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# The beauty of algebra

Why the abstraction of mathematics is so fundamental. Created by Sal Khan.

Video transcript

Before we get into
the meat of algebra, I wanted to give you a quote
from one of the greatest minds in human history, Galileo
Galilei, because I think this quote encapsulates
the true point of algebra and really mathematics
in general. He said, "Philosophy is written
in that great book which ever lies before our eyes--
I mean the universe-- but we cannot understand
it if we do not first learn the language and grasp the
symbols in which is written. This book is written in
the mathematical language, without which one wanders in
vain through a dark labyrinth." So very dramatic, but very deep. And this really is the
point of mathematics. And what we'll see as we start
getting deeper and deeper into algebra is that we're going
to start abstracting things, and we're going to start
getting to core ideas that start explaining really how
the universe is structured. Sure, these ideas can be
applied to things like economics and finance and
physics and chemistry. But at their core,
they're the same idea, and so they're even more
fundamental, more pure, than any one of
those applications. And to see what I mean by
getting down to the root idea, let's go with a-- I guess we
started with the very grand, the philosophy of
the universe is written in mathematics--
but let's start with a very concrete,
simple idea. But we'll keep
abstracting, and we'll see how the same idea
connects across many domains in our universe. So let's just say
we're at the store, and we're going
to buy something. And there is a sale. The sale says that it
is 30% percent off, and I'm interested. I don't shop at
too fancy a store. So let's say I'm interested
in a pair of pants. And the pair of pants before
the sale even is about $20. And that is about how
much I spend on my pants. So I'm interested in
a $20 pair of pants. But it's even better, there's
a 30% off sale on these pants. Well, how would I
think about how much I'm going to get
off of that $20? And this isn't algebra yet. This is something that you've
probably had exposure to. You would multiply
the 30% times the $20. So you would say your
discount is equal to-- you could write
it as 30% times $20. I'll do the $20 in purple. Or you could write it, if
you wanted to write this as a decimal, you could
write this as 0.30 times $20. And if you were to do the
math, you would get $6. So nothing new over there. But what if I want to
generalize it a little bit? That's the discount on this
particular pair of pants. But what if I wanted to know
the discount on anything in the store? Well, then I could
say, well, let x be the price-- let me do
this in a different color. So I'm just going
to make a symbol. Let x be the price
of the product I want to buy, price,
the non-discount price of the product in the store. So now, all of a
sudden, we can say that our discount is
equal to 30% times x. Or if we wanted to
write it as a decimal, if we wanted to write
30% as a decimal, we could write 0.30 times x. Now, this is interesting. Now you give me the price
of any product in the store, and I can substitute
it in for x. And then I can essentially
multiply 0.3 times that, and I would get the discount. So now we're starting
to, very slowly, we're starting to get into
the abstraction of algebra. And we'll see that these will
get much more nuanced and deep and, frankly, more
beautiful as we start studying more and more
kind of algebraic ideas. But we aren't done here. We can abstract this even more. Over here, we've said
we've generalized this for any product. We're not just saying
for this $20 product. If there's a $10 product, we
can put that $10 product in here for x. And then we would
say 0.30 times 10, and the discount would be $3. It might be $100 product, then
the discount would be $30. But let's generalize even more. Let's say, well, what is
the discount for any given sale when the sale is
a certain percentage? So now we can say
that the discount-- let me define a variable. So let's let m equal-- or I'll
say p just so it makes sense. p is equal to the
percentage off. Now what can we do? Well, now we can say
that the discount is equal to the percentage off. In these other examples,
we were picking 30%. But we can say now it's p. It's the percentage off. It's p. That's the percentage off
times the product in question, times the price, the
non-discount price of the product in question. Well, that was x. The discount is
equal to p times x. Now, this is really interesting. Now we have a general
way of calculating a discount for any given
percentage off and any given product x. And we didn't have to use
these words and these letters. We could have said let
y equal the discount. Then we could have written
the same underlying idea. Instead of writing
discount, we could have written y is equal to
the percentage off p times the non-discount price
of the product, times x. And you could have defined these
letters any way you wanted. Instead of writing
y there, you could have written a
Greek letter, or you could have written
any symbol there. As long as you can
keep track of it, that symbol represents the
actual dollar discount. But now things get
really interesting. Because we can use this type
of a relationship, which is an equation--
you're equating y to this right over here, that's
why we call it an equation-- this can be used for
things that are completely unrelated to the price,
the discount price, at the store over here. So in physics,
you'll see that force is equal to mass
times acceleration. The letters are different,
but these are fundamentally the same idea. We could've let y is equal to
force, and mass is equal to p. So let me write p
is equal to mass. And this wouldn't be an
intuitive way to define it, but I want to show
you that this is the same idea, the
same relationship, but it's being applied to two
completely different things. And we could say x is
equal to acceleration. Well, then the famous force
is equal to mass times acceleration can be rewritten. And it's really
the same exact idea as y, which we've
defined as force, can be equal to
mass, which we're going to use the
symbol p, which is equal to p times acceleration. And we're just going to happen
to use the letter x here, times x. Well, this is the
exact same equation. This is the exact same equation. And we could see that we
can take this equation, and it can apply to
things in economics, or it can apply to
things in finance, or it can apply to things in
computer science, or logic, or electrical engineering,
or anything, accounting. There's an infinite
number of applications of this one equation. And what's neat
about mathematics and what's neat about
algebra in particular is we can focus on
this abstraction. We can focus on
the abstract here, and we can manipulate
the abstract here. And what we discover
from these ideas, from these
manipulations, can then go and be reapplied to all
of these other applications, to all of them. And even neater, it's
kind of telling us the true structure
of the universe if you were to strip away all
of these human definitions and all of these
human applications. So for example, we could say,
look, if y is equal to p times x-- so literally, if someone
said, hey, this is y, and someone says, on the
other hand, I have p times x, I could say, well, you
have the same thing in both of your hands. And if you were to divide
one of them by a number, and if you wanted them
to still be equal, you would divide the
other one by that number. So for example, we know that
y is equal to p times x. Well, what if you wanted
to have them both be equal? And you say, well,
what is y divided by x going to be equal to? Well, y was equal to p
times x, so y divided by x is going to be
the same thing as p times x divided by x. But now this is interesting. Because p times x
divided by x-- well, if you multiply by something and
then divide by that something, it's just you're going to
get your original number. If you multiply by
5 and divide by 5, you're just going to start with
p or whatever this number is. So those would cancel out. But we were able to manipulate
the abstraction here and get y over x is equal to p-- and
let me make that x green. And now this has implications
for every one of these ideas. One is telling us
a fundamental truth about the universe,
almost devoid of any of these applications. But now we can go and take
them back to any place that we applied. And the really
interesting thing is we're going to find there
are an infinite number of applications,
and we don't even know, frankly, most of them. We're going to discover new ones
for them in a thousand years. And so hopefully this
gives you a sense for why Galileo said
what he said about really mathematics is really the
language with which we can understand the philosophy
of the universe. And that's why people tell us
that if a completely alien life form were to ever
contact humans, mathematics would probably
be our first common ground, the place that we can start
to form a basis that we can start to communicate from.