00:01:53.500
by something of a
certain arc length.

00:01:53.500 --> 00:01:56.405
And we measure the
arc length in-- well,

00:01:56.405 --> 00:01:57.780
the way I like to
think about it,

00:01:57.780 --> 00:02:01.170
is the angle is in radians
and the arc length itself

00:02:01.170 --> 00:02:04.507
is in radiuses, which
isn't really a word.

00:02:04.507 --> 00:02:05.840
But that's how I think about it.

00:02:05.840 --> 00:02:09.770
How many radiuses
is this arc length

00:02:09.770 --> 00:02:13.080
that subtends the
angle in radians?

00:02:13.080 --> 00:02:15.380
So let me show you
what I'm talking about.

00:02:15.380 --> 00:02:19.600
So this arc length right
here, if the radius

00:02:19.600 --> 00:02:23.540
is r, what is the length
of this arc length?

00:02:23.540 --> 00:02:26.090
Well, we know from
basic geometry

00:02:26.090 --> 00:02:27.830
the entire
circumference over here

00:02:27.830 --> 00:02:29.730
is going to be 2 pi r, right?

00:02:29.730 --> 00:02:31.710
This entire
circumference, that's

00:02:31.710 --> 00:02:34.420
really by definition,
this entire circumference

00:02:34.420 --> 00:02:35.850
is going to be 2 pi r.

00:02:35.850 --> 00:02:37.580
So what is just this
arc length here?

00:02:37.580 --> 00:02:41.210
And I'm assuming this is
a fourth of the circle.

00:02:41.210 --> 00:02:44.100
So it's going to
be 2 pi r over 4.

00:02:44.100 --> 00:02:47.940
So this arc length
over here is going

00:02:47.940 --> 00:02:55.650
to be 2 pi r over 4, which is
the same thing as pi over 2 r.

00:02:55.650 --> 00:02:57.700
Or you could say this
is the same thing

00:02:57.700 --> 00:02:59.430
as pi over 2 radiuses.

00:03:03.510 --> 00:03:05.734
One of those-- you
know, not a real word,

00:03:05.734 --> 00:03:07.400
but that's how I like
to think about it.

00:03:07.400 --> 00:03:12.920
Or you could say it subtends
an angle of pi over 2 radians.

00:03:12.920 --> 00:03:18.710
So over here, theta
is pi over 2 radians.

00:03:18.710 --> 00:03:21.900
And so really, when you're
measuring angles in radians,

00:03:21.900 --> 00:03:24.420
it's really you're saying,
OK, that angle subtended

00:03:24.420 --> 00:03:29.340
by an arc that has a length of
how many radiusi, or I don't

00:03:29.340 --> 00:03:32.442
even know what the
plural of radius is.

00:03:32.442 --> 00:03:35.090
Actually, I think it's
radii but it's fun

00:03:35.090 --> 00:03:36.575
to try to say radiuses.

00:03:36.575 --> 00:03:38.700
Radii, actually let me do
that just so no one says,

00:03:38.700 --> 00:03:42.790
Sal, you're teaching people the
wrong plural form of radius.

00:03:42.790 --> 00:03:44.170
Radii.

00:03:44.170 --> 00:03:47.290
So this arc length
is pi over 2 radii

00:03:47.290 --> 00:03:52.590
and it subtends an angle
of pi over 2 radians.

00:03:52.590 --> 00:03:54.940
We could do another
one just for the sake

00:03:54.940 --> 00:03:56.850
of making the point clear.

00:03:56.850 --> 00:04:01.780
If you went all the
way around the circle

00:04:01.780 --> 00:04:05.100
and you got back to the
positive x-axis here,

00:04:05.100 --> 00:04:06.240
what is the arc length?

00:04:06.240 --> 00:04:09.000
Well now, all of a
sudden the arc length

00:04:09.000 --> 00:04:11.310
is the entire circumference
of the circle.

00:04:11.310 --> 00:04:18.420
It would be 2 pi r, which is
the same thing as 2 pi radii.

00:04:18.420 --> 00:04:22.260
And we would say that the angle
subtended by this arc length,

00:04:22.260 --> 00:04:23.770
the angle that we
care about going

00:04:23.770 --> 00:04:26.700
all the way around the
circle, is 2 pi radians.

