An important property of the integers is that you can factor them into primes in a unique way. This is called the "fundamental theorem of arithmetic." For example, from the last class, 28=2x2x7. The prime factors of 28 are 2 (twice) and 7. This is unique: 28 has no other factorization. The problem about friendly numbers required you to find all distinct, proper factors of 220, and to do this you should probably start by factoring 220 into its prime factors: 220=2x2x5x11. From here you can find the distinct proper factors 1, 2, 5, 11, 4, 10, 22, 55, 110, 44, 20, by taking products of all proper subsets of (2,2,5,11) (and remembering the factor 1), etc.
The point is, integers can be factored into primes, in a unique way. That is often the starting point for problems involving the integers. The Fundamental Theorem of Arithmetic may seem obvious to you, based on your experience with integers and how they multiply, but it is actually not easy to prove. We will not prove it now, but might return to it later. It was proved by the time of Euclid (300 BC), and for now we will just use it, as something that is known about the integers.
You probably know a representation of ALL the usual numbers, as most people conceive of them, in the form of a line, the number line:
This organizes all the usual numbers, including the negative numbers, in a nice way. The integers are all there, and the fractions are also represented, in between the integers. Every number x therefore represents a position on the number line, and conversely, every position on the number line represents some number. This is such a familiar idea, and so simple, seemingly, that it is hard to realize what an achievement it is. This idea, fully worked out, solves a problem that bedeviled mathematics for literally thousands of years! It was only in the 19th century that a satisfactory understanding of the number line was achieved: that is VERY close to our own time. Understanding this ancient problem, and why it was a problem at all, is today's subject. The number line is now built into mathematical notation that everyone learns in grade school, so that it seems obvious. It requires a conscious act of imagination to see the problem for what it once was.
The problem starts with a practical observation: one needs more numbers than just the integers. This was undoubtedly true even in Neolithic economies. To keep track of animals in flocks you can use integers, because animals come in whole units, but many things come in a form that is readily divisible into parts, allotments of grain, beer, and land, for example, just to name commodities which show up in the most ancient records of Mesopotamia and Egypt. When you divide something in half, or into thirds, etc., you have to express this somehow. There is no obvious notation for this! It has to be invented, and it is possible to make very bad choices for how to do it.
One simple solution to the problem of handling fractions is just to invent new, smaller units. This is a very familiar idea. We probably would not say, for example, "1-1/4 feet," which seems a bit clumsy; rather we would say "1 foot, 3 inches," or perhaps just "15 inches." By introducing a new smaller unit, the inch, we avoid having to express fractions of a foot. The number of inches in a foot, i.e., the size of the new unit, is chosen to make various fractions of a foot easy to express in whole number of inches. You can express 1/2, 1/3, 1/4, 1/6, 1/12 as 6, 4, 3, 2, and 1 inch respectively. This introduction of new, smaller units, must have happened again and again, in every kind of measure, the choices differing from culture to culture and commodity to commodity. And it IS a kind of solution to the problem of notating fractions. The fractions just turn into integers in the new unit of measure!
The Greeks before Alexander (d. 323 BC) did not have a good notation for fractions. They used an Egyptian system, notating it with Greek letters. It was quite ineffective for computation, but the Greeks neatly sidestepped this difficulty by using lengths to represent numbers. As we have noticed above, this idea of associating lengths with numbers, via the number line, is something we do too. Here is an example, in a picture, of the way they thought about numbers that weren't simply integers:
To multiply two numbers a and b, represent them as lengths and form the rectangle with these lengths as sides. The product is the area. On the right I have reminded you what it would mean for integers, but the multiplication on the left is more general, because it makes sense even if a and b are not integers, and even if we have no notation to express these numbers at all. (To get a length representing ab you would have to do one more step, namely construct a second rectangle, having the same area but with height 1, a standard construction in Euclidean geometry, which still does not require any notation.)
To show that you can actually do something
with this idea, here is a proof of the distributive law of algebra:
The sides of the rectangle on the left are a+b and c+d, and the usual distributive law of multiplication is clear to see. On the right I remind you in an example how it works in case the numbers are integers, but the statement on the left is more general, because a, b, c, and d are not necessarily integers.
