The beauty of algebra

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 whichever 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 of mathematics and what we'll see as we start getting deeper into 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 the i guess we could we you know we started with the very grand you know 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's a sale the sale says that it is 30 off 30 percent off and i'm interested i don't shop at my two fancy stores so let's say i'm interested in a pair of pants and the pair of pants before the sale even is is about 20 bucks 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 percent 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 percent times the 20 so you would say your discount your discount is equal to you could write it as thirty percent times twenty dollars i'll do the twenty dollars in in purple or you could write it if you wanted to write this as a decimal you could write this as zero point three zero times 20 dollars and if you were to do the math you would get six dollars so nothing nothing new over there but what if i want to generalize a little bit that's the discount on this particular pierre pair of pants but what if i wanted to know the discount on anything in the store well then i could say well let let x let x be the price let me just in a different color so i'm just going to make a symbol let x be the price of the the product i want to buy price the non-discount price non-discount non-discount price of the product of the product in the store so now all of a sudden we can say that our discount we can say our discount is equal to thirty percent thirty percent times x thirty percent times x or if we wanted to write it as a decimal we could write if we were to write 30 percent 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 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 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 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 twenty dollar product if there's a ten dollar product we can put that ten dollar product in here for x and then we would say point three zero times ten and the discount would be would be three dollars it might be a hundred dollar project then the discount would be thirty 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 let's let let's let m equal or i'll say p just so it makes sense let's p is equal to is equal to the percentage off percentage percentage off now what can we do well now we can say that the discount the discount is equal to the percentage off in these other examples we were picking 30 percent 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 in 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 isn't 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 that that symbol represents the actual dollar discount but now things get really interesting because we can use this type of an exp 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 at the store over here you might have 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 have let y is equal to force we could 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 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 we could say x we could say x is equal to acceleration well then the famous at 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 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 we can see that we can take this equation and it can apply to things in economics 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 all of these other applications to all of them and even neater is 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 it says and someone says on the other hand they say 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 you if you wanted them to still be equal you would divide the other one by that number so for example for example let's say we know that y is equal to p times x what if you wanted to have them both be equal and you say well what is y divided by x is 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 to say you're going to get your original number if you 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 let me 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 these applications but now we can go and take them back to any place that we applied and what they're really interesting is we're going to find you they're an infinite number of applications and we're going to 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 is really the language with we can understand the philosophy of the universe and that's why people tell us that if a completely alien life form would ever contact humans it probably mathematics would probably be our first common ground the place that we can start to to form a basis that we can start to communicate from