00:04:30.960 --> 00:04:34.170
And so, out of this
comes all of the things

00:04:34.170 --> 00:04:37.810
that we know about how to
graph trigonometric functions

00:04:37.810 --> 00:04:40.100
or at least how we measure
the graph on the x-axis.

00:04:40.100 --> 00:04:42.640
And I'll also touch on
Euler's formula, which

00:04:42.640 --> 00:04:46.520
is the most beautiful formula,
I think, in all of mathematics.

00:04:46.520 --> 00:04:48.910
And let's visit those right
now, just to remind ourselves

00:04:48.910 --> 00:04:51.260
of how pi fits in
to all of that.

00:04:51.260 --> 00:04:53.510
So if I think about our
trigonometric functions.

00:04:53.510 --> 00:04:56.530
Remember, if this was-- so
on trigonometric functions,

00:04:56.530 --> 00:04:58.750
we assume we have
a unit circle here.

00:04:58.750 --> 00:05:02.615
So in the trig functions, this
is the unit circle definition

00:05:02.615 --> 00:05:03.490
of the trig function.

00:05:03.490 --> 00:05:06.530
So this is a nice
review of all of that.

00:05:06.530 --> 00:05:10.080
You assume you have a unit
circle, a circle of radius 1.

00:05:10.080 --> 00:05:12.510
And then the trig
functions are defined as,

00:05:12.510 --> 00:05:16.020
for any angle you have
here, for any angle, theta,

00:05:16.020 --> 00:05:21.660
cosine of theta is how
far you have to move in--

00:05:21.660 --> 00:05:26.780
or the x-coordinate of the
point along the arc that

00:05:26.780 --> 00:05:28.080
subtends this angle.

00:05:28.080 --> 00:05:29.870
So that's cosine of theta.

00:05:29.870 --> 00:05:35.785
And then sine of theta is
the y value of that point.

00:05:38.302 --> 00:05:39.260
Let me make that clear.

00:05:39.260 --> 00:05:46.310
Cosine of theta is the x value,
sine of theta is the y value.

00:05:46.310 --> 00:05:50.210
And so if you were to graph
one of these functions,

00:05:50.210 --> 00:05:53.380
and I'll just do sine of
theta for convenience.

00:05:53.380 --> 00:05:55.640
But you could try it
with cosine of theta.

00:05:55.640 --> 00:05:57.520
So let's graph sine of theta.

00:05:57.520 --> 00:06:00.840
Let's do one revolution
of sine of theta.

00:06:00.840 --> 00:06:02.840
And we tend to label
it-- so let's do sine.

00:06:02.840 --> 00:06:05.896
When the angle is 0,
sine of theta is 0.

00:06:05.896 --> 00:06:08.020
Let me draw the x- and
y-axis just so you remember.

00:06:08.020 --> 00:06:12.065
This is the y-axis and
this is the x-axis.

00:06:16.500 --> 00:06:20.630
So when the angle is 0, we're
right here on the unit circle.

00:06:20.630 --> 00:06:22.860
The y value there is 0.

00:06:22.860 --> 00:06:25.440
So sine of theta is going
to be right like that.

00:06:25.440 --> 00:06:27.160
Let me draw it like this.

00:06:27.160 --> 00:06:30.110
So this is our
theta, and this is--

00:06:30.110 --> 00:06:32.590
I'm going to graph sine
of theta along the y-axis.

00:06:32.590 --> 00:06:36.610
So we'll say y is equal to
sine of theta in this graph

00:06:36.610 --> 00:06:38.670
that I'm drawing
right over here.

00:06:38.670 --> 00:06:40.730
And then we could
do-- well, I'll

00:06:40.730 --> 00:06:43.200
just do the simple points here.

00:06:43.200 --> 00:06:44.830
And then if we
make the angle go--

00:06:44.830 --> 00:06:46.440
if we did it in
degrees, 90 degrees,

00:06:46.440 --> 00:06:49.880
or if we do it in radians,
pi over 2 radians.

00:06:49.880 --> 00:06:51.280
What is sine of theta?

00:06:51.280 --> 00:06:52.461
Well, now it is 1.

00:06:52.461 --> 00:06:53.460
This is the unit circle.

00:06:53.460 --> 00:06:54.480
It has a radius 1.

00:06:57.100 --> 00:07:03.740
So when theta is equal to pi
over 2, then sine of theta

00:07:03.740 --> 00:07:04.460
is equal to 1.