Whether the numbers in the above geometric multiplication are integers or not depends on what the unit of length is. We could always invent a unit equal to length a, for example. Then a would represent the integer 1. But if we took a different unit of length, a would represent some other number, perhaps a fraction. What the Greeks did with these arguments was show how arithmetic works, even for numbers they were incapable of writing down! This approach is sometimes called "geometrical algebra," and it is the subject of Book II of Euclid's Elements (~300 BC). One often hears that this material is "Pythagorean," generalizing, as it does, the arithmetic of the integers, by means of geometry.
The most spectacular example of this idea in action is without question Pythagoras' theorem, which is the climax of Book I of Euclid's Elements (~300 BC). This amazing theorem says that if a, b, and c are the sides of a right triangle ( c being the hypotenuse) then
Conversely, if a, b, and c satisfy this relation, then they are the sides of a right triangle. Of course, those squares in the relation meant actual squares! The left side is the sum of two smaller square areas, which together equal a larger square area:
Since the proof is by geometry, the sides
a, b, and c can be any numbers, subject only to
the restriction that they form a right triangle, or, equivalently,
. They certainly need not be integers! It
is an interesting fact that there are integers which work,
(3,4,5) for example, since 9+16=25, and hence a triangle with
sides (3,4,5) is a right triangle. But Pythagoras' theorem
is not restricted to such cases.
There is a nice animated proof of Pythagoras' theorem on the web which only requires you to create a right triangle with the mouse, then push "NEXT" to see the steps of the proof, following Euclid Book I, in effect. Another site invites you to create your own proof, (this one is aimed at Japanese middle school children, and it is charming). You can also find information about Pythagoras. Almost nothing is known about Pythagoras with certainty. Most sources estimate he died around 500 BC -- but as you will see if you look, even so elementary a thing as that is disputed.
A simple example of a right triangle
is the isosceles right triangle with sides 1, 1, 
.
The hypotenuse has length between 1 and 2, so it should be expressed
as some fraction, like 3/2, or 7/5, only it isn't exactly either
of those. We have called it here, since its
square is 2. It is a perfectly good length. Since it is not an
integer, we think of introducing a new smaller unit by dividing
up the original unit 1 into q equal pieces, thus getting
a new unit of size 1/q small enough to "measure"
the hypotenuse, i.e., small enough that the hypotenuse is an integer
number of these small units. Suppose it takes p of these
small units, each of size 1/q, to measure .
Then p/q=. But, as proved by the Pythagoreans,
probably by Pythagoras himself, this is impossible! There is NO
unit which can measure both 1 and . The
argument is as follows: Suppose there were integers p and
q such that
.Then by squaring we find that
Both sides are integers, so by the Fundamental Theorem of Algebra both sides have a unique factorization into primes. How many times does the prime number 2 occur in the factorization of the left side? It must be an even number of times*. How many times does the prime number 2 occur in the factorization of the right side? It must be an odd number of times*. But that would be two different factorizations of the same integer into primes, and there is only one, unique factorization. Thus our supposition that p and q could be integers has led to a contradiction, and cannot be true. The method of introducing new small units to solve the problem of notating fractions does NOT solve the problem. There are numbers between the integers which are not fractions in this sense. In this example, the square root of 2 ought to be some kind of fraction, but it is not of the form p/q. The Greeks called these numbers "irrational." The existence of irrational numbers led them to distrust arithmetic, since apparently arithmetic could not represent all the "numbers" (lengths) of geometry. If you took the numbers of arithmetic, meaning all fractions of the form p/q, you would not get the number line. There seemed to be "holes" or "breaks" in the line, corresponding to the irrational lengths. This problem of the irrationals was still a central problem in the Renaissance, over 2000 years after Pythagoras.
*Whatever the prime factors of an integer p may be, they all occur an even number of times in the factorization of p squared. For example, 39=3 x 13. Thus 39 x 39 = 3 x 3 x 13 x 13. Obviously every factor of 39 is repeated, i.e., occurs an even number of times.
In Plato's dialogue Theaetetus we hear that the Pythagorean mathematician Theodorus is investigating whether other square roots, of the numbers 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, and 17, are irrational, i.e., not expressible as ratios of integers in the form p/q. This passage is taken as evidence that by Plato's time the irrationality of the square root of 2, was known (otherwise Theodorus would have started the list with 2, not 3). Choose one of those numbers and prove, by the method used above for the square root of 2, that your chosen square root IS, in fact, irrational. Then take one of the numbers 4, 9, 16, which of course have integer square roots, and explain clearly why the proof does NOT apply, i.e., why we cannot use the same method to "prove" that the square root of 9 is irrational, for example, (which would be a clear sign that we were doing something wrong!)