Our Playground: The Real Numbers and Their Development

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Welcome to your first course in calculus! Help with exercises and general questions will be given on the Discussion Board for this page, which can be found on the top of this page. Suggestions are welcome.

In this lesson, we are going to talk a little about the "false history" of calculus and we are going to explicitly define the development of the numbers that we have used to do arithmetic and science everyday, but that we never really looked at too closely before. You may have been mystified by weird proofs that 0.999... repeating indefinitely is actually a valid expression for the number 1, but by the end of this lesson, though you might be amazed at the hidden properties of our everyday numbers, you'll be able to handle them and understand statements like the one above with confidence. To understand why mathematicians developed calculus to the point that it is today, we first need to look at our number system.

The Development of Numbers[edit]

What we will be looking at in this section is a pseudo-history of mathematics; it is very straightforward and every development follows logically from the one before. This is not how things actually occured, but to cover the true history of rises and falls and concepts that never made it would fill volumes and would use up all of our remaining time on Earth (for example, it is rather the case that calculus was developed first, and the analytical properties of the real number system were discovered/created to account for holes in our concept of continuity). We'll stick with the pseudo-history for now.

The Counting Numbers[edit]

It may be cool to think of a culture which has never developed a need to count anything, and thus has no need for numbers. Perhaps this culture consists of a single organism which receives no input from its environment. After all, any input would inspire the ability to differentiate between, for example, "I just ate." and "I just ate again." Even simpler animals have some ability to count. Experiments with birds have shown that they have some ability to account for whole quantities less than 5. The experiment was along the lines of having a certain number of people enter a building containing grain (that attracted the bird) and then having them leave one at a time over a long period of time. The bird would fly into the building to eat the grain whenever it thought there was no-one left in the building. Whenever the number of people was greater than 5, the birds would get confused and fly into the building even though there were still people there.

Similarly, there are certain primitive cultures in existence today that have the words in their language to separate small quantities of objects, but resort to words akin to "lots" for more than a few objects. Interestingly enough, some of these cultures have never separated the concept of quantity from quality, a basic step in the birth of mathematics. Their word for "two boats" would have nothing in common with their word for "two bowls". The concept of two is not separated from the quality (boat or bowl) of the object. We have similar primitive words left over in our language today, ie. a gaggle of geese or a pack of dogs, but we have long since abandoned their ties to quantitative quality.

We still use the base 60 system for describing locations using latitude and longitude.

Anyway, eventually humans came to embrace exact large numbers by using place value systems. (This is not historically accurate either. The Roman numeral system was not a place value system and survived for an incredibly long time, given the fact that calculations done in this system could only be done by highly skilled accountants and philosophers. The common person could not do much of anything with numbers.) We would assign one position to multiples of one, another position to multiples of some number b called the base, and so on. Today, we use ten as the base of our number system, but other bases were used throughout history. An artifact of this is most of our time concepts being based in base 60 (Babylonians). Every time more than 60 seconds passes, we increment the next place value (the minute) and reset the "seconds place" to 0.

The Rational Numbers[edit]

Eratosthenes measured the circumference of the Earth using the power of ratios.

The ability to count a whole number of objects was great for money management (no-one cares about half a penny) or keeping track of your livestock (you don't really care about ratios when butchering; each part has its own whole number, ie. 4 legs go to Mrs. Brown and so forth), but philosophers needed a more flexible concept of quantity. They needed a way to compare two quantities to each other beyond simply saying one was larger and the other was smaller. How much larger one was than the other was very important to natural philosophy (which became the physical sciences in modern times). These ratios and the inevitable need to do arithmetic with them evolved into the system of rational numbers. Note that they are not called rational because they "make sense", even though they may make perfect sense to you. This is to forswear the abuse the poor irrational numbers, which we are about to introduce, have borne for years.

Egyptian surveyors used the 3-4-5 triangle to create right angles.

The study of geometry profited greatly from the application of rational numbers. Now, one could use any fraction of an existing measurement necessary to describe the length or size of a geometric object. For years, it had been known that one could easily construct a right angle by using a closed rope with 12 equally spaced knots and stretching out a triangle from this rope so that the hypotenuse had exactly 5 spacings between the knots and one of the legs had 3 spacings. The Egyptians made use of this simple tool to construct right angles in their architecture and fields.