00:07:04.460 --> 00:07:08.282
So if this is 1 right here,
sine of theta is equal to 1.

00:07:08.282 --> 00:07:12.060
If theta-- and then if we
go 180 degrees or halfway

00:07:12.060 --> 00:07:16.580
around the circle--
theta is now equal to pi.

00:07:16.580 --> 00:07:19.320
Let me do this in
a color, orange.

00:07:19.320 --> 00:07:21.070
I have already used orange.

00:07:21.070 --> 00:07:23.630
Theta is now equal to pi.

00:07:23.630 --> 00:07:27.230
When theta is equal to pi, the
y value of this point right

00:07:27.230 --> 00:07:28.770
here is once again 0.

00:07:28.770 --> 00:07:29.860
So we go back to 0.

00:07:29.860 --> 00:07:32.350
Remember, we're talking
about sine of theta.

00:07:32.350 --> 00:07:35.400
And then we can go
all the way down here,

00:07:35.400 --> 00:07:38.320
where you can see
there's 270 degrees.

00:07:38.320 --> 00:07:46.780
Or you could view this
as 3 pi over 2 radians.

00:07:46.780 --> 00:07:51.610
So this is in
radians, this axis.

00:07:51.610 --> 00:07:54.010
So 3 pi over 2
radians, sine of theta

00:07:54.010 --> 00:07:56.549
is the y-coordinate on the
unit circle right over here.

00:07:56.549 --> 00:07:57.840
So it's going to be negative 1.

00:07:57.840 --> 00:08:02.320
So this is negative 1.

00:08:02.320 --> 00:08:05.370
And then finally, when you go
all the way around the circle,

00:08:05.370 --> 00:08:12.840
you've gone 2 pi radians, and
you're back where you began.

00:08:12.840 --> 00:08:14.710
And the sine of theta,
or the y-coordinate,

00:08:14.710 --> 00:08:16.710
is now 0 once again.

00:08:16.710 --> 00:08:19.590
And if you connect the dots, or
if you'd plotted more points,

00:08:19.590 --> 00:08:22.880
you would see a sine
curve over just the part

00:08:22.880 --> 00:08:24.875
that we've graphed
right over here.

00:08:24.875 --> 00:08:26.125
So that's another application.

00:08:26.125 --> 00:08:27.280
You say hey, Sal,
where is this going?

00:08:27.280 --> 00:08:29.120
Well I'm showing you--
I'm reminding you

00:08:29.120 --> 00:08:30.619
of all of these
things because we're

00:08:30.619 --> 00:08:34.549
going to revisit it with a
different number other than pi.

00:08:34.549 --> 00:08:37.380
And so I want to do
one last visit with pi.

00:08:37.380 --> 00:08:40.980
You say, look, pi is powerful
because-- or one of the reasons

00:08:40.980 --> 00:08:43.570
why pi seems to have some
type of mystical power,

00:08:43.570 --> 00:08:46.090
and we've shown this in
the calculus playlist--

00:08:46.090 --> 00:08:51.550
is Euler's formula,
that e to the i theta

00:08:51.550 --> 00:08:55.710
is equal to cosine of
theta plus i sine of theta.

00:08:55.710 --> 00:08:57.370
This, by itself,
is just a crazy--

00:08:57.370 --> 00:08:59.410
it's just one of those
mind boggling formulas.

00:08:59.410 --> 00:09:01.680
But it sometimes looks
even more mind boggling

00:09:01.680 --> 00:09:06.090
when you put pi in for theta
because then, from Euler's

00:09:06.090 --> 00:09:09.880
formula, you would
get e to the i pi

00:09:09.880 --> 00:09:12.510
is equal to-- well,
what's cosine of pi?

00:09:12.510 --> 00:09:14.840
Cosine of pi is negative 1.

00:09:14.840 --> 00:09:16.630
And then sine of pi is 0.

00:09:16.630 --> 00:09:17.860
So 0 times i.

00:09:17.860 --> 00:09:20.010
So you get this formula,
which is pretty profound,

00:09:20.010 --> 00:09:21.730
and then you say,
OK, if I want to put

00:09:21.730 --> 00:09:25.700
all of the fundamental numbers
together in one formula,

00:09:25.700 --> 00:09:27.790
I can add 1 to
both sides of this.

