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## Logarithmic scale (Algebra 2 level)

# Logarithmic scale (with Vi Hart)

## Video transcript

VI HART: All right. So I'm Vi Hart and I'm
here with Sal Khan and-- SAL KHAN: Hello. VI HART: Yeah. We're talking about just
how we think about numbers and what is the most
natural way to think about them in our
everyday lives. SAL KHAN: And Vi said that she
was going to test me right now. VI HART: Yeah. All right, can I borrow the pen? SAL KHAN: Yes. Yes. VI HART: I get to use the
official pen and screen-- oh wait. [INAUDIBLE] SAL KHAN: No, that's all screen. Yes. VI HART: OK. OK. SAL KHAN: You need training, Vi. VI HART: I need training. Yay. SAL KHAN: It looks like a pizza. VI HART: It's a triangle. SAL KHAN: Right. Where's your test, Vi? VI HART: OK. SAL KHAN: You're diverging. VI HART: I'm sorry. All right, so here's
a number line. Regular old number line. No, wait. I want to start at one. All right, we're
going to start at one, and we're going to go
all the way to a million. And I'm going to give you the
pen now, and I'm going to ask, where is 1,000? SAL KHAN: Where is 1,000? Where is 1,000? I see. I see what you're doing. VI HART: So you can think
about this logically. SAL KHAN: Yes. Yes. So I'll tell you what
went through my brain. My first knee-jerk
reaction was to put 1,000 like right over here. That's what I was tempted to do. VI HART: Mhm. SAL KHAN: And then
my brain kicked in. VI HART: Right. Yeah. You think of it-- SAL KHAN: My
highly-analytical mind. VI HART: Yeah. Because you know there is a
correct answer to this problem. We can think, all
right, where is 1,000 on the number line
related to a million? Well, a million
divided by 1,000. SAL KHAN: Is 1,000. VI HART: You want
one thousandth. SAL KHAN: Thousandth. Right. So it's not there. I was going to draw
it a tenth of the way. No. 1,000 is like there. You barely notice the
difference between that and-- VI HART: Yeah. You couldn't even see the
difference in thousandths. SAL KHAN: Yes. So this is fascinating. What is this about? Why did I do that? VI HART: Yeah. Why do we think of 1,000 as
being much closer to a million than it is? And we do this,
actually, all the time. We're not so used
to having to think about the difference
between 1,000 and a million. But when we're thinking about
the difference between 1 and 2, or the difference between
2 and 3, or 1 and 10, we think, 1 and 2. There's a big difference there. 2 is twice what 1 is. SAL KHAN: Yes. It's twice 1. Right. VI HART: And the
difference between 9 and 10 is the same distance
when you're looking at it at the usual scale. It's 1. SAL KHAN: Right. Right. VI HART: But when we're thinking
about real-life things, well, the difference
between 9 and 10 isn't so big in any real
life situation. SAL KHAN: No. But the difference
between 1 and 2 is huge. VI HART: Yeah. SAL KHAN: In
real-life it's double. VI HART: Yes. SAL KHAN: Right. VI HART: So now we have to
think on a logarithmic scale is what it is. SAL KHAN: Oh. Yes. The old logarithmic scale. So what you're saying
is that we, as humans, even though everything
we're taught is these linear scales, where
we want to say this is 1, and then maybe this
is 10, and then this is 20-- even though
that's what we're taught, and that's what most
of our mathematics, we plot lines and stuff like that. VI HART: Yeah. That's how we draw it
out on paper, usually. SAL KHAN: Yes. VI HART: But that doesn't
make sense, usually, for how we think about things. Because the difference
between 5 gazillion and 5 gazillion and 10 is-- SAL KHAN: Is nothing. Is nothing. VI HART: Nothing. Whereas the difference
between 1 and 10 is huge. SAL KHAN: Right, right, right. And so that's why
almost the multiple matters more than the absolute
distance between the numbers. VI HART: Yeah. SAL KHAN: Absoluately. And that's what the
logarithmic scale captures. VI HART: It is. And that's why we see
the logarithmic scale in so many things in real life. As a mathemusician, on
the piano we see it. SAL KHAN: Yes. VI HART: It's actually
