So let's start with some algebra. So there are nice simple equations like x plus 1 equals 5; course it's not too hard to solve that equation, we get x equals 4. Now this is an equation in whole numbers and we solved it in whole numbers; pretty soon if you write down equations you get things you can't solve in whole numbers. If the equation had been x plus 6 equals 5 there's no whole number solution to x, I would need minus 1 for that so that's x equals minus 1. So I need to introduce negative numbers to do that. That's good, but after all we might face an equation like 2x plus 1 equals 0, so we need some fractions as well to keep going on solving these equations. After a while you begin wondering why you didn't multiply xs together and
you get polynomial equations. So a polynomial is something like a constant times x to the n plus another number times x to the n
minus 1 plus - blah blah blah. (Brady: These are equations that
have got things) (to the power?)
- Right, they're not even equations they're really functions. If I have a function f of x then I talk- I could solve an equation but an equation needs an equal sign. So f of x equals 0 is an equation and I talk about a solution, but if I have just a function f of x then I can talk about a root of the function, that's the same thing as a solution. So I have have the habit of alternating between talking about roots and and solutions so I don't want to confuse anybody. Let's look at an equation, if we had x squared minus 1 equals 0, x could be plus 1, that's a solution, but there's another solution 2x equals minus 1. If I have other polynomial equations though it's not so easy to find solutions; for instance a simple one would be x squared equals 2. Well, there's no rational number that solves that equation, that's something the Greeks discovered they
considered it a deep philosophical mystery. But now of course we just write x equals square root of 2. And there's another one, if you'll believe that there's a square root of 2 then you surely believe there's a minus square root of 2, so that's another solution. So again I have two solutions which is pretty nice. So I could go on that way, unfortunately I quickly hit equations that don't have solutions in real numbers, something like x squared plus 1 equals 0 has no real number solutions because if I put in a real number x this is always has a positive square and when I add 1 it's even more positive so I can never get 0. So at this
point is begin- nice to begin graphing equations. For instance if f of x is x squared plus 1 then when x is 0 I get the value of 1, so I put a 1 there, and when x is for example 1 then this would be 2 so I'd be up here. When x is 2 it's 4 so it's up here somewhere; and everybody probably has seen this as a parabola. So it doesn't have any roots, it doesn't have any any values of x which make the value of f of x be 0. You can see that from the picture because a solution would be a place where the graph crossed the x-axis, in other words where y was 0, this line is where y equals 0. So for a function to have a root or an equation to have a solution the graph of the function has to cross the x-axis.
- (It's like an electric fence or a) (mag- special magic line.)
- A magic line, right. There is one kind of polynomial equation that always has a root in real numbers and that's the case when F has odd degree. So every polynomial has a degree, in this case it's n - exponent on the highest power. If you have a polynomial of odd degree; well this one was of even degree and that turns out in some way to be why it didn't have a root. If- let's take one of odd degree. If we took for instance the function f of x equals x- well f of x equals x of course is of odd degree and the graph of that is just a straight line that goes through the origin. You can see it has a solution when x equals 0 but I could do fancier things, I could say f of x equals x- maybe x squared times x minus 1, what does that look like? Well I can see from the way I wrote it that f of x equals 0 has two roots; at x equals 1 and the other one is at x equals 0. And the graph is going to look, it turns out, something like this. And I'll explain how I knew that in a second, but I see the root and I see that there's a really robust root there. This one, I sort of almost missed if I'd move down just a little bit it wouldn't have been a root anymore. For instance if I put here minus 1 then this whole picture would move down by 1, but it would still have a root here. And this is really a robust picture somehow. So that that's there's a theorem behind that; mathematicians love to write the word theorem, especially when they can prove what they're about to write. Any polynomial, and here I mean with real coefficients, of odd degree.
- (That's not) (the coefficients that are of
odd degree, this is) (the polynomial?)
- The polynomial has odd degree that's right; begins something like x to the fifth plus 7x to the fourth plus something else. -Of odd degree has at least one real root. This odd exponent is going to be the key, that's right. Just from that I'm going to show you that there's some w which when you put it in plug it in here it gives zero as the answer. That's pretty amazing that you can say that just from knowing that little piece of the polynomial. Anyway this is a proof that really explains something I think. I'm going to graph the polynomial, you might ask me how can you graph the polynomial if you don't know anything about it? But I'm going to do it anyway. I'm going to assume that a is positive just for the sake of argument, to make make one picture, but there would be an exactly corresponding picture if a were negative. And I want to find a value of x where y is zero, and of course that means a place where the graph will cross the x axis. The x axis is really the graph of y equals zero; it's that magic point on the y axis called zero and we want to get there.
