Hmm, I think this is a fine essay for people who have already internalized the notion of algebra per se, but it's not great for a student who has to yet to do so and is asking the titular question "What is algebra?"
Here's my attempt for, say, a relatively smart high school student. Consider this a rough draft; I'm writing it from start to finish without editing. I'd love, love, love feedback, though.
Today even schoolchildren understand arithmetic. We have these things called "numbers." There are different types of numbers like natural numbers, integers, rational numbers, and irrational numbers. We have rules for manipulating these numbers like addition, subtraction, multiplication, and division.
This wasn't always the case, however. I don't mean that humans couldn't always add, but I do mean that it took humans thousands of years and multiple false starts to come up with a sensible way to represent numbers and these operations. Think about trying to do division with Roman numerals, for example. It'd be a nightmare!
There was a time when numbers like "4/5", "-2", "0", and "√5" made people freak out because we didn't have the symbols to represent them and it wasn't obvious what they corresponded to "in real life," if anything. Imagine yourself living in a world like the ancient Greeks, for example, where you represented numbers by talking about lines of a given length. It was very hard for you to talk about numbers per se without drawing a shape that somehow encoded that number. Now, pop quiz: how do you represent something like "0" or "-2" in this world? If you were a Roman using Roman numerals, how would you represent "4/5"?
This is just a story to highlight that although we take our numbering system and arithmetic for granted, this was not the case for most of human history, even most of recorded human history. The way we do arithmetic was an invention that both helped us do arithmetic and helped us understand WTF a number even is.
By the way, if you freak out at the idea of imaginary numbers or the idea of a number i which satisfies the property i^2 = -1, this is no different than the kind of freaking out the ancient Greeks did when they first encountered √2 or other cultures tried to make sense of negative numbers.
Now, let's think about what we really did by inventing arithmetic as we understand it today. You can't point to the number "5" anywhere in the world, right? Even the symbol 5 isn't five per se, any more than "five", "fünf", "|||||", "V", or "五" are five per se. But with one symbol "5" we can represent this abstract thing.
Then something like "5 + 4" might represent the length of a line segment made from concatenating a line segment of length 5 and a line segment of length 4, the age of a 5-year-old in 4 years, the volume of water made from pouring five buckets of water into a pool and then four buckets of water, the number of apples shared between a 5-apple basket and a 4-apple basket, and so on. So, these abstract things we call "numbers" and "arithmetical operations" can represent many more concrete things.
Let's call this process "abstracting." Algebra is what you get when you treat numbers as the concrete thing and apply this same process. With arithmetic, we want to talk about numbers divorced from a particular concrete realization. That is, we want to talk about the number 5 without having to talk about a basket of five apples. With algebra, we want to talk about numbers per se divorced from a particular concrete number.
Remember, when we invented arithmetic we had to invent a bunch of symbols to represent the abstract thing. We do that in algebra, too. Often we use single-letter symbols like x and y, but we could use anything like ☃, ☂, or zorpzop. These symbols "stand in" some number in the same way that the symbol 5 "stands in" for all the things 5 could possibly represent in the world. We can talk about 5 without talking about the things it might represent.
So, we say things like "let x be a number" or "let x be a positive number" or "let x be a rational number." What can we say about x in each of these situations?
For example, if x, y, and z are all standing in for some number, we can say the following:
x + 0 = 0 + x = x regardless of what number x is
x + y = y + x regardless of what numbers x and y are
x + (y + z) = (x + y) + z regardless of what numbers x, y, and z are
x*1 = 1*x = x regardless of what number x is
x*(y + x) = x*y + x*z regardless of what numbers x, y, and z are
These are true because of what we mean when we say "number" and what we mean when we say "addition." It's not as if these are true for some numbers and not all, nor is it as if we know these are true because we've "checked all the numbers." That's impossible because there are an infinitude of numbers.
So, now we might ask things like, "Are there any numbers x such that x^2 + 1 = 0? How about x^2 + x - 1 = 0?" These questions might be hard to answer, but we have now at least invented a language where we can ask them, whereas before "abstracting" arithmetic into algebra we had no easy and succinct way of asking them.