00:09:27.790 --> 00:09:35.750
And you get e to the i
pi plus 1 is equal to 0.

00:09:35.750 --> 00:09:38.920
Sometimes this is called Euler's
Identity, the most beautiful

00:09:38.920 --> 00:09:41.230
formula or equation
in all of mathematics.

00:09:41.230 --> 00:09:42.490
And it is pretty profound.

00:09:42.490 --> 00:09:45.790
You have all of the fundamental
numbers in one equation.

00:09:45.790 --> 00:09:50.620
e, i, pi, 1, 0, although
for my aesthetic taste

00:09:50.620 --> 00:09:53.250
it would have been even
more powerful if this

00:09:53.250 --> 00:09:54.264
was a 1 right over here.

00:09:54.264 --> 00:09:55.680
Because then this
would have said,

00:09:55.680 --> 00:09:59.880
look, e to the i pi,
this bizarre thing,

00:09:59.880 --> 00:10:01.590
would have equaled unity.

00:10:01.590 --> 00:10:04.130
That would have been super
duper profound to me.

00:10:04.130 --> 00:10:08.540
It seems a little bit of a
hack to add 1 to both sides

00:10:08.540 --> 00:10:10.820
and say, oh, look,
now I have 0 here.

00:10:10.820 --> 00:10:13.100
But this is pretty darn good.

00:10:13.100 --> 00:10:15.770
But with that, I'm
going to make-- well,

00:10:15.770 --> 00:10:17.020
I'm not going to argue for it.

00:10:17.020 --> 00:10:19.740
I'm going to show an
argument for another number,

00:10:19.740 --> 00:10:22.410
a number different than pi.

00:10:22.410 --> 00:10:25.160
And I want to make it clear
that these ideas are not my own.

00:10:25.160 --> 00:10:27.390
It comes from--
well, it's inspired

00:10:27.390 --> 00:10:30.410
by-- many people are on this
movement now, the Tau Movement,

00:10:30.410 --> 00:10:33.490
but these are kind
of the people that

00:10:33.490 --> 00:10:35.050
gave me the thinking on this.

00:10:35.050 --> 00:10:37.629
And the first is Robert
Palais on "Pi is Wrong."

00:10:37.629 --> 00:10:39.670
And he doesn't argue that
pi is calculated wrong.

00:10:39.670 --> 00:10:42.880
He still agrees that it is
the ratio of the circumference

00:10:42.880 --> 00:10:46.330
to the diameter of a
circle that is 3.14159.

00:10:46.330 --> 00:10:48.540
But what he's saying is
that we're paying attention

00:10:48.540 --> 00:10:49.900
to the wrong number.

00:10:49.900 --> 00:10:52.570
And also, you have Michael
Hartl, "The Tau Manifesto."

00:10:52.570 --> 00:10:54.940
All of this is available online.

00:10:54.940 --> 00:10:57.545
And what they argue for
is a number called tau,

00:10:57.545 --> 00:10:59.530
or what they call tau.

00:10:59.530 --> 00:11:03.090
And they define tau, and it's
a very simple change from pi.

00:11:03.090 --> 00:11:06.030
They define tau not as the
ratio of the circumference

00:11:06.030 --> 00:11:08.520
the diameter, or the ratio of
the circumference to 2 times

00:11:08.520 --> 00:11:09.340
the radius.

00:11:09.340 --> 00:11:11.980
They say, hey, wouldn't
it be natural to define

00:11:11.980 --> 00:11:15.990
some number, the ratio of the
circumference to the radius?

00:11:15.990 --> 00:11:19.930
And as you see here, this
pi is just one half times

00:11:19.930 --> 00:11:21.020
this over here, right?

00:11:21.020 --> 00:11:24.250
Circumference over 2 r, this
the same thing as one half times

00:11:24.250 --> 00:11:25.550
circumference over r.

00:11:25.550 --> 00:11:27.850
So pi is just half of tau.

00:11:27.850 --> 00:11:34.020
Or another way to think about it
is that tau is just 2 times pi.

00:11:34.020 --> 00:11:36.802
Or, and I'm sure
you probably don't

00:11:36.802 --> 00:11:38.760
have this memorized,
because you're like, wait,

00:11:38.760 --> 00:11:40.770
I spent all my
life memorizing pi,

00:11:40.770 --> 00:11:46.680
but it's 6.283185 and keeps
going on and on and on,

00:11:46.680 --> 00:11:48.640
never repeating just like pi.