You might ask why zero? Why not take y equals seven as your goal? And we could do that too, exactly the same proof would do that too. I want to graph it, well unfortunately I don't even know where to start. How can I graph a function that I know so little about? But I do know one thing: a positive number to an odd power is positive, but a negative number to an odd power is a negative number. And if I make the absolute value of the number really really big then I get a really really big number; and if the number is positive it's a really really really positive number, if it's negative it's a really really negative. Let's say a and b and c and all the constants of this are pretty
small, they're bounded by something anyway. And I take x to be 100 times as big as the biggest constant. Then x- and let's say the exponent's 5 just for the sake of argument - then x to the fifth power will be 100 times larger than x to the fourth power because I had- I multiplied it an extra time by x. So this first term will be enormously larger than anything else. And in fact if I graph this function you won't even see the influence of the other- other terms on the scale that I'm trying to draw here. Basically this first term is going to dominate, it's going to be the most important term. And so I can at least approximate what I'm doing, especially when x is large, just by thinking about the first term. Of course when x is small all bets are off, I can't
say a thing; but when x is large I understand. So let's take x large, let's take x 100 times larger than all the other coefficients - I'll just pretend it's 100 for the sake of argument.
- (So a big x.) A big x, x big. And I'm going to say absolute x big, and this is the way a mathematician would write that, absolute x much bigger than zero; so minus 100, plus 100, those are good candidates. So if x is big, x is 100, then x to this positive power will be much bigger than any of the other exponents and it'll be about 100 to the 2n plus 1; that's a big number. I'm going to draw it up here and you have to think that the y-axis is is a really different scale than the x axis. That point I can put in with some confidence, I don't know exactly where it is, pretty close.
- (It's way out there in the-) It's way out there, right. Now if I take x to be minus 100 then in- for the same reason, now I get enormous absolute value for x to the fifth or x to the 2n plus 1; but 2n plus 1 is odd so I'll get an enormous negative value. So that was the first point of the proof was that I can- I can say something about these way-out values. The second point of the proof - and there are only two - is that a graph of a function like this, which is called a continuous function, can be drawn without lifting my pencil. So I'm going to start drawing. I don't know where the graph really goes but I'm going to draw as if I did. It goes somehow. And when I get to this point - oh my gosh - it has to cross the x-axis to get up there. Well, all right, here we go. It just crossed, that's a root of the equation; and it goes on somehow. I don't have any idea really where the root is; it might be on the left, it might be on the right, but in order to get from here to here without lifting your pencil you simply have to draw a line that crosses. (Like you can't walk from
Mexico to Canada) (without going through the US?)
- Right, right, of course you could swim around and you- or you maybe who could dig a tunnel. And so we might have a little worry - how do we really know that there wasn't a little tunnel just at that place? Maybe the real numbers just opened up like the Red Sea and let the red thing through? Well, that's a really basic point about the real numbers and it wasn't completely understood until Dedekind I think in the nineteenth century, but you have to decide what you think the real numbers are. If I took the rational numbers- for instance, suppose I said I don't know what the real numbers are but let's do with the rational
numbers instead; this wouldn't have worked because there- for example x cubed minus 2 has no
real roots, that's an odd degree polynomial, and that's because this line would slip through the rational numbers where there's no rational number. And even adding square roots and cube roots wouldn't be enough, you'd have to add a lot of roots to make this work. So how do we know there isn't a hole there? And this is one of the great ideas in mathematics that's due to Dedekind and it was a definition of real numbers in terms of rational numbers. And it's so nice that I have to tell you about it even though you might already be convinced that there's no holes in the real numbers. So I'm going to draw the rational numbers like this, there are a lot of rational numbers but they're also lots of holes as we've seen. And Dedekind had the idea that what a real number really is is a way of dividing the rational numbers into two parts. So here- let's say this is going to be a real number, a and Dedekind said that what a really is is the set of rationals- rational numbers less than a, together with the set of rational numbers greater than a. So these two sets tell where a is in the real numbers. And the reason there are no holes left is if there were a hole I would just find- I would just define it to be the number which is the two sets, all the numbers the rational numbers less than that number and all the numbers greater than that number. So I've defined holes out of existence this way. (It's a really funny definition-)
- It's a great trick.
- (That's like me saying the) (definition of David is the left side of) (his body plus the right side of his body.) It's worse than that, it's saying it's the space on the left of me together with the space on the right of me; and it tells you just as little about me as this tells you about the real number that's sitting there. But it's a- it's a good existence kind of proof, or really construction, this is the sort of thing that people study in maybe the junior year course in mathematics.
- (Or philosophy by the sound of it.) Or philosophy, yes. These are called Dedekind cuts and it's a really important idea in mathematics. I want to graph this function, I want to tell you what happens when I apply x and I want to find a point here which - or at least to convince you that there is a point - a blue point which will go to zero on the red side.
