Section 1: Number Systems
© 1996 by Karl Hahn
Calculus deals with properties of the real numbers. In order to understand calculus you must first understand what it is about the real numbers that separates them from other kinds of numbers we use from day to day.
These are the first numbers we learn. When we count, we start from one and list off the names of numbers in sequence. And this simple description clues us in on what the crucial properties of the counting numbers are: There is a first counting number, and for each counting number, there is a next counting number, or in other words, a successor. No counting number is its own successor. No counting number has more than one successor. No counting number is the successor of more than one other counting number. Only the number 1 is not the successor of any counting number.
And there is one more important property. That is, the counting numbers are like a ladder. If you know how to step onto the first rung of the ladder, and from any rung you know how to step to the next rung, then you can get to every rung. You probably remember learning about proof by induction back when you took algebra in high-school. This is the principle on which it is based. The way textbooks usually state it is that if you have a collection of counting numbers, and 1 is in that collection, and the successor of each counting number in the collection is also in the collection, then the collection contains all the counting numbers there are. But if you remember only the ladder analogy, you will still have the basic idea.
We learn early in life that we can always add two counting numbers to get another counting number. Then we learn about subtraction, and that the same is not always true for it. For example, I can't subtract 20 from 15 and get a counting number. Later we learn that we can always multiply two counting numbers and get another counting number. Then we learn about division, and that the same is not always true for it. For example I can't divide 3 into 10 and get another counting number.
So in these ways, regarding subtraction and division, the counting
numbers seem somehow deficient.
To remedy the situation with subtraction, we invented negative numbers
and zero and tacked them onto the counting numbers starting just
before 1. We called the result the integers. Addition, subtraction,
multiplication, and division extend easily to be applicable to the
expanded number system. Every number still has a successor, but no
longer is there a first number. But you can certainly subtract any
integer from any other integer and still have an integer as a
result. And although it's hard to have -5 apples, anybody who has
ever been on a shopping spree with a credit card knows very well how
it is possible to have -1000 dollars.
To remedy the situation with division, we invented the rational numbers, which we learned are fractions that reside in vast multitudes between the integers. Again addition, subtraction, multiplication, and division extend easily to the further expanded system. We have to do without the idea of every number having a successor. But in return, we can divide any rational by any other rational (except zero) and get a rational as a result.
There is something else we get in return as well. From the beginning we have had a concept of one number being greater than another. Like the four operations, this concept started with the counting numbers and extended easily into the integers and the rationals. And with the concept of one number being greater than another comes the concept of between-ness. In the counting numbers and the integers, two numbers, one of which is the successor to the other, have no number in between them. But in the rationals there is an in-betweenster between any two distinct numbers. In fact there is a whole raft of them.
Closely related to this property is a property that mathematicians
call density. It is that you can choose two distinct rationals to be
as close as you would like them to be. With the integers, two
numbers can't get any closer than one unit apart before they become
the same. But at anywhere among the rationals, I can name how close
I would like a pair of rationals to be and you will be able to find
two such rationals.
The density property described above leads to a concept that is
fundamental to calculus, and that is the concept of a limit. If I
gave you 2 gallons of milk today, one and a half gallons tomorrow, one
and a third gallons the next day, one and a fourth gallons the next
day, and so on, and continued in that manner for the rest of
eternity, what can I say about how much milk you might get on a
typical day? Well, there is certainly a formula for it. If I label
the days, starting with today, as 1, 2, 3, 4, and so on, I can say
that on day number n, I will give you
If you tried to argue that 1.01 gallons is a lower bound, I could disprove it by noting that on the 101st day you would be getting less than that. But if you argued that any amount of one gallon or less is a lower bound, I would be unable to find any day in the future on which I would be giving you less than that amount of milk. So all amounts of one gallon or less are lower bounds. But of all of those, the amount of one gallon exactly is special. It is the greatest of all the lower bounds. And even though there is no day that I will give you exactly one gallon of milk, there will come a day on which the amount I give you will be as close to one gallon as I would like. Not only that, on all subsequent days the amount will be that close or closer.
