 So by now you are familiar and skilled at algebra, but what is algebra? We might call it a method of solving equations involving real or complex numbers. Now as you get to higher mathematics, it becomes useful to call this school algebra, and so this allows us to talk about abstract algebra. And abstract algebra tries to generalize this idea to equations involving other things. So let's think about that. Algebra involves equations to be solved. School algebra uses many types of equations. Linear equations, polynomial equations, radical equations, rational equations, and logarithmic exponential and trigonometric equations. Ideally we'd want to develop an algebra to solve all of these, but that's too hard. To simplify our problem let's begin by considering the operations involved. Linear, polynomial, radical, or rational equations can be transformed into equations that involve only the four basic operations of arithmetic, plus, minus, times, multiply, whole number exponentiation, which is really a form of multiplication, and all of these operations are applied a finite number of times. And so we define an expression as algebraic if it uses only the four basic operations of arithmetic, including whole number exponentiation, applied a finite number of times. An equation is algebraic if it can be transformed into an equality of algebraic expressions. For example, maybe cube root of x plus 5 equals 12 shows that this is algebraic. And let's think about this. If we were to solve this, our first step would be to eliminate the radicals. And we could raise both sides to the third power, and we get an equality of algebraic expressions. And so our equation is algebraic. And in fact we can show that a much more frightening equation like this is algebraic. So again let's think about this. If we were to solve this, we'd eliminate the quotients by multiplying by the denominators and eliminate the radicals by raising both sides to powers. And we could do this, but we won't do it on screen because it's quite a process. Now you might remember from calculus that we can express functions like log, sine, and e to the axis of power series. These require us to apply the basic operations an infinite number of times. And as a result, these are not algebraic functions. We say that these are transcendental functions. Now we could allow equations involving transcendental functions, but that would be too hard. And we could allow equations involving all powers and all arithmetic operations, but that would be too hard at least as a first step. And so to begin with, let's allow one operation and let's consider equations like, I don't know, x plus 7 equals 15 or 5x equals 12. Now we're only allowing one operation, but mathematical operations fall into several types. First, there are what we would call unary operations that take only one argument. These are things like square root, tangent, and so on. You only need one number and you can take the square root of one number, you can take the tangent of one number. But there are operations that require two numbers to make sense. And so these are the binary operations. These are things like divide plus. If you don't have two numbers, you can't do a division or an addition. And for completeness sake, we'll note that there are ternary operations that take three arguments. For example, the definite integral is a ternary operation. And we can imagine operations that require four, five, six, or five billion arguments. But again, let's go down to basics. The basic operations of arithmetic are all binary operations. And we call the arguments the operands. So in 5 plus 3, the operation is plus and the operands are 5 and 3. Now, there's three common ways to represent the binary operation itself. First, we can use a symbol between the operands. So when we write a plus b, the plus is the symbol that represents the operation. We can also use function notation f of a and b. And so here's a function with two arguments. It's a binary operation. And lastly, we can juxtapose the operands. So a, b gives us the operands a and b. And the juxtaposition indicates that there's going to be some binary operation that acts on them. In this case, we read the decimal multiplication. Now, this does impose a little bit of a problem. We should use function notation, but we often use a symbol or juxtaposition. And here it's very important to remember there are only so many symbols. So a symbol like plus or times or an expression like a, b might not mean what you're used to. And that's one of the most important things to keep in mind when studying abstract algebra. Context is everything. So the study of abstract algebra revolves around what are called algebraic structures. And so we'll define the following. An algebraic structure involves some set of operands, g, some things that we can apply an operation to, and one or more binary operations that act on the elements of g itself. Now, again, we have many ways of representing the binary operation, but for now, if we represent the binary operation as star, then the simplest algebraic structure can be represented as the two-tuple g star. g is our set of operands, and star is the symbol we're using for the binary operation. So, for example, let's express the algebraic structure consisting of the natural numbers with the operation of addition. So here's an important idea to keep in mind as you work your way through advanced mathematics. Definitions are the whole of mathematics. All else is commentary. And so since we want to talk about the algebraic structure, let's pull in our definition of the algebraic structure. And let's see an algebraic structure involves some set of operands, g, one or more binary operations that act on the elements of g, adi, adi, adi. Okay, so let's see what we want to do here. So the first thing we want to do is to identify the operands. These are the things that our binary operation will act on. And since this is the operation of addition on the natural numbers, then the set of operands is going to be the set of natural numbers. Meanwhile, the operation is addition, which means we can express our algebraic structure as the two-tuple n plus. Or maybe we have the algebraic structure consisting of continuous functions with the operation of composition. So if I have the juxtaposition fg, what I really mean is f applied to g of x. So again, definitions are the whole of mathematics. All else is commentary. Our definition of algebraic structure requires us to identify the operands and the binary operation. We'll call the set of continuous functions c0. That's actually something we're borrowing from real analysis. File that under coming attractions. And if our binary operation is function composition, the symbol we often use for function composition is the open circle. And so our structure is c0 open circle. And that specifies both the operands, the continuous functions, and the operation function composition.