00:11:48.640 --> 00:11:50.570
It's 2 times pi.

00:11:50.570 --> 00:11:53.090
And so you're
saying, hey Sal, pi

00:11:53.090 --> 00:11:57.760
has been around for
millennia, really.

00:11:57.760 --> 00:11:59.437
Why mess with such a
fundamental number,

00:11:59.437 --> 00:12:01.270
especially when you
just spent all this time

00:12:01.270 --> 00:12:03.320
showing how profound it is?

00:12:03.320 --> 00:12:04.900
And the argument
that they'd make,

00:12:04.900 --> 00:12:06.800
and it seems like a
pretty good argument,

00:12:06.800 --> 00:12:09.880
is that actually things seem
a little bit more elegant when

00:12:09.880 --> 00:12:13.160
you pay attention to this number
instead of half of this number,

00:12:13.160 --> 00:12:15.460
when you pay attention to tau.

00:12:15.460 --> 00:12:18.920
And to see that, let's revisit
everything that we did here.

00:12:18.920 --> 00:12:22.720
Now, all of a sudden, if
you pay attention to 2 pi,

00:12:22.720 --> 00:12:23.850
as opposed to pi.

00:12:23.850 --> 00:12:26.630
Or we should call it--
if you pay attention

00:12:26.630 --> 00:12:29.130
to tau instead of tau over 2.

00:12:29.130 --> 00:12:31.090
What is this angle
that we did in magenta?

00:12:33.640 --> 00:12:36.490
Well first of all, let's
think about this formula right

00:12:36.490 --> 00:12:36.990
over here.

00:12:36.990 --> 00:12:39.360
What is the circumference
in terms of the radius?

00:12:39.360 --> 00:12:41.026
Well, now we could
say the circumference

00:12:41.026 --> 00:12:43.500
is equal to tau times
the radius because tau

00:12:43.500 --> 00:12:45.096
is the same thing as 2 pi.

00:12:45.096 --> 00:12:47.470
So it makes that formula a
little bit neater, although it

00:12:47.470 --> 00:12:50.240
does make the pi r squared
a little bit messier.

00:12:50.240 --> 00:12:52.720
So you could argue
both sides of that.

00:12:52.720 --> 00:12:56.850
But it makes the measure of
radians much more intuitive

00:12:56.850 --> 00:13:00.180
because you could say that
this is pi over 2 radians,

00:13:00.180 --> 00:13:03.870
or you could say that this
is pi over 2 radians is

00:13:03.870 --> 00:13:07.810
the same thing as to
tau over 4 radians.

00:13:07.810 --> 00:13:09.190
And where did I get that from?

00:13:09.190 --> 00:13:12.760
Remember, if you go all the
way around the circle, that

00:13:12.760 --> 00:13:13.640
is the circumference.

00:13:13.640 --> 00:13:15.389
The arc length would
be the circumference.

00:13:15.389 --> 00:13:18.010
It would be tau
radii, or it would

00:13:18.010 --> 00:13:21.991
be to tau radians would be the
angle subtended by that arc

00:13:21.991 --> 00:13:22.490
length.

00:13:22.490 --> 00:13:23.980
It would be tau radians.

00:13:23.980 --> 00:13:26.109
All the way around
is tau radians.

00:13:26.109 --> 00:13:27.400
So that by itself is intuitive.

00:13:27.400 --> 00:13:30.670
One revolution is
one tau radians.

00:13:30.670 --> 00:13:32.630
If you go only one
fourth of that,

00:13:32.630 --> 00:13:35.470
it's going to be tau
over four radians.

00:13:35.470 --> 00:13:37.260
So the reason why
tau is more intuitive

00:13:37.260 --> 00:13:39.297
here is because it
immediately-- you

00:13:39.297 --> 00:13:40.880
don't have to do
this weird conversion

00:13:40.880 --> 00:13:44.130
where you saying, oh, divide
by 2, multiply by 2, all that.

00:13:44.130 --> 00:13:47.152
You're just like, look, however
many radians in terms of tau,

00:13:47.152 --> 00:13:48.860
that's really how many
revolutions you've

00:13:48.860 --> 00:13:51.670
gone around the circle.

