Alex's Adventures in Numberland - Alex Bellos [63]
Ferguson was the last of the manual digit-hunters and the first of the mechanical. Using a desk calculator he added almost another 200 places in just over a year, so in September 1947 pi was known to 808 decimal places. Computers then changed the race. The first of them to battle pi was the Electronic Numerical Integrator and Computer, or ENIAC, which was built in the final years of the Second World War at the US Army’s Ballistic Research Laboratory in Maryland. It was the size of a small house. In September 1949 the ENIAC took 70 hours to calculate pi to 2037 digits – smashing the record by more than a thousand decimal places.
As more and more digits were found in pi, one thing seemed pretty clear: the numbers obeyed no obvious pattern. Yet it was only in 1767 that mathematicians were able to prove that the higgledy-piggledy sequence of digits would never repeat itself. The discovery followed from considering what type of number pi might be.
The most familiar type of number is the natural numbers. They are the counting numbers starting at one:
1, 2, 3, 4, 5, 6…
The natural numbers, however, are limited in scope because they expand in only one direction. More ul are the integers, which are the natural numbers together with zero and the negatives of the natural numbers:
… –4, –3, –2, –1, 0, 1, 2, 3, 4…
The integers cover every positive or negative whole number from minus infinity to plus infinity. If there were a hotel with an unlimited amount of floors and an unlimited amount of lower and lower basements, the buttons in the elevator would be the integers.
Another basic type of number is the fraction, which are the numbers written when a and b are integers but b is not 0. The top number in a fraction is the numerator and the bottom number is the denominator. If we have several fractions, the lowest common denominator is the lowest number that can be divided by all the denominators without leaving a remainder. So, if we have and , the lowest common denominator is 10, since both 2 and 10 divide into 10. What about the lowest common denominator of ,, and ? In other words, what is the smallest number that 3, 4, 9 and 13 divide into? The answer is surprisingly big: 468! I mention this to make a semantic rather than a mathematical point. The phrase ‘lowest common denominator’ is often used to describe something basic or unsophisticated. It sounds evocative, but misrepresents the arithmetic. Lowest common denominators can often be big and unconventional: 468 is quite an impressive number! A more arithmetically meaningful phrase for something mainstream and cheap is highest common factor – which is the largest number that can be divided into every one of a group of numbers. The highest common factor of 3, 4, 9 and 13, for example, is 1, and you can’t get much lower or more unsophisticated than that.
Since fractions are equivalent to ratios between integers, they are also called rational numbers, and there is an infinite amount of them. In fact, there is an infinite number of rational numbers between 0 and 1. For example, let’s take every fraction where the numerator is 1 and the denominator is a natural number bigger than or equal to 2. This is the set composed of:
We can go further and prove that there is an infinite number of rational numbers between any two rational numbers. Let c and d be any two rational numbers, with c less than d. The point halfway between c and d is a rational number: it is . Call this point e. We can now find a point halfway between c and e. It is . It is rational and also between c and d.