Simon Singh =-= "A further five numbers"
Five more mathematical biographies, looking at the histories, uses and idiosyncracies of the most special numbers.

Programme 1:   1 – the most popular number!
Literally, the most popular number, as it appears more often than any other number. More specifically, the first digit of all numbers  is a 1 about 30% of the time, whereas it is 9 just 4% of time. This was accidentally discovered by the engineer Frank Benford. It works for all numbers – mountain heights, river lengths, populations, etc. 

Why? A deep law of mathematics declares that the universe of numbers cannot help avoid the popularity of 1. What is the point? This is a little known law of mathematics, so people faking tax returns will not follow the law and their numbers will start with an even mix of 1s through to 9s. The US tax dept uses Benford’s Law to spot frauds. And sharp-eyed teachers can spot children who cheat when they are asked to toss a coin 100 times and note the sequence. Children try to fake the data, but they don't know what the data is supposed to look like. Simon explains how pupils can fake this data successfully and how you can fake your tax return without being caught. (Well, not for this reason at least).

Programme 2: 2 - At the double.
We all remember the story of the Persian who invented chess and who asked to be paid with 1 grain of rice on the first square, 2 on the second, 4 on the third and so on, doubling all the way to the 64th square. He bankrupted the state! 

This doubling is a form of exponential growth, which appears in everything from population growth to financial inflation to the inflation theory that supposedly caused the Big Bang. Moore’s Law (invented by Gordon Moore of Intel) showed that computers double in speed every 18 months, and this has held true for the last 40 years, which is why your desktop has more power than the computer on board Apollo 11. 

But can Moore’s Law continue? What are the limits of computing? What happened when American railroad growth stopped obeying its version of Moore’s Law. Can we beat Moore's Law, and new computer technologies? As Moore himself said: “If the automobile industry advanced as rapidly as the semiconductor industry, a Rolls Royce would get a million miles per gallon, and it would be cheaper to throw it away than to park it". Unfortunately the Rolls Royce would also crash every 10 miles!

Programme 3: 6 degrees of separation 
Six is often treated as 2x3, but has many characteristics of its own. Six is also the "pivot" of its divisors (1+2+3=6=1x2x3) and also the centre of the first five even numbers: 2, 4, 6, 8, 10. Six seems to have a pivoting action both mathematically and socially. How is it that everyone in the world can be linked through just six social ties? As Simon discovers, the concept of “six degrees of separation” emerged from a huge postal experiment conducted by the social psychologist Stanley Milgram in 1967. Milgram asked volunteers to send a package by mail to one of a hundred people chosen at random. But they could only send mail to people they knew on first name terms.

More recently Duncan Watts and colleagues at Columbia University in New York conducted a massive email experiment with more than 60,000 people from 166 different countries taking part. Participants were assigned one of 18 target people. They were asked to contact that person by sending email to people they already knew and considered potentially "closer" to the target. The targets were chosen at random and included a professor from America, an Australian policeman and a veterinarian from Norway. 

The researchers found that it in most cases it took between five and seven emails to contact the target. Watts says this shows that email has not fundamentally changed the way social ties are created. We explain why six is key to understanding the nature of social networks – and how it is having important implications in helping scientists model the rapid spread of infectious diseases.
 

Programme 4:  6.67 x 10^-11 – the number that defines the universe. 
Newton’s equation of gravity included a number G, which indicates the strength of gravitation. It took 100 years before the shy Englishman Henry Cavendish (he left notes for his maids because he was too shy to talk to women) measured G to be 6.67 x 10^-11 Nm²/Kg². It allowed him to weigh the Earth itself. There has been an ever-greater desire to measure this number with accuracy, which even implied an antigravity at times. How did we measure this tiny number and what does it mean for the universe? The Astronomer Royal Martin Rees explains that a large value for G would mean that stars would burn too quickly and a low value would mean that the stars would not form in the first place, so is G perfectly tuned for life? Is God a mathematician?


Programme 5: 1729 – the first taxicab number
Curious properties sometimes lurk within seemingly undistinguished numbers. 1729 sparked one of maths most famous anecdotes: a young Indian, Srinivasa Ramanujan, lay dying of TB in a London hospital. G.H. Hardy, the leading mathematician in England, visited him there. "I came over in cab number 1729," Hardy told Ramanujan. "That seems a rather dull number to me."  

"Oh, no!" Ramanujan exclaimed. "1729 is the smallest number you can write as the sum of two cubes, in two different ways." Most of us would use a computer to figure out that 1³ + 12³ = 9³ + 10³ = 1729. Ramanujan did it from his sickbed without blinking. 

Mathematicians have mined his theorems ever since. In the last in the current series Simon Singh examines their impact and how mathematicians have proved them and put them to use. Far more than just another number theory, 1729 is the first of the “Ramanujan numbers” or taxicab numbers. Mathematicians are competing to search for more of them (with higher powers) and testing the strength of new computing technology. The search is seen as mathematics’ current greatest challenge. Only recently, a lost bundle of Ramanujan’s notebooks turned up in a Cambridge library setting maths off on a new voyage of discovery.