00:13:51.670 --> 00:13:53.970
And so, if you've gone
one fourth around,

00:13:53.970 --> 00:13:55.420
that's tau over four radians.

00:13:55.420 --> 00:13:58.460
If you've gone halfway around,
that'd be tau over two radians.

00:13:58.460 --> 00:14:02.580
If you go 3/4 around that'd be
three tau over four radians.

00:14:02.580 --> 00:14:05.750
If you go all the way around
that would be tau radians.

00:14:05.750 --> 00:14:09.640
If someone tells you that
they have an angle of 10 tau

00:14:09.640 --> 00:14:13.487
radians, you'd go
around exactly 10 times.

00:14:13.487 --> 00:14:14.820
It would be much more intuitive.

00:14:14.820 --> 00:14:18.220
You wouldn't have to do this
little mental math, converting,

00:14:18.220 --> 00:14:20.040
saying, do I multiply
or divide by 2

00:14:20.040 --> 00:14:22.040
when I convert to
radians in terms of pi?

00:14:22.040 --> 00:14:24.870
No, when you do it in terms of
tau radians, it's just natural.

00:14:24.870 --> 00:14:27.080
One revolution is
one tau radians.

00:14:27.080 --> 00:14:29.680
So that makes-- and it makes
a sine function over here.

00:14:29.680 --> 00:14:33.214
Instead of writing
pi over 2-- well,

00:14:33.214 --> 00:14:34.630
when you look at
a graph like this

00:14:34.630 --> 00:14:36.220
you're like, where was
this on the unit circle?

00:14:36.220 --> 00:14:37.803
Was this one fourth
around the circle?

00:14:37.803 --> 00:14:38.617
Was this one half?

00:14:38.617 --> 00:14:40.700
And this is actually one
fourth around the circle,

00:14:40.700 --> 00:14:41.557
right over here?

00:14:41.557 --> 00:14:43.640
But now it becomes obvious
if you write it in tau.

00:14:46.800 --> 00:14:50.380
Pi over 2 is the same
thing as to tau over 4.

00:14:50.380 --> 00:14:53.510
Pi is the same thing
as to tau over 2.

00:14:53.510 --> 00:15:02.300
3 pi over 2 is 3 pi-- oh,
sorry, 3 tau over 4, 3/4 tau.

00:15:02.300 --> 00:15:04.200
And then one revolution is tau.

00:15:04.200 --> 00:15:07.010
And then immediately, now
when you look at it this way,

00:15:07.010 --> 00:15:09.250
you know exactly where you
are in the unit circle.

00:15:09.250 --> 00:15:10.750
You're one fourth
around the unit circle,

00:15:10.750 --> 00:15:12.333
you're halfway around
the unit circle,

00:15:12.333 --> 00:15:14.390
you're 3/4 of the way
around the unit circle.

00:15:14.390 --> 00:15:17.290
And then you're all the
way around the unit circle.

00:15:17.290 --> 00:15:20.030
And so the last thing
that I think the strong pi

00:15:20.030 --> 00:15:22.100
defenders would say
is well, look Sal,

00:15:22.100 --> 00:15:26.330
you just pointed out one of
the most beautiful identities

00:15:26.330 --> 00:15:28.300
or formulas in mathematics.

00:15:28.300 --> 00:15:31.670
How does tau hold up to this?

00:15:31.670 --> 00:15:34.380
Well, let's just try it
out and see what happens.

00:15:34.380 --> 00:15:41.490
So if we take e to the i tau,
that will give us cosine of tau

00:15:41.490 --> 00:15:44.520
plus i sine of tau.

00:15:44.520 --> 00:15:47.210
And once again, let's just
think about what this is.

00:15:47.210 --> 00:15:51.580
Tau radians means we've gone all
the way around the unit circle.

00:15:51.580 --> 00:15:54.570
So cosine of tau--
remember, we're

00:15:54.570 --> 00:15:56.290
back at the beginning
of the unit circle

00:15:56.290 --> 00:16:01.200
right over here-- so cosine of
tau is going to be equal to 1.

00:16:01.200 --> 00:16:04.115
And then sine of
tau is equal to 0.

00:16:07.340 --> 00:16:14.430
So e to the i tau is equal to 1.

00:16:14.430 --> 00:16:16.230
And I'll leave it
up to you to decide

00:16:16.230 --> 00:16:20.970
which one seems to be more
aesthetically profound.