This is no different than not being able to easily ask, "Can we construct an equilateral triangle with side lengths of π?" before abstracting from more concrete things into numbers. Without a symbol for π we have to say things like "the constant that is the ratio formed between the circumference and diameter of a circle." This is how mathematics was done for thousands of years. It was tough going, as you can imagine, and we missed many things that would seem "obvious" to people using our notation.
You can continue this process further, by the way, and abstract further from algebra. This is what mathematicians call abstract algebra (http://en.wikipedia.org/wiki/Abstract_algebra). In this context there are multiple algebras and the "algebra of numbers" becomes the concrete thing in this new system. Linear algebra is a different algebra, for example, with a different sets of "numbers" and a different set of "arithmetical operations" that don't always correspond 1-to-1 with the numbers and operations we find in arithmetic.
Often, when presented with a new physical system of objects that interact in a certain way, we can try to abstract these objects and operations into symbols and derive rules about these abstract symbols and operations that correspond to the workings of the physical system.
We might call this symbolic system "an algebra." For example, Claude Shannon invented an algebra for relay and switching circuits that allows us to understand how they operated and how to combine them without actually building physical circuits. See http://www.cs.virginia.edu/~evans/greatworks/shannon38.pdf
I like this explanation. High school students are surprisingly smart and a little perspective -- that "algebra" is just manipulations in a symbolic system (and not even a special one, just very widely used) -- is sometimes all it takes to help them "get it" a little more.
It's kind of funny, because students are taught how to use these things like numbers and operations and variables, but it's never explained in an abstract way what "algebra" is or why it's even called that. Hell, I didn't know until I took linear algebra in college.
When my sister (who is currently taking algebra) asked me what "algebra" meant, I told her:
An algebra is a combination of a set of objects and a set of operations. The algebra you're learning has real numbers as the set of objects. You know what the operations are: they're things like multiplication, addition, sqrt, whatever. There are other algebras too that use things other than the reals.
As an aside, this is a great way to teach fractions as well (or at least, I like it a lot). People try to give intuitive explanations with pieces of pie or whatever but students tend to find them very confusing. You would be surprised how fucking confused students can get about fractions, I mean some just can't wrap their heads around it ever. Even as adults. They're not dumb, they try to learn fractions by "intuition" and, frankly, fractions aren't intuitive. The simplest route is to just lay it out: these are objects with one expression and another expression and a line between them, and you can multiply them like this, and add them like this, and "move" expressions from one side of the line to the other by taking the reciprocal like this, and that's it.
Division with Roman numerals being a nightmare was not obvious to me. That was my first friction point here :-)
> There was a time when numbers like "4/5", "-2", "0", and "√5" made people freak out because we didn't have the symbols to represent them
Who freaked out, and how could they freak out if they could not behold these numbers in the first place (i.e. no symbolic representations)?
> without drawing a shape that somehow encoded that number
Recommend you use "represented that number" because encoding has a strong, separate type of meaning to me.
> Now, pop quiz: how do you represent something like "0" or "-2" in this world? If you were a Roman using Roman numerals, how would you represent "4/5"?
For "0" I would have made an empty box, out of strings if necessary. For the -2 I would have placed two of the strings in a different location. For the Roman numeral conundrum I would have placed IV and then a line and then a V below that. Now, I just solved your conundrums. Or didn't I? To really speak to beginners it's important to delve into this stuff. So far I have not discovered why our number system is so great, and I don't feel like I'm freaking out about anything in particular. :-)
Thanks for your writeup, though. I hope you can turn it into a book for people like me.
> Who freaked out, and how could they freak out if they could not behold these numbers in the first place (i.e. no symbolic representations)?
The story goes that Hippasos of Metapontum discovered the existence of irrational numbers like √2 while at sea, and his fellow Pythagoreans threw him overboard, because it proved that there were aspects of the world that could not be represented with rational numbers. That said, the notion of a square root had been established for at least a thousand years by that point (by the Egyptians and Sumerians), so the freaking out was less about their representation and more about their properties.
> For "0" I would have made an empty box...
