The Beauty of Algebra Why the abstraction of mathematics is so fundamental
The Beauty of Algebra
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- Before we get into the meat of algebra,
- I want 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 it 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 gonna 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 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 could,
- we started with the very grand philosophy of the universe
- as written in mathematics.
- but let's start with a very concrete simple idea.
- but we'll keep abstracting and see how the same idea connects
- across many domains in our universe.
- So lets 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% off and I'm interested --
- and I don't shop at too fancy stores --
- so let's say I'm interested in a pair of pants
- and the pair of pants before the sale event is about $20.
- 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 is a thirty percent off sale on these pants.
- 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 have probably had exposure to.
- You would multiply thirty percent times the twenty dollars.
- so you would say your discount --your discount-- is equal to,
- you could write is as 30% times $20.
- We'll write twenty dollars in purple.
- Or you could write it --if you wanted to write-- this is a decimal
- you could write it as 0.30*20 dollars
- And if you were to do the maths
- you would get 6 dollars.
- So nothing, nothing new over there.
- But what if I wanted to generalize 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, let x be the price
- -- let's do it in another color --
- I'm just gonna make a symbol.
- Let x be the price of the product I want to buy,
- the nondiscount price of the product in the store.
- So now all of a sudden, we can say our discount,
- is equal to 30%, 30% times x,
- or if we wanted to write it as a decimal,
- we could write 30% as a decimal.
- we could write 0.30 times x --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 it 0.3 times that
- and I would get the discount.
- Now we're starting to, very slowly,
- starting to get into the abstraction of algebra.
- We'll see that they would get much more nuanced and deep
- and frankly more beautiful
- as we start studying more and more 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 dollar product.
- If there's a 10 dollar product,
- we can put that 10 dollar product in here for x,
- and then we would say 0.30 times 10.
- The discount would be 3 dollars.
- It might be a $100 product,
- then the discount would be 30 dollars.
- But let's generalize even more.
- Let's say : "What is the discount for any given sale,
- --when the sale has a certain percentage-- ?"
- Now we can say that the discount
- --let me define a variable--
- Let's let m = ... I'll say p, just so it makes sense,
- p is equal to the percentage off.
- Now what can we do ?
- Now we can say that the discount
- is equal to the percentage off.
- In these other examples we were picking 30 percent,
- but now we can say 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"
- Let y is equal to 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 anyway you wanted.
- Instead of writing y there,
- you could have written a greek letter,
- as long as you can keep track of the fact
- that the 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 thing right over here,
- that's why we call it an equation.
- This can be used for things
- that are completely unrelated
- to the discount price at the store over here.
- You might have...
- so in physics you will see
- that force is equal to mass times acceleration.
- The letters are different
- but these are fundamentally the same idea.
- We could have let y is equal to force
- and m is equal to... or 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 wanna show you that the same idea, the same relationship
- that's being applied to 2 different things.
- And we could say x is equal to acceleration.
- We could say x is equal to acceleration.
- Well then the famous "force is equal to mass times acceleration"
- can be rewritten and it is really the same exact idea
- as y -- which we've defined has force --
- can be equal to mass
- -- for which we are going to use the symbol p --
- which is equal to p times acceleration.
- We are 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.
- We can 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 is 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,
- 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 those human applications.
- So for example we can say :
- "Look, if y is equal to p times x".
- So litterally if someone said
- "Hey, this is y."
- and someone says on the other hand "I have p times x."
- I can say "Well, you have the same thing in both of your hands."
- And if your were to divide one of them by a number
- and you wanted them to still be equal,
- you'd divide the other one by that number.
- So for example,
- let's say -- we know that y is equal to p times x --
- what if you want to have them both be equal.
- What is y divided by x gonna be equal to ?
- Well y was equal to p times x
- so y divided by x
- is gonna be the same 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 divide by that something
- you're gonna get the original number.
- If you multiply by 5 and divide by 5,
- you're just gonna get 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
- -- we make that x green --
- y over x is equal to p
- and now this has implications,
- this has implications for every one of these ideas.
- One is telling us a fundamental truth about the universe,
- almost devoid of any of those applications,
- but now we can go and take them back
- to any place where we applied it.
- The really interesting thing is we're gonna find new...
- -- there are an infinite number of applications
- and we don't even know, frankly, most of them.
- We're gonna 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
- mathematics is really the language
- with which we can understand the philosophy of the universe.
- That's why people will tell us
- that if a completely alien life form would 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.
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