In other words, you tell me how close to one gallon I will be giving you, and I can name the day on which the amount I deliver daily to you will be, forever after, at least that close. That is what makes one gallon the limit. And that concept of something coming as close as you'd like it and staying there -- the concept of a limit -- lies at the heart of calculus. Which is why you will be going over it so much in the first few weeks of any beginning calculus course, and why you should take the time and effort to understand it well.
With the rationals we have a nice little number system. You can combine any two rationals by any of four operations, and with the exception of dividing by zero, you will get another rational. What more could you want from a number system? Well nothing, actually, until you encounter the idea of a limit.
I know you've probably studied irrationals long before you were aware of limits. I know that in high-school you learned that the square root of two was irrational (perhaps the teacher even proved it for you, if you were lucky) and that you had to write it as an infinite, non-repeating decimal. But did any of you bother to ask what was meant by an infinite non-repeating decimal? If the decimal is infinite, you can't write it, now can you? So what on earth does it mean?
Well let's start out by asking what a finite decimal means. If I carry
out a decimal to one place beyond the point, then I am specifying some
number of tenths:
4 1.4 = 1 +
eq. 1.5-1 10
If I carry out a decimal to two places beyond the decimal point, then I am specifying some number of hundredths:
41 1.41 = 1 +
eq. 1.5-2 100
Similarly with three places beyond the decimal point:
414 1.414 = 1 +
eq. 1.5-3 1000
and so on. In general if a decimal is carried out to n places beyond the decimal point, we do a division using the digits to the right of the decimal place as the numerator and 10n as the denominator. So the meaning of finitely long decimals is quite clear and unambiguous. They all represent rationals.
But note that not all rationals can be expressed as finite decimals. The number 1/3 is quite clearly a rational, but can never be expressed as any counting number divided by a power of 10 (this is because no power of 10 can ever be divisible by 3). In high-school you were taught that you had to write this as an infinite repeating decimal. Again that nasty problem of human beings, having a finite life span, not being able to write infinite decimals, whether they repeat or not. Of course, with the repeating variety, you can write one period of a repeat sequence, then wave your arms and say, "and this continues in exactly this way forever." But even granting that that is a satisfactory way of writing the infinite pattern, you are still left with the problem of what does it mean. There simply are not n convenient decimal places, so you can't divide some counting number by 10n. The explanation we used of what a finite decimal means fails here miserably.
But enter now, the concept of a limit. As you probably recall, the decimal expansion of 1/3 is a decimal point followed by an endless string of 3's. But I can truncate the string at any point and end up with a finite decimal to which I can apply our unambiguous meaning. In fact, I can make a sequence beginning with truncating after 1 digit, then after 2, then after 3, and so on. And applying our rule for going from finite decimals to quotients, I get:
3 33 333 3333
, , , , ... eq 1.5-4 10 100 1000 10000
This sequence has a limit according to the notion we discussed earlier with the milk. And its limit is exactly 1/3. You tell me how close a fraction you need to 1/3, and I can tell you how many terms you need to go into the sequence given in 1.5-4 to find nothing but terms that are at least that close. And this applies no matter how close you tell me I have to be. There is no rational number besides 1/3 that the above sequence comes arbitrarily close to and stays there. We say that the sequence converges to 1/3.
And by the way, it is not the only sequence of rationals that converges to 1/3 (although it is the only one that can be written as an infinite decimal). Here are two others that also converge to 1/3:
4 34 334 3334
, , , , ... eq 1.5-5a 10 100 1000 10000 1 5 21 85 341 , , , , , ... eq 1.5-5b 4 16 64 256 1024
Note that in 1.5-5b the next denominator is obtained by multiplying the previous one by 4. The next numerator is obtained by multiplying the previous one by 4 and then adding 1.
It is easy to show that for every rational that cannot be written as a finite decimal, you can find an infinite repeating decimal that when interpreted as we now know to do, converges to that rational. And you can also show that no other infinite decimal, repeating or not, will converge to that same rational.
So what about the infinite decimals that don't repeat? What do they represent? Well we have just shown that they certainly don't converge to any rational. In addition, since the infinite non-repeating decimals are just as susceptible to the converging sequence interpretation as the infinite repeating decimals are, what can we say that their sequences converge to?