The First Sign of Trouble: Irrational Numbers in Geometry[edit]

Philosophers during this period (and beforehand) explored many applications of the relationship that is now known as the Pythagorean theorem. It made perfect sense to them that every possible length should be able to be expressed as, at the least, a fraction of some whole amount. This can easily be argued without looking too deep. If you take a pencil and draw a line, then label that line with equally spaced whole numbers (we'll call this line "the number line"), then no matter where you place a point on that line, you should always be able to label it as being x times closer to 1 than to, say 2. It might not be a nice fraction like \frac{1}{4}, but it should still be a fraction. After all, between every two rational numbers there was an infinitude of rational numbers. What other type of length could there be?

Exercise 1.1: Show that if a and b are rational numbers with a < b, then there exists a rational number c such that a < c < b.

At this time, the Pythagoreans, followers of the philosopher Pythagoras, were seeking out number patterns everywhere, due to the success of rational numbers in music. They were very interested in calculating triples, sets of 3 integers that satisfied the Pythagorean theorem a^2 + b^2 = c^2. Unfortunately, there was a very simple triangle that could actually be made, yet did not seem to fit into the Pythagorean theorem. If you measured out a square plot of land, no-one at this time could tell you exactly how far it was to walk across the diagonal of that square. While they could tell you that the distance was between two numbers, it was maddening that they could not assign a workable exact quantity to that length. Occurences like this were actually kept very secret, and even up to the nineteenth century, irrational numbers, numbers that could not be expressed as the ratio of two whole amounts, were viewed with suspicion as in some way not being "real" numbers.

Exercise 1.2: If a number is even, it can be written in the form 2q for some whole number q. Similarly, if a number is odd, it can be written in the form 2p + 1 for some whole number p. Show that if a number is even, then the square of that number is also even. Show the same for odd numbers.

Exercise 1.3: Remember from your knowledge of arithmetic (or at least accept it as a given for now) that two distinct numbers always have distinct prime factors. Said differently, two numbers with identical prime factors must be identical. This is the Fundamental Theorem of Arithmetic. Suppose 1 + 1 = c^2 where c = \frac{m}{n} where m and n are whole numbers. Using the Fundamental Theorem of Arithmetic, show that this cannot be the case (Use algebra and reasoning to deduce a contradiction to the statement that c = \frac{m}{n} for all such whole numbers, assuming the first part of the supposition is true).

Tips for first proof: Remember that you're trying to convince a peer through logic that the statement above is false. This means you may have to write a few arguments, in addition to showing the mathematics. There are many ways of proving a statement. Can you find another method to prove the above statement?

Somehow, these irrational numbers have filled in gaps that we could not see in our number line. As a matter of fact, it may be disturbing to realize that there are, in a sense that will be made clearer if you continue on to study real analysis, "more" irrational numbers than rational numbers on our number line.

Irrational numbers disturbed early mathematicians, and they developed creative methods of getting as close as they needed to the true value of the irrational quantity using rational numbers. Methods such as these are present in the work of Eudoxus and Archimedes, and are actually not that far away from modern calculus.

Continuity Saves the Day[edit]

We are now going to fast-forward quite a bit to the modern day. When you punch 2 into your calculator and press the square root function, you will most likely get quite a few digits, but a finite amount of them. You can always just put that amount over the proper power of ten to write it as a rational number, so what your calculator is giving you cannot be the square root of two. It is instead approximating the square root of two up to a certain amount of decimal places using methods that will become apparent in these very lessons. How does your calculator (or rather your calculator's programmers) know that the value it calculates is accurate? It's obvious that it is, since we depend on computers to calculate physical interactions dealing with things a lot more involved than simple square roots! This is the theoretical framework that requires numbers of completely arbitrary precision. Using this theory, we should be able to, at any moment, get a decimal, binary, or what-have-you equivalent of the abstract numbers that we will be dealing with. These abstract numbers that completely fill in all possible lengths on the number line, and including their negatives, are what we call the real numbers.

Wait, you may say. How do we know there aren't even more exotic and weird numbers hiding out on our number line that we just don't have the experience to see yet? This is where the notion of what mathematicians call continuity makes its first appearance. Continuity is the heart of calculus, and is actually what you already expect, as it helps define the real numbers to behave as if we had drawn the number line with an imaginary pencil. Informally, we say that a line is continuous if you can draw the line without lifting your pencil from the paper. Suppose you freeze the action of drawing the number line at a particular point in time. The pencil is currently in the act of drawing some point after 3 but before 4. In order to insure that the line is continuous, we intend to make sure that the pencil is always drawing some point. This means that no matter how close we get to the point of the pencil, there will always be a line leading up to the point the pencil has stopped at.

Allow the pencil to draw the line and place your finger on the line. Your finger definitely starts touching the line at some point: we define that point to be a real number. In technical terms, the real number system ensures that if you can name a finite number greater than the length you're wondering about, then that length exists as a real number. In this way, we include every possible length on what we have idealized as a continuous line.

