An excursion into integers, rationals and `real' numbers (Part 1)
The mathematics just described deals almost exclusively with whole numbers or integers. The area of study devoted to discovering their properties is renowned for being very difficult. Despite this, it has captivated the minds of some of the greatest mathematicians over the ages. There are numbers beyond the integers, the simplest and most familiar of which are the fractions. In case you have forgotten, a fraction has a numerator and a denominator. The numerator of the fraction 5/13 is 5 while its denominator is 13. The fraction 5/13 can be expressed in ratio form as 5:13.
So, speaking a little loosely, you could say that fractions are expressible as the ratio of two whole numbers. For this reason, they are often called the rational numbers or rationals for short. Thus 1/2 and minus 3/4 are rational numbers though they are not integers. Integers, on the other hand, are rational numbers because they can be thought of as fractions whose denominators are 1.
For a long time it was believed that there were no types of number other than rationals. The ancient Pythagorean proclamation that "all is number" was an expression of this belief. But there are other species of number.
Beyond the rationals are the real numbers, including such wonders as the square root of two. the golden-ratio phi, the base of the natural logarithms e and, of course, the ubiquitous. These numbers are considered "real" because they convince us of their existence by being geometrical ratios or other tangible quantities. There are also numbers which are "not real", known as complex numbers and built around the mysterious square root of minus one. Unfortunately, these exotic numbers play no part in the tale to be told so I must refrain from digressing for a couple of hundred pages to tell you all about them - not that I could. However, I cannot resist saying just a little about some of them.
Dad gave one three-hour lecture (less 20 minutes for a tea break) on the four numbers, square root of two, phi, e and. I knew that is an interesting number, I had never heard of e and I thought I had heard about phi but I was mistaken. I fancied that there was nothing much to be said about the square root of two, other than saying that it's that number which when squared gives two. Of course, I should have known better!
I was to learn that there is much, of an elementary nature, that can be said about the square root of two. It was with its very definition that we started our excursion, teasing out what is really meant by it. We learned how upset the Greeks were when they first realised the "different" nature of this number - upset to the point, apparently, of drowning some poor devil, just because he blabbed that all was not well with the accepted dogma that every positive number could be expressed as a ratio of two natural numbers.
The number square root of two makes its presence felt by being the length of the diagonal of a square whose side measures one unit in length. Ironically, it is the famous theorem of Pythagoras that tells us this. So what is new or different about square root of two? Well, try as they might the Greeks couldn't find a rational number which when squared gave 2 exactly. No approximations, like 17/12 (which incidentally doesn't do too bad a job), would do. No sir! It was to be all or nothing:
"Most wanted: a fraction whose square is 2.00000000000000000... with zeros all the way to infinity."
They never found one. Being clever fellows, it gradually dawned on them that perhaps there is no such fraction, but how could they possibly prove such a thing? Well, I won't go on about it, but as you probably have guessed they did finally (about 300 years after first suspecting that there was more to the square root of two than meets the eye) find an argument which showed beyond doubt that there is no fraction which when squared gives 2. So the square root of two was a new specimen.
The proof that the square root of two is irrational (not expressible as a ratio of two whole numbers) so enthralled the English mathematician G.H. Hardy (1877-1947) as a boy that from the moment he read it he decided to devote his life to mathematics. He did and he became one of England's greatest mathematicians. He wrote a book called A Mathematician's Apology. It was Hardy who said "Beauty is the first test. There is no permanent place in the world for ugly mathematics."
The proof that the square root of two is irrational is another of those proofs by contradiction. It's a classic.
Dad showed us how the ancient Babylonians obtained a formula for finding rational approximations to the square root of two which is equivalent to one that was obtained later (after 1700) using a procedure known as Newton's method. He also showed how to derive the next term in the following sequence: 1/1, 3/2, 7/5, 17/12, 41/ 29, 99/70 ...
The successive terms of this sequence give better and better fractional approximations to the square root of two. Try a few on a calculator to see. Just divide the numerator by the denominator and square the result. Have another look at the fractions above to see how you might get the next one from the previous one before reading on.
To get the denominator of the next term in the sequence you simply add the denominator and numerator of the previous fraction. To get the numerator you simply add twice the denominator of the previous fraction to its numerator.
Let us see how 7/5 generates the next fraction according to this procedure. The new denominator is 5+7=12, while the new numerator is 2 x 5+7=17. So the next fraction should be 17/12. It is.