First, it is only fair to point out that we shouldn't have an expectation that all sequences converge to anything at all. For example:
1 2 3 4 5
, , , , , ... eq. 1.5-6 10 10 10 10 10
where each term is 1/10 bigger than the term before it. We have no expectation that this sequence converges because it clearly has no upper bound. But the sequence 1.5-4 is different, as is any sequence that derives from an infinite decimal. They possess the special property in that you can tell me as tiny a range as you like, and I can tell you how far into the sequence you have to go so that all the subsequent terms are within that range of one another.
To illustrate this, let's say I am learning to pitch a baseball. I practice by throwing one pitch each minute at a barn door. Each pitch leaves a mark on the door. But as I get better at throwing, the marks the pitches leave begin to group together in a bunch. As I get better and better, the marks I leave group tighter and tighter. In fact, you can draw a circle around the area where my pitches appear to be grouping, and eventually every ball I throw hits inside the circle. If you draw a smaller circle, eventually I get good enough to place every ball inside that circle as well. And no matter how small you draw the circle, with enough practice, I can hit the inside of that circle every time. If every smaller circle is inside all the bigger ones, then the pattern of my pitching is analogous to the special property I am describing.
A sequence that has this property is called a Cauchy sequence. Any sequence that derives from any infinite decimal, repeating or not, results in a Cauchy sequence (Optional: Click here for brief biography of Cauchy) For proof of this, click here.
But the rationals seem somehow incomplete. Why? Because there are all those Cauchy sequences deriving from non-repeating infinite decimals that don't converge to any rational number. To remedy that, we invent the real numbers.
So what is a real number? Mathematicians have come up with several equivalent definitions, but since we have already discussed Cauchy sequences, I will use the definition that bases itself on them. A real number is the limit of a Cauchy sequence of rationals. It's that simple. It means that any time we have something that looks like it should converge to a limit, we say it does. And we always say it should if it meets the requirements that Cauchy gives us. Even if the sequence does not converge to any rational number, we still say it converges to a real number.
Since every infinite decimal represents a Cauchy sequence, whether it repeats or not, every infinite decimal represents a real number. Even the finite decimals represent real numbers, since you can append an infinite string of zeros onto them to make them infinite decimals.
To further clarify the difference between rationals and reals, let's suppose I drive down the road. Let's say my odometer is calibrated out to any number of digits beyond the decimal point. So I drive some number (less than 10) of tenths of a mile. Then I drive some number (again less than 10) of hundredths of a mile. Then I do the same in thousandths, ten thousandths, and so on. At each time, I have gone some number of miles that can be represented with some rational number. But, if the road has only points on it that are rational numbers of miles, then after I do this infinitely many times, I could end up nowhere. No rational number of miles could possibly indicate where I am on the road. There are rational numbers that are close to where I end up -- as close as I'd like. But none is completely accurate. Yet where I end up is a very real place. And that is why we adopt the real numbers in preference to the rationals for doing calculus.
The remainder of the discussion of real numbers in this section is optional. You may skip ahead to Section 2 if you like. But you may find this discussion interesting. You might also like to have a look at a deeper and more rigorous treatment of what the real numbers are all about. If so, click here to see an essay on the subject by Dr. Eric Schechter at Vanderbilt University.
Earlier, we showed two Cauchy sequences that converged to the same value. What gives? Well, sometimes two different Cauchy sequences can indicate the same real number. We have a test for that. Simply take the difference, term by term, of the two sequences. If the resulting sequence heads for a limit of zero, then the two Cauchy sequences represent the same real number. For example, if we take the sequences given by 1.5-4 and 1.5-5a:
4 3 34 33 334 333 3334 3333
- , - , - , - , ... eq 1.5-7 10 10 100 100 1000 1000 10000 10000
Clearly, the nth term of this difference sequence is:
which you can bring as close to zero as you'd like by making n big enough. So that means that the two Cauchy sequences represent the same real number.
