The beauty of algebra
Why the abstraction of mathematics is so fundamental. Created by Sal Khan.
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- Why in algebra do you use letters instead of numbers?(425 votes)
- You use letters to represent different numbers. This makes it easier when you are doing the same operation over and over. Rather than coming up with a new equation each time, you get one equation and just put in different numbers for the letter.
This video explains variables: http://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/v/what-is-a-variable(346 votes)
- What is the best reason for learning Algebra?(122 votes)
- Algebra is the fundimental language of mathematics. Mathematics is fundimentally "why things are" - as we currently understand them to be. It is a science, but it is also an artform of logic - in a sense much like any spoken language. So much of our understanding of science is based on formulae and equations.. problems, variables..
Not only does Algebra offer us the universal syntax to understanding the language(s) of mathematics, it offers us the most basic toolkits to understanding the most complicated languages, proses, and poems of science and technology itself.
It is the most basic language of science from mathematics, to quantum phyisics to economics to .. well it can even be applied to psychology. It is logic in its almost purest form - dare i say the most important language we can ever have the opportunity to teach ourselves.(67 votes)
- why do we have to have algebra?(210 votes)
- Because it will make you better at life, if only you'll use it. "The human mind has never invented a labor-saving device equal to algebra." --J. Willard Gibbs.
Algebra creates abstractions which apply to tens, hundreds, or even thousands of scenarios. In other words by solving ONE algebra problem, you are solving hundreds of "regular" problems, all at once. Sounds nice, doesn't it? For example "maximization" and "minimization" problems (both involving quadratic equations) will help you, for example, fence in the most area with the least amount of fencing material (thus saving you money).(354 votes)
- what jobs would i use algebra for?(179 votes)
- Engineer (pretty much all types of engineers), physicist, economist, accountant, architect, chemist, teacher and professor (if one teaches math), pharmacist (to add in chemicals and/or drugs to each other), and a lot of other jobs. I think I'm missing some...but i think your getting the idea?
Hope this helped. =)(277 votes)
- Why do you find the discount by multiplying the discount by the price?(64 votes)
- When you get a 30% discount on a $20 item, that's saying that you save 30¢ for every $1. If we were only buying a $1 item, then we would save 30¢ in total, right?
But we're actually buying an item worth $20. So we get to save 20 times the amount that we would save if we were only buying a $1 item. That's because for each of the 20 dollars we save 30¢.
We could write the expression as: 30¢ + 30¢ + 30¢ + ... + 30¢ (20 times)
Or we could use multiplication...
Since 30¢ = 30% of a dollar = $0.30, we can write the multiplication expression as: $20 * 0.30.
That means you save $20 * 0.30, or $6.(159 votes)
- If people take Algebra so seriously, then why did they invent the calculator?(39 votes)
- "If people take Algebra so seriously, then why did they invent the calculator?"
well, let me ask you some questions of my own:
if people take walking so seriously, then why did they invent the car?
if people take writing so seriously, then why did they invent the typewriter?
if people take talking so seriously, then why did they invent the telephone?
if people take X so seriously, then why did they invent the Y?
The answer to all these questions are the same, Y is a tool that makes doing X easier, faster, more efficient, optimal, convenient, and so on and so forth. Just because they are tools, however, doesn't make learning X any less important. Using Algebra, we can abstract the core idea of the question and answer any question with the same basic principle in place.(51 votes)
- I want to become an interpreter (foreign languages are my passion!) Would I ever need to use Algebra in my day-to-day life? If I'm translating words and phrases for someone, aren't numbers universal? Why would I have to learn Algebra if I want to go into a field that deals with spoken language?
In all my years of school, nobody has ever given me an answer besides "it's required by the state." (I'm in the United States in my Junior year of High School.)(11 votes)
- Algebra isn't just used in jobs and careers. It can be applied to your everyday life. As an interpreter, you'll probably be making some good money. Any smart person budgets and keeps track of their money. I hope you do! Most people use Excel spreadsheets for their personal finances. A lot of algebra is used in that application.
So, don't assume you'll never have to use algebra in your day-to-day life. A lot of students complain about how "useless" it is... but don't even know they could very well be using it when they're older!(14 votes)
- Sal makes it seem like y = px is the answer to the universe.
So is algebra the answer to the universe?(6 votes)
- No, the answer to life, the universe, and everything is 42.(13 votes)
- Why is sal putting a dot instead of a multiplication symbol. Does that mean in algebra you can write multiplication in different ways like that dot.(5 votes)
- Yes! And sometimes we don't even need the multiplication sign, as in xy. In this case it is understood that x is being multiplied by y.
We also use parenthesis, for example (5)(4) is the same as 5x4 is the same as 5·4.
They all mean 5 times 4!.(7 votes)
- What is the divisibility rule for the numbers 4 and 9(5 votes)
- The divisibility rule for 9 is that the digits must add up to a multiple of 9..
For example 171 is divisible by 9 because 1+7+1 =9
3,924 is divisible by 9 becaue 3+9+2+4=18
5,667 isn't divisible by 9 because 5+6+6+7=25
The rule for 4 is that the last two numbers have to be a multiple of 4
For example 3,512 is divisible by 4 because 12 is divisible by 4.
79,532 is divisible by 4 because 32 is divisible by 4
4,326 isn't divisible by 4 because 26 isn't divisible by 4.(5 votes)
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.