Calculus for Cranks: Lecture 1: How to add and multiply real numbers

Most calculus books begin with a description of the real numbers. Often, this description is not very useful. We are told that the reals lie on a line amongst the rationals, that they obey certain rules, and that they can be understood through an appeal to our intuition.

Later in the book, mysterious theorems are introduced without proof. (Beyond the scope of this book, it says.) Complicated definitions are introduced for limits and continuity and then are never used. It is difficult to take the whole thing seriously. Perhaps, the author doesn't really want us to understand but just to learn some incantations for "solving" prearranged "problems". Regardless of whether he wants this, we oblige him.

This is the beginning of a series of hub pages which will serve as a set of lecture notes on calculus for the idealized crank. The crank doesn't want credit for a calculus course. He just wants a rapid introduction to the ideas contained there which will allow him to completely master them. The crank feels somewhat confused by the unclear references to real numbers and their properties contained in most texts. For his convenience, we begin by describing real numbers precisely.

A real number is given by a sign, followed by an integer, followed by a decimal point followed by an infinite number of digits between 0 and 9.

A positive real number:

+N.a₁a₂a₃ ...

A negative real number:

-M.b₁b₂b₃ ...

It requires a certain amount of faith in set theory to even believe in infinite strings of numbers after the decimal point, but we will neglect this point (which is a definite weakness of the real number system.) We will treat set theory naively and pretend these objects exist. However we are still not finished because not every real number has a single decimal expansion. Some special real numbers have two. There are certain pairs of decimal expansions which we identify as a single real number. We should say precisely what are these pairs.

First, +0.000000000... is identified with -0.000000... (There is only one zero.) Secondly if a_m is a digit other than 9, then N.a₁a₂a₃...a_m9999... is identified with N.a₁a₂a₃...(a_m+1)00000....Finally N.9999999.... is identified with (N+1).000000... That's it. To sum up, the only real numbers with two representations are those with repeating 0's or repeating 9's.

We have now defined the real numbers as a set within the rules of naive set theory. They are either decimal expansions, or pairs of identified decimal expansions. That was easy enough. It all agrees with the intuition that we got in grade school. But one of the things that we are taught in grade school is that real numbers can be added and multiplied (and subtracted and divided). But how? What is the definition of addition and multiplication.

We see this is a much deeper question when we begin to think about it. In grade school, we are always taught algorithms for adding and multiplying numbers with finitely many places after the decimal point. These algorithms tell us that we should begin from the right. However, with real numbers, this is impossible. Real numbers have infinite decimal expansions. We can never get to the right!

How we resolve this difficulty will be the subject of this hub. The solution contains a concept which underlies all the ideas of the calculus. However, even the problem, as we have stated it here is concealed from most beginning calculus students.

Why are mathematicians so enamored of the real numbers if they are so unwieldy to work with? One might be forgiven for thinking that part of the reason is in the advertising. A person would have to be a genius to name a set of objects that are so exotic: the real numbers. With a name like that you've got to believe in them.

One answer is that mathematicians like axioms and the real numbers are the only object satisfying a particularly elegant system of axioms. (For experts, they are the only complete, totally ordered field.) Still, one should not lose sight of how weird they are. In the late author Madeleine L'Engle's book A Wind in the Door, L'Engle writes that the only things which can be loved are those which are named. One may wonder how she felt about real numbers.

People often tell me that Chomsky is a great linguist because he discovered that there are infinitely many sentences (and names for that matter.) I laugh because I think this is something that set theorists understood long before. The point is that though the names are finite, they may be arbitrarily long, and so there are infinitely many of them. This still places a limitation on how many names there are. The names are countable. We may assign to each a natural number. (The1st name, the 2nd name etc.) How do we do this? We just write them in alphabetical order. Real numbers can't be alphabetized, mainly because you can't get all the way to the right.

Cantor was a great set theorist who proved that the real numbers are uncountable. Here was his proof (a proof by contradiction): Suppose there is a list of all the real numbers, associated to the natural numbers. There is number whose first place after the decimal is different from that of the first number, whose second place after the decimal is different from that of the second number, and so forth. (The nth place after the decimal is different from that of the nth number.) Therefore the number we have produced differs from any number on the list. This contradicts the assumption that we made a list of all the real numbers in the first place. Hence the real numbers are not countable.

So what? It means that we base all of our mathematics on a system of numbers so vast that most of its elements can't be named. No matter what language we speak, the vast majority of the real numbers can never be mentioned. Then in what sense do they exist? What would Madeleine L'Engle say?

Why then base mathematics on something so outlandish. In part, the answer is that it doesn't actually matter. If we are sensible about what questions we ask, they will still have meaning without ridiculous infinitary constructions like the real numbers. Another answer is that the real numbers provide a useful idealization, letting us treat objects much more complicated than numbers as numbers. The main reason these notes will deal with real numbers is just to be consistent with standard mathematical practice. The crank reader might want to leave his mind open to other possibilities, however.

In this text capsule, we will consider the question of how to add two positive real numbers

x=N.a₁a₂a₃...

and

y=M.b₁b₂b₃...

We leave as an exercise to the reader to define addition of negative numbers and subtraction.