This, of course, leads to the question of how do you carry out the four arithmetic operations on real numbers? Well, you know how to add, subtract, multiply, and divide rationals. Do the same procedure, term by term, to the Cauchy sequences, and you are doing the operations on real numbers. The only problem is division, where the divisor sequence might contain zeros. If the divisor sequence ends in all zeros, then the divisor is zero, and you can't divide by it (the same is true if the divisor converges to a limit of zero). But if it has zeros but converges to something other than zero, simply drop the zero terms from the divisor sequence and do a term by term division on what remains.
This still leaves the nasty philosophical problem of humans not being able to carry out infinite operations. And, in fact, humans cannot do arithmetic on real numbers. Neither can computers. But we can talk about the reals and their properties based upon the definitions given above. And based upon what we find out about real numbers, we can do computations on them, not entirely accurately, but with as much accuracy as you have patience to get.
I hear you saying, "These real numbers are a real pain in the butt for something you can't even do arithmetic with. What's the point?" The point is that the reals have a property on which calculus depends. Calculus is about limits. Remember that some Cauchy sequences of rational numbers did not converge to any rational number. That means that, among rationals, limits do not exist everywhere you would like them to.
Clearly we can make Cauchy sequences of real numbers as well. That would be a sequence of sequences. Every Cauchy sequence of real numbers converges to a limit that is a real number. You can demonstrate the truth of this for yourself. Out of the sequence of sequences, take the first term of the first sequence, the second term of the second sequence, the third of the third, and so on. You end up with another Cauchy sequence of rational numbers (so it too is a real number), and it is precisely what the sequence of sequences converges to. [This is not the entire story on sequences of sequences. You can concoct one that requires a little more effort than the first of the first, the second of the second, and so on, in order to show convergence. Yet by going deep enough into each sequence, you can prove convergence of any Cauchy sequence of real numbers. If you want to see that proof, click here. But remember it is optional material and you won't need to know it for class. Furthermore it is not at all an easy proof for a beginner. So if you are curious about it, you might want to put it aside for now and come back to it when your skills have grown. And if you're not curious about it, don't feel the least bit guilty about skipping it.]
Since infinite decimals always represent Cauchy sequences, we can use them to fabricate an example of the above. We learned in algebra that both the square root of two and the value of pi are irrational, and must be represented using infinite decimals. Suppose I construct an infinite decimal that contains the first digit of the square root of two followed by the digits of π. That represents my first Cauchy sequence. I construct the next one by taking the first two digits of the square root of two, followed by the digits of π. The third is the first three digits of the square root of two followed by the digits of π, and so on:
s1 = 1.31415926... = 1, 1.3, 1.31, 1.314, 1.3141, 1.31415, 1.314159, ... s2 = 1.43141592... = 1, 1.4, 1.43, 1.431, 1.4314, 1.43141, 1.431415, ... s3 = 1.41314159... = 1, 1.4, 1.41, 1.413, 1.4131, 1.41314, 1.413141, ... s4 = 1.41431415... = 1, 1.4, 1.41, 1.414, 1.4143, 1.41431, 1.414314, ... s5 = 1.41423141... = 1, 1.4, 1.41, 1.414, 1.4142, 1.41423, 1.414231, ... . . . . . . . . .
As we know, each of the sj's above represents a Cauchy sequence of rational numbers. So we have a sequence of sequences, that is, a sequence of real numbers. Not one of the individual sj's converges to the square root of two. But the sequence of sequences does. Why? Because each sequence converges to something closer to the square root of two than the last, and because you can find one that converges to something as close to the square root of two as you'd like by going down the rows deeply enough.
Using the prescription of taking the first term from the first sequence, the second term from the second sequence, the third from the third, and so on, we get a "diagonal" sequence, sdiag, of:
sdiag = 1, 1.4, 1.41, 1.414, 1.4142, 1.414213, ...
which is precisely the Cauchy sequence of rationals that is well known to converge to the square root of two. Of course an example is not a proof, but it does illustrate the idea.
The point is, a sequence of real numbers that meets Cauchy's conditions always has a limit and that limit is always a real number. That is a foundation on which we can do some calculus ...
You get a break. No exercises given for this section. But rest up and be prepared. I'll be working you hard on the next section to make up for it.
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