Making new numbers from old. Integer numbers, rational, real, and complex numbers. Negative numbers. Having the initial numbers is all very well, but quite quickly you run into troubles. For instance, when solving equations. If you have the equation x +2 = 3, then it is clear that x must be 1. But if you've got the equation x + 3 = 2, then there is no natural number which solves this equation. Now this may seem a rather abstract mathematical equation, but in the moment you are a trader, and you're dealing with money. And as an account, and you need to balance the accounts. So the account of your other trader is x, and he needs to pay you 2. To end up at 0, you very much are interested in the value of x. Now, with the natural numbers, you can't express x, so you need to introduce new numbers, the negative numbers. So by definition, The number, Minus a, where a is a natural number. Is the solution To the equation x + a = 0. Now negative numbers are a rather unnatural concept, because every child can count, one marble, two marble, but then what is minus one marble? Cannot exist. It cannot be a negative marble. So one of the reasons for that is, is that these numbers have been for a long time represented in a different color, typically in red. So if the black numbers, which are the good numbers, and the red numbers, which are the negative numbers. But which are basically seen as positive numbers of a different kind. And it has taken us surprisingly long, until the end of the 18th century, until people actually accepted negative numbers and worked with them. We need more numbers, we also need the rational numbers. Also, these numbers have been introduced basically from a need from economics. And its need to divide things. For instance, if you've got 6 items, which you want to divide over 2 persons. Then every person will get 3. But if you've got 2 items, which you want to divide over 6 persons. Then again, there's no natural number, and even not an integer number. x which can express the amount every person gets. So in order to do that, we have to introduce rational numbers. And rational numbers, which I denoted by Q for quotients, are introduced as again, ordered pairs of numbers. Where the first element can be just an integer number, the German word for number. And a second q can also be an integer number, except for 0. That one's forbidden. And you need to tell something about these pairs. The pair (p,q), should represent the same number as the pair (kp,kq). Now of course, we never write these pairs like this. We write them like this. We write kp divided by kq, and we've got a familiar equation that kp over kq is p over q. And we have, have to define addition and multiplication for these numbers. In the normal way, p over q + r over s, again things we've learned at school. We write ps divided by qs + qr divided by qs, and then we say is ps+qr divided by qs. In ordered pair notation, this would of course be (p,q) +(r,s) is (ps+qr, qs). Well if you give this as a definition, it looks rather incomprehensible. But if you look first at this computation, then it is quite clear where this definition comes from. And even more easily, if we have multiplication, p over q times r over s is (pr over qs) is (pr, qs), And in this way multiplication and addition of rational numbers are defined. Now, how does this solve our problem? Well, If we give an example. We've got equation 6x = 2 and then our solution would be the pair x = (1,3). Well how do we compute this? 6, the number 6 can be represented as the number (6,1). And then we have to multiply 6 by x, which is (1,3) multiplied by (6,1), which is then (6,3). Which we can reduce, or can write as (3 times 2, 3 times 1) and by the reduction. This is (2,1) which is by definition equal to the integer number 2. So we see that x = (1, 3). This new rational number solves this equation. And we write it as 1 over 3, and say, yes, that's a third, we know about one third. But rational numbers are not enough. And the reason they're not enough is partly due to the theorem of Pythagoras. So remember, if you've got a triangle with one straight angle, and its sides are called a, b, c then theory of Pythagoras states that c squares = a squared + b squared. Which means that if a is 1 and b is 1, then c squared is 1 squared + 1 squared. Is 2 and c is what we call the square root of 2. Now you can show, or we will show it in the proof section. You can show that the square root of 2 is not a rational number. So there exists, except between the rational numbers, there exists more numbers. And the way to think about it is to put all the numbers in a line. All right, so this is the line, the number line. And then here's 0, here's 1, here's 2 etc. The negative numbers are the other direction -1, -2, 1/2 is over there, 3/2. -1/2 is over there and it can fill up, here's 1/4. I can fill up the line with rationals but still there are numbers missing. And for instance one of the numbers that is missing is here, square root of 2, For every rational number I can decide whether it's bigger or smaller than square root of 2. Because if p/q is smaller than the square root of 2 that would imply that p squared over q squared is smaller than 2. But for 2 integers I can always decide whether p squared over q squared is smaller than 2 or bigger than 2. So I can make the set of all rational numbers p/q, Which are such that p squared over q squared is smaller than 2. This set, Has no largest element. Because that would be the largest element of the third would be a rational number, with this property and no other rational number having the same property. But you can imagine that if I've got a rational number which has a finite distance, which squared has a finite distance to 2, I can always tuck in even bigger rational number. So this set has no largest element. So we define, Square root of 2 as the supreme of this set. So basically I am saying while there is no largest element, but the set doesn't extend of all the reals I know that. I know only the rational numbers, which are here in the red set are in it, but over here there are real numbers which are definitely not in the set. So, I just say, well let there be a larger number of the set, and I call this, define this larger number as being square root of 2. In this way, every element of the real line corresponds to a real number. So we should ought be there yet, shouldn't we? Because we've can now hear the real line. And every line, every point on the real line, corresponds to a real number. So surely, these must be all the numbers. Well, no, there are still such things as complex numbers. And they arise again from the need to solve a certain