We are now going to state what was said above in mathematical terms, only because we need a precise definition independent of arbitrary things like the thickness of the pencil, how shaky our hand is, the nature of atoms, and so forth in order to prove things logically within the framework of calculus. Calculus is not about any of those things above; it is about applying algebra to analyze geometry, so the language of these two subjects is what the definition will be given in. We will need to introduce some language first. This type of language will be very useful to give concrete logical notions that we can use to prove the statements we will be making. It is language used in many areas of mathematics, so it is not exclusive to calculus. You may already know some of these terms.

Set: A set is a collection of objects. We can explicitly define sets by writing what they contain in curly braces, such as {1, 2, 3}. '1' is called an element of the set.

Upper Bound and Lower Bound: A set of numbers has an upper bound if there exists some number q that is greater than (or equal to) each element in the set. Similarly, a lower bound p is a number that is less than (or equal to) each element of the set.

We, logically, say that a set of real numbers is bounded if it has both an upper and lower bound. This is easily illustrated by placing an upper and lower bound somewhere on the number line and noting that if a set can claim to have those upper and lower bounds, then it must be inside the bounded region.

Back to the notion of finding the number c such that 1 + 1 = c^2. We know that c must be less than 2 and greater than 1. Show this. Thus, c must be a real number (in respect, it must be on the imaginary number line we drew above)! We call this real number \sqrt{2}, and denote it by dividing the real number line into two sets: those numbers less than \sqrt{2}, and those numbers greater than \sqrt{2}. If we take a look at the former set, we see that the set of numbers greater than \sqrt{2} is a set of upper bounds for our set. After a moment's thought, we can see that what we mean by real numbers can be defined by placing a simple requirement on all possible sets of real numbers.

The Construction of the Real Numbers: If a set of numbers has an upper bound, then it has a lowest possible upper bound that is a real number. Similarly, if a set of numbers has a lower bound, then it has a greatest possible lower bound that is a real number.

These least upper bounds (called supremums) and greatest lower bounds (called infimums) are what Dedekind used to define the system of real numbers. Thus, every possible set of numbers that you can name, given that it has at least one bound, represents at least one real number. This formulation is exactly what Eudoxus and Archimedes were counting on as existing when they used their methods of exhaustion to find things like the area of a circle or the length of a section of a parabola. Their methods produced lists of numbers that they were able to prove had finite upper and lower bounds.

A Short Tour of the Real Numbers[edit]

As an introduction to the strangeness of real numbers, note that the irrational numbers have some oddities and monsters hiding within them, just out of reach of various tools. It is interesting to find out that each of the following sets of weirder and weirder numbers is actually "larger" than the sets preceding it, in a sense that will be made clear in a course on real analysis.

Unconstructible Numbers[edit]

The number \sqrt{2} as explored before can easily be constructed using the old geometer's tools of a straight edge (an unmarked ruler) and a compass (a device for drawing circles centered at a point of your choice). It was proven using group theory that there exist unconstructible numbers, lengths which cannot be constructed on paper using a finite amount of steps. These include numbers as simple as the fifth root of 5.

Transcendental Numbers[edit]

While the fifth root of five can easily be described as being the root of the equation x^5 - 5 = 0, there also exist numbers that are not the root of any polynomial equation with rational coefficients. These are called transcendental numbers; they have transcended the need for any reference to rational numbers. These include, but are not limited to, the popular numbers e and \pi. They are the jewels of real numbers.

Uncomputable Numbers[edit]

Here is where the monsters start coming out of the dark forest of irrationals. You can use a computer to calculate what you believe to be many reals to any precision you need. Unfortunately, most reals are not computable to arbitrary precision by any finite algorithm! Interestingly enough, this does not affect our ability to do calculus, but it is a part of the study of complexity theory. e and \pi are, lucky for us, computable numbers. One example of an uncomputable number is Chaitin's constant.

Undefinable Numbers[edit]

Horrors! Using the notion of countability you will learn in real analysis, it is easy to prove that almost all real numbers are not even definable. We can at least talk about a few uncomputable numbers, as was seen by Chaitin's constant above. These strange reals lurking in the shadows can never actually be described or even referred to directly! Luckily, once again, we don't have to worry about these shadowy characters, as they don't affect the behavior of calculus.

Hmm...A coffee break. I need to regroup, add some more motivation and reference links to some of the things I mentioned, as well as give a good closure to open the floor to both calculus and analysis. This page will be cross-posted to real analysis, as it is also a good impetus to analysis.