Abstraction: limits

Gowers necessarily avoids the details of the concept of "limit" and since the real numbers are intricately related to the idea of "limit" this means that he cannot give a particularly full picture of what a real number is. His choice of simplifying assumptions or conventions is correct, or at least satisfactory, but perhaps not the first choice for specialists. Since they may appear strange, I will say a few words about them.

To Gowers, at least for the purposes of this book, a real number is an infinite decimal expression possibly with a minus sign such as $-234.38475683745\dots$, $0.0003849999999999\dots$, and so on. There must be a rule that tells you, given $n$, what the $n$th digit here is, but that rule can be given in any way you like. (The decimal doesn't have to be recurring, and the rule doesn't have to be a formula of any particular kind.) However there is an important proviso in this view of real numbers, which is that a number ending with a recurring $9999999999999\dots$ is actually equal to the corresponding decimal with recurring zeros and the previous digit increased by 1. So for example the number $0.0003849999999999\dots$ is actually equal to $0.0003850000000000\dots$.

This really is equality, and not something else, such as approximate equality. Some students find it a shock to learn that $0.9999999999\dots$ and $1.00000000000\dots$ are actually equal. Gowers gives some arguments why this must be the case and why this convention is the only sensible one. (I suggest you ignore the comment about Abraham Robinson and nonstandard analysis on page 60. It is correct, though rather misleading as stated because it is lacking in so many details.)

Thus, for this book at least, a real number is a decimal expression as above with the convention that expressions with recurring 9's are actually equal to equivalent rounded expressions, and this equality is the strongest possible equality: it is a non-negotiable equal-by-definition.

This does, for sure, raise difficulties, which are not insurmountable, but these difficulties are the reason why most mathematicians usually prefer a different definition of real number. (A large number of candidate definitions exist, all of which "do" the same things and are therefore equivalent in the sense of the abstract structuralist view that "numbers are what they do, not what they are".) The immediate difficulties are in definitions of addition and multiplication.

Gowers gives a sketch definition for addition, based on adding from the left and revising or correcting previously calculated digits as needed. This will indeed work perfectly well, because of the (provable) fact that no digit will ever have to be corrected more than once.

Gowers does not give a sketch definition for multiplication, but the idea would be similar: multiply from the left and revising or correcting previously calculated digits as needed. This will work, but is much more difficult to prove that it works.

There is a third and much more subtle problem in both of these, to do with the convention on non-negotiable equal-by-definition of some numbers. Given decimals $x,y,x\prime ,y\prime ,z\prime$, suppose it happens that $x=x\prime$ and $y=y\prime$ by the convention (i.e. they might possibly be the same digits or one ends $9999999999999\dots$ and the other one is as above) and that the definition of addition is followed in each case to give $x+y=z$ and $x\prime +y\prime =z\prime$. Then it had better be the case that our convention on "equals" already gives $z=z\prime$. (Note: this is not some new convention for "equals". It must be the old one!) In other words following two calculations, one possibly done with a decimal with recurring 9s and the other with an equal decimal with recurring 0s, should give equal decimals in all possible cases.

For a couple of quick examples, going through the working of "addition from the left",

using a great deal of revision of digits, and

(which needs no such revision of digits). This is OK because $1.9999999999\dots =2.0000000000\dots$ by our convention. Similarly

and

by revision of digits, which is also OK (of course). But it has to be proved that this works for all possible combinations $x,y,z,x\prime ,y\prime ,z\prime$ and for both addition and multiplication.

A more usual point of view would be to define real numbers in some other equivalent way and then define decimal notation by

Of course the problem now is to make sense of the infinity symbol $\infty$ in the summation. That what the first year course on limits is about (Remember that this symbol is not really a number but some tag that has been added to the formula to denote that there is a process that "goes to infinity" going on.) It is also why Gowers chooses to define numbers in the way he does so that he can "hide" some of these problems and the $\infty$ symbol.

To sum up the section on the square root of 2, what Gowers says is that we would accept a decimal expression $E$ as being a square root of 2 provided for any small number of the form

there is a number of digits $k$ so that the first $k$ digits of $E$ squares to give a number between $2+\epsilon$ and $2-\epsilon$. This is precisely the limiting form that is studied in a typical undergraduate module on limits, and you should carefully check that the fictitious number "infinity" is no longer present.

The typical first year module might go on to prove in a similar way that every positive real number $x$ has $n$th roots for all positive integers $n$.

The example "the instantaneous speed of my car is 40 m.p.h." shows another kind of limit, in which the concept of infinity is present, but buried a bit deeper.

The main problem is that speed is defined for steady speed or average speed over a distance of a length of time, and the car is accelerating. (At this point you might want to review the idea of model and decide what parts of the discussion refer to the model and what part to "real life". Indeed, does instantaneous speed mean anything in "real life" or not?)

Since speed is distance divided by time, the classical mathematical solution is to say that the "the instantaneous speed of my car is 40 m.p.h." means that given any small number of the form

I can always choose a tiny fraction of time and measure the speed using distance gone in that time divided by the amount of time so that the speed I measure is within $\epsilon$ of 40 m.p.h.

Note that the interval of time I choose depends on the accuracy required and the situation is ideal in the sense that I assume I can always measure small times and distances to any accuracy needed.

You may know that speed is the derivative of distance travelled by time. Differentiation is also defined in just the same limiting process, so limits are essential for understanding calculus.

One final comment on this section: you may know from Einstein's special relativity or something similar that the idea of "instantaneous events" does not make sense in the phyicial universe we believe that we exist in. This is not what Gowers is talking about at all, and you should ignore such considerations and pretend that "instantaneous events" do make sense when reading this section. In other words, Gowers is using a slightly naïve mathematical model of reality here, and this is all we need in this section.

This is an elegant calculation using limits. Make sure you are clear what is being argued. We start from a definition (of $\pi$) and then use that number in something related but different.

The starting point is that $\pi$ is defined to be the ratio of circumference of a circle to its diameter (and this of course doesn't depend on the size of the circle). It is not even obvious that this number $\pi$ will enter into the formula for the circle's area. Maybe a different constant is needed? (If you think it is obvious then you have been indoctrinated too heavily by your teachers in school maths.) But it does. I suggest you read the section carefully and draw your own conclusions.

The area of a circle is a limit of a summation. Limits of summations like this are central in the idea of integration which is the other key concept in calculus. Thus all of the main ideas in calculus have something to do with limits.

title: Explain the notion of limit and give examples.

Conclusions: the idea of "limit" arises anywhere where real numbers are needed (and rationals are not enough) or where calculus is needed. Indeed the concept of real number is intricately linked with that of "limit", and we can see this by understanding what we mean by a decimal expansion by re-writing it as the limit of an infinite summation.