If I were to bring up the subject of defining x+y in a calculus class, there would always be some world-wise student who would suggest, "Why not just plug x and y into our calculators?" Students really believe that their calculators work with real numbers. In fact, the calculators work with floating point numbers - a small piece of a decimal expansion together with a register saying where this expansion should live relative to the decimal place. The point is, in order for calculators to be able to say anything about addition, there must be something that can be said using only a small number of decimal places.

Let's try this intuitively. What can we say about x+y if we know just N and M? We know that there are just two possibilities for the integer part. The integer part will be either N+M or N+M+1. (The astute reader might object that if x is actually N.999999.... and y is actually M.999999... then we can get the answer N+M+2 but we view this answer as N+M+1.999999.... If we know the result in a certain digit to with two possibilities, we will say that it has been bound. If we know it precisely, we say that it has been fixed. We have shown that knowing just the integer parts of x and y, we have bound the integer part of x+y. Can we ever fix it? Suppose we know a₁and b₁as well. If a₁+b₁is 8 or less then the integer part of x+y is fixed as N+M. If a₁+b₁is 10 or greater, then we carry the 1 and the integer part of x+y is fixed at N+M+1. If however a₁+b₁is 9, we don't know yet whether we will carry a 1. It depends on later digits. It may take a long time to fix the integer part if we keep getting 9 every time we add two digits. However if we always get 9 then we have a well defined sum N+M.99999... which is identified with N+M+1.00000... The point is that in these cases some digits may never be fixed. But we just need to show that each digit can be bound in finitely many steps in order to get the sum of the real numbers to be defined as a real number.

To be more precise, given x and y as above,we let x_jand y_j denote their decimal expansions up to the jth place. We may view these as rationals with denominator 10^j. Thus we know how to add them as rationals. Then we have that (x+y)_j is bound to be x_j+y_j or x_j+y_j+1. Eventually either it will be fixed or the sum x+y will actually be defined to be x_j+y_j+1. Thus we have actually defined addition of real numbers.

In this capsule, we confront the problem of how to multiply our real numbers x and y from the previous capsule. Again we will use the jth decimal approximations x_jand y_j. We can multiply these because they are rationals. We will try to figure out how many digits of xy we can bound just by knowing the product x_jy_j. The answer is more complicated than for addition.

We would like to think that x_j ≤x≤x_j+10^-j and that y_j≤y≤y_j+10^-j. Now we multiply these two inequalities to get upper and lower bounds for xy. We get

x_jy_j ≤xy≤x_jy_j + 10^-j(x_j+y_j) + 10^-2j

Since we are dealing with inequalities, we can afford a slight over estimate and use the fact that x_j+y_jis less than or equal to M+N+2 and that 10^-j is less than or equal to 1. Thus we get

x_jy_j≤xy≤x_jy_j+ 10^-j(M+N+3)

This is a slightly weaker inequality. The advantage of it is that M+N+3 is independent of j and therefore fixed throughout this multiplication problem. We use it to define another number k which is fixed throughout the problem. We let k be the smallest integer so that M+N+3 < 10^k. Then

x_jy_j≤xy≤x_jy_j + 10^k-j

What does that inequality mean? It means that once we have multiplied the jth approximations of x and y we have bound the j-kth digit of the product. We may never fix it, but if we don't, the product is defined anyway. Thus we have defined multiplication of positive real numbers.

Exercise: Define division of positive real numbers

 Mathematicians generally don't like the presentation of addition and multiplication given above. It gives a special status to the number 10. Of course, there no reason we had to do things base 10. The Babylonians did their calculations in base 60. The late author Douglas Adams pointed out that the universe does its calculations base 13. Of course, it doesn't matter which base we use, but that requires a proof.

Exercise: Define addition and multiplication for reals in base 13. Then show that you have actually defined the same operations I just did.

 The operations of addition and multiplication defined above are very friendly for the makers of calculator. The reason is that the calculator maker always knows how many digits of input  he needs to specify a certain number of digits of output almost perfectly (to bound them).

Let's make a definition out of that. An operation that inputs some real numbers and output a real number is calculable in base b if for every positive integer j there is an integer l (possibly depending on the input numbers and on j) so that specifying the inputs up to digit l (in base b) binds the output up to digit j.

Read that definition carefully. It has a lot of quantifiers. (For every ... there exists ...) In the example of addition l is j. In the example of multiplication l is j+k where k depends on the inputs. Basically the definition says that an operation is calculable in base b if there is a base b floating point calculator that can compute it to any degree of accuracy.

In calculus, a great deal of emphasis is placed on functions that are continuous. We will give the definition in the next hub of this series. Every continuous function is calculable in any base. Every function calculable in any base is continuous. We will prove that next time. (In fact, it is sufficient to be calculable in two relatively prime bases.)

 The ulterior purpose of much of this lecture has been to give a concrete point of view on continuity. We will use this viewpoint in much of what follows.

I would like to close this hub by pointing out that it is possible to be calculable in one base without being continuous. To illustrate, I provide links to two rather nice papers of my student Michael Bateman where this fact is used to quite dramatic effect.