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  • 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,