the logarithmic scale. So let's get our
piano picture out. SAL KHAN: Oh, look at that. There's a piano. VI HART: Yeah. OK. OK. Can I have the pen? SAL KHAN: Here you go. Yes. VI HART: All right. Let's see if I can
figure this out. All right. So here we have
this C. Let's call it middle C. And
here we have this D. And there's a certain
distance between these. And then here is
this C and this D. And when we're listening
to these notes, we think, all right,
they're one note apart. This is the same distance
here between here and here and here and there. SAL KHAN: Yes. VI HART: But if you look
at the actual frequencies, the distances are not the same. This was maybe a bad
example, because I don't know the
frequency of D, but-- SAL KHAN: No. Well, we could-- VI HART: Well I'll give you an
example I can give numbers to, which is maybe the difference
between this octave and the difference
between this octave. Right? If this is C-- SAL KHAN: Call it x. VI HART: x. SAL KHAN: Whatever
the frequency is. VI HART: Right. Yes. This is great. SAL KHAN: It's
like 440 kilohertz. I don't know what it is. VI HART: No. A is 440. SAL KHAN: A is 440. VI HART: C is-- SAL KHAN: Well we'll call it x. VI HART: More like, I
don't know, let's say 300. SAL KHAN: Yes. VI HART: All right. So if this is 300
or 300x or just x, then this frequency
would be 600. SAL KHAN: 600. Right. It doubles. VI HART: It doubles when
you go up an octave. And this would be 1,200 up here,
this C. We're in a weird scale here. But the difference
between here is 300. And the difference
between here is 600. But when we're
listening to octaves, we feel like the difference
between this octave shouldn't be half as much as
the difference between these two notes. Right? The distance from an
octave should be an octave. Right? SAL KHAN: Right. So fundamentally the way we
perceive pitch is logarithmic. VI HART: It's just
fundamentally logarithmic. If you want to have all
your notes on the piano be right next to
each other, instead of having a piano where you
have one key for C here, and one key for C
here, and the next C is going to be over here. Right? SAL KHAN: Yeah. Twice as far. Yeah. VI HART: And the next C on the
piano would have to be like-- SAL KHAN: So if
piano manufacturers-- they innately made it based
on a logarithmic scale, whether they knew it or not. VI HART: Yeah. Because we think about it
on a logarithmic scale. SAL KHAN: They could have
made it on a linear scale, and then the keys would
just get fatter and fatter as we went to the right. VI HART: Oh, yeah. Fatter keys instead of
making them farther apart. SAL KHAN: Someone
should make that. VI HART: A fat key piano. SAL KHAN: A linear scale piano. Yes. VI HART: That would be awesome. But that's not how
we think of pitch. SAL KHAN: No. It might be hard to play that. VI HART: It would
be [? awesome. ?] SAL KHAN: And it's
just not pitch. It also even be how we perceive
a magnitude of the frequencies. Because we have
the decibel scale, which is a logarithmic scale. VI HART: Yeah. There's a lot of natural,
intuitive, logarithmic scales. So when we're looking at
how loud something is, that's also the difference
between how I'm talking now and how I'm talking if
I'm a little louder. And we feel like distances
between loudness also-- SAL KHAN: Right. VI HART: It's harder to explain. SAL KHAN: We perceive it a
lot-- it is harder to explain. But we'll leave it there. VI HART: I don't
have any pictures of how loud something is. SAL KHAN: We don't. No. And we don't want
to bother people by just getting
louder and louder. VI HART: By screaming. SAL KHAN: Yes. VI HART: I can scream
at you some more. I think that would be great. SAL KHAN: Right. No. But that's fascinating. Especially this
little game here. I'm going to start doing this
at the next party I go to. VI HART: Mhm. It's good. And it makes sense. When we're looking at things,
the difference between how much $1 million
and $10 million is. SAL KHAN: Yes. VI HART: The world follows
these kind of rules. SAL KHAN: Right, right, right. Very cool. VI HART: Yeah. SAL KHAN: Awesome.