Your representations are parasitic on the fact that you already have a deeply ingrained representation for those concepts and are comfortable manipulating those concepts. It's like suggesting that you'd reinvent the wheel if you lived in a civilization with no wheels merely because it seems obvious to you looking at it today. Really it's not at all obvious, as no New World civilization ever developed the wheel, and most Old World civilizations borrowed it rather than inventing it independently.
Think of it this way: if you understood fractions, but nobody else did, how would you represent them symbolically such that everyone else would? I'm of the opinion that our typical notation for calculus is spectacular (especially compared to Newton's original notation—the d/dx notation used today comes via Leibniz) but if you don't understand calculus, how does that help you? You understand negative numbers, but to a person with no concept of debt, a person for whom numbers represent 'how many sheep you have', what does that mean, and how do you symbolize it so that they do understand?
> For "0" I would have made an empty box, out of strings if necessary.
That's 4. Because you used four strings. Oh, you do it with one string? It looks like 1. You do it with N strings? That's N. Oh, you're trying to talk about what's inside? Nothing's inside. That's silly. We both know there's nothing in there. Why are you trying to show me nothing?
Are you trying to multiply by the inside? That doesn't make any sense. Why would you multiply by nothing, or add by nothing? What's the point? No, I don't know what it does. Why should I care about your nonsense?
> For the -2 I would have placed two of the strings in a different location.
Now do "square root of negative nine".
> For the Roman numeral conundrum I would have placed IV and then a line and then a V below that.
Oh, so there's 4 on the top and 5 on the bottom? So, 9 total, right? IX?
According to story, Pythagoras freaked out about it. You don't need a symbolic representation of √2 to ask the question "what is the distance between opposite corners of a square with side length 1"?
Great explanation, worth an article! My thinking was more of a struggle a student has to take to correlate with a abstract concepts like algebra while in school. It is almost impossible for a kid to really look meaningful context out her daily maths class. However, all this concepts are born out of "need". If that "need" magically does not gets shown to kids "in their own language" there is going to be lot of trouble in bringing up right math power.
I guess the teaching style has to change. Was also wondering, folks in this thread, talking great things about math are teachers? Should be less, I guess!
I'm confused about what algebra actually means. Wikipedia states "It follows that algebra, instead of being a true branch of mathematics, appears nowadays, to be a collection of branches sharing common methods."
Like any word, it has evolved into meaning a bunch of different things and it's not easy to define what it actually means. The original article "What is algebra?" seems to talk about elementary algebra.
Forget about the Wikipedia article, which is trying to define algebra "in general." I know of no definition that encompasses all uses, both technical and colloquial and I'm not sure why we'd want one. A better question than "What is algebra?" might be "What does it mean when Person X says algebra?"
For most people it means high school algebra, which is what I was trying to explain until the very end. (High school) algebra is to arithmetic as arithmetic is to the more concrete and often physical things I mentioned. Put another way, algebra is what we get when we go through the same process we applied to invent arithmetic as we understand it today, but treat particular numbers as the more concrete thing.
Speaking precisely high school algebra means "the study of the real numbers under the operations of addition and multiplication as we typically understand them." Linear algebra and the algebra of the complex numbers sometimes make an appearance, too, which often confuse students because we're now equivocating and calling all these things just "algebra."
From the article: “(I should stress that in this article I’m focusing on school arithmetic and school algebra. Professional mathematicians use both terms to mean something far more general.)”
Here's my attempt for, say, a relatively smart high school student. Consider this a rough draft; I'm writing it from start to finish without editing. I'd love, love, love feedback, though.
Today even schoolchildren understand arithmetic. We have these things called "numbers." There are different types of numbers like natural numbers, integers, rational numbers, and irrational numbers. We have rules for manipulating these numbers like addition, subtraction, multiplication, and division.
This wasn't always the case, however. I don't mean that humans couldn't always add, but I do mean that it took humans thousands of years and multiple false starts to come up with a sensible way to represent numbers and these operations. Think about trying to do division with Roman numerals, for example. It'd be a nightmare!