equation. In this case, the equation square root of x = -1. Well what's the problem? Of course we can draw the graph of the square root of x squared here. So this is y = x squared, and here's the graph. Y is -1 and clearly those graphs don't have an intersection point. Well yes that is true, but in general if we have the equation x squared = a squared then we can transform this as follows, x squared- a squared = 0. So (x- a) (x + a) = 0. So we find two roots. And in general, a quadratic equation has two roots. Now when people find out that in general a third order equation has three roots, or in most cases. Then it turned out that it might be interesting to postulate that any quadratic equation has two roots. So they introduced a number i, Such that, i squared would be -1. And then you can write down x squared = -1 = i squared and with the solution formula you have that (x- i)(x + i ) = 0 so that means that x = i or x = -i. But coming back to the real line, there's a problem. Because where should this number i be? There's no space left. Well the solution to that was rather ingenious I have to say. There's no space left in r so let's put it up here. Here's a number i. Which means that a complex number is a pair, again, a pair, Of real numbers. So The complex number a, b is also written as a plus ib. And the set of complex numbers is the set of odd pairs (a, b) such that a and b are real numbers. All right, but how to compute? With real numbers and with complex numbers. Well, It's very simple. Just as, with real numbers, but you have to keep in mind that i square is minus 1. So, for instance, 1+2i, sort of complex number. If I wanted to add to 3 plus 4i, I add 1 and 3, that's a 4. I add 2 and 4, that's a 6i. If I want to multiply these numbers, then I do the multiplication just as I would expect it. That's 1 * 3 + 1 * 4i + 2i * 3 + 2i * 4i. Which is 3 + 4i + 6i + 8i squared. And now I have to keep in mind that i squared is equal to -1, so that would be 3 + 10i- 8, which would be -5 + 10i. Division needs a little trick. So, if I want to divide 1 + 2i by 3 + 4i, you need to remember a little trick, which goes as follows, that you multiply numerator and denominator by 3-4i. So you take the number in the denominator and you repeat it, but you replace that number i by minus i. And there's a special construction, I'm going to say something later about it. If you do that, well, in the numerator, there's a product that is 3 + 6i- 4i- 8i squared. And in the denominator, we've got 9- 12i + 12i- 16i squared. So we go on a little, the i squares become -1s. So I'll get 11 + 2i. And in the denominator, you see, and that was the whole point of the exercise, that you've got a minus 12i and a plus 12i. Those two cancel, and the minus 16i squared becomes a plus 16, so that gets us 25. Now we know how to divide by a real number. So this will be 11 divided by 25 plus 2 divided by 25i. So this is the way you divide by a complex number. Now what about that trick? The trick, use something which is known as a complex conjugate. So if I've got a number, complex number, a + ib, so a set of complex numbers. Then the complex conjugate, of z is, z bar, is a- ib. It's another complex number, and the way to think about it, is probably best to draw the number again. So if I've got here my number a plus ib, so there's that. That is a plus ib. Then this corresponds to the point with coordinates a and b. Now z bar corresponds to the point with coordinates a and minus b, which would be the mirror image of that point with respect to the real xs. Now what is so great about complex conjugates? Well, the point of the complex conjugate or one of these points is that if I multiply a number with its complex conjugate, then I'll get a real number. So if I have a + ib and a- ib, then I'll get a squared + abi -abi -i squared b. Then again, those two terms cancel out against each other, and I have a squared + b squared. Right, now, if you have a look again at this little diagram, then you see that this line has a length a. Up here, this little line has length b. And this angle here is a right angle, which means that, by Pythagoras, the length of the hypotenuse is a squared plus b squared. So a squared plus b squared, and the square root of that, is exactly the distance of z to the origin. So that means that the square root of z, z bar, is the distance of z to the origin, which is by definition the absolute value of z. Recall, the absolute value of a number is the distance of that number to the origin. So I can write this also like this. If we To find the absolute value of z this way, then I've got that z times z-bar is the absolute value of z squared. Which is not only a real number, it's even a non-negative real number. This absolute value of a complex number is something which we use rather often when dealing with complex numbers. Now are there any more numbers? I mean, we've gone from natural numbers. To integers, To rational numbers, Then we got the real numbers, The complex numbers. And each step we had to take in order to be able to solve equations here, it was an equation x plus 2 is 3. Here we wanted to solve an equation like 6x is 2. Here we wanted to be able to talk about square roots. Here we wanted to solve the equation x squared is minus 1. Are there anymore equations we might want to solve? And do they give rise to more numbers? The answer is no, but it's a qualified no. There are such things as quaternions, Which are of the form a + bi + cj + dk. Where the i squared is j squared is k squared is all -1. And i times j is k, and j times k is i, and k times i is j. Which have been found by an Irish mathematician, Hamilton. But they are not commutative. Commutative means that the product of two numbers is always the product with the factors in inverse order. And that doesn't hold always. And you can even go one step further, the octonions, Which are even more complicated. But since quaternions already not commutative, the octonions are also not commutative. So yes, there are, in a way, you can generalize the concept of numbers. But these aren't useful numbers, so we don't need to know them. So the story ends at the complex numbers. Even the complex numbers have a drawback of the real numbers because the real numbers we can order. We can say one real number is bigger than another real number. Complex number we cannot order because there are number of complex numbers which are all the same distance to the origin for instance. But the complex numbers are very useful because we need them in order to model periodic behavior. But that is something you will encounter later. So these are all the numbers you'll ever need, I think.