There was a time when numbers like "4/5", "-2", "0", and "√5" made people freak out because we didn't have the symbols to represent them and it wasn't obvious what they corresponded to "in real life," if anything. Imagine yourself living in a world like the ancient Greeks, for example, where you represented numbers by talking about lines of a given length. It was very hard for you to talk about numbers per se without drawing a shape that somehow encoded that number. Now, pop quiz: how do you represent something like "0" or "-2" in this world? If you were a Roman using Roman numerals, how would you represent "4/5"?
This is just a story to highlight that although we take our numbering system and arithmetic for granted, this was not the case for most of human history, even most of recorded human history. The way we do arithmetic was an invention that both helped us do arithmetic and helped us understand WTF a number even is.
By the way, if you freak out at the idea of imaginary numbers or the idea of a number i which satisfies the property i^2 = -1, this is no different than the kind of freaking out the ancient Greeks did when they first encountered √2 or other cultures tried to make sense of negative numbers.
Now, let's think about what we really did by inventing arithmetic as we understand it today. You can't point to the number "5" anywhere in the world, right? Even the symbol 5 isn't five per se, any more than "five", "fünf", "|||||", "V", or "五" are five per se. But with one symbol "5" we can represent this abstract thing.
Then something like "5 + 4" might represent the length of a line segment made from concatenating a line segment of length 5 and a line segment of length 4, the age of a 5-year-old in 4 years, the volume of water made from pouring five buckets of water into a pool and then four buckets of water, the number of apples shared between a 5-apple basket and a 4-apple basket, and so on. So, these abstract things we call "numbers" and "arithmetical operations" can represent many more concrete things.
Let's call this process "abstracting." Algebra is what you get when you treat numbers as the concrete thing and apply this same process. With arithmetic, we want to talk about numbers divorced from a particular concrete realization. That is, we want to talk about the number 5 without having to talk about a basket of five apples. With algebra, we want to talk about numbers per se divorced from a particular concrete number.
Remember, when we invented arithmetic we had to invent a bunch of symbols to represent the abstract thing. We do that in algebra, too. Often we use single-letter symbols like x and y, but we could use anything like ☃, ☂, or zorpzop. These symbols "stand in" some number in the same way that the symbol 5 "stands in" for all the things 5 could possibly represent in the world. We can talk about 5 without talking about the things it might represent.
So, we say things like "let x be a number" or "let x be a positive number" or "let x be a rational number." What can we say about x in each of these situations?
For example, if x, y, and z are all standing in for some number, we can say the following:
These are true because of what we mean when we say "number" and what we mean when we say "addition." It's not as if these are true for some numbers and not all, nor is it as if we know these are true because we've "checked all the numbers." That's impossible because there are an infinitude of numbers.So, now we might ask things like, "Are there any numbers x such that x^2 + 1 = 0? How about x^2 + x - 1 = 0?" These questions might be hard to answer, but we have now at least invented a language where we can ask them, whereas before "abstracting" arithmetic into algebra we had no easy and succinct way of asking them.
This is no different than not being able to easily ask, "Can we construct an equilateral triangle with side lengths of π?" before abstracting from more concrete things into numbers. Without a symbol for π we have to say things like "the constant that is the ratio formed between the circumference and diameter of a circle." This is how mathematics was done for thousands of years. It was tough going, as you can imagine, and we missed many things that would seem "obvious" to people using our notation.
You can continue this process further, by the way, and abstract further from algebra. This is what mathematicians call abstract algebra (http://en.wikipedia.org/wiki/Abstract_algebra). In this context there are multiple algebras and the "algebra of numbers" becomes the concrete thing in this new system. Linear algebra is a different algebra, for example, with a different sets of "numbers" and a different set of "arithmetical operations" that don't always correspond 1-to-1 with the numbers and operations we find in arithmetic.
Often, when presented with a new physical system of objects that interact in a certain way, we can try to abstract these objects and operations into symbols and derive rules about these abstract symbols and operations that correspond to the workings of the physical system.
We might call this symbolic system "an algebra." For example, Claude Shannon invented an algebra for relay and switching circuits that allows us to understand how they operated and how to combine them without actually building physical circuits. See http://www.cs.virginia.edu/~evans/greatworks/shannon38.pdf