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Alex Through The Looking-Glass

How Life Reflects Numbers and Numbers Reflect Life

Alex Bellos

Numbers have baggage. $7.99 feels lot cheaper than $8.00 bc it seems like a $7 item rather than an $8 one. But we have been so conditioned by this that we see anything ending in 9 as a bargain. So experiments found a dress priced at $39 sells better than when priced at $34. But don't want to associate your restaurant meal with "cheap" so menu prices are in whole numbers. And they usually leave off the $ sign, bc that reminds you of the pain of paying.

Benford's Law: any list of numbers will be more 1's than 2's which will be more than 3's and so on. As in 30%, 20%, 12% etc. Quick diagnostic tool for fraud - if a large set of numbers don't obey this rule, there's a good chance the data has been faked. Usually applied to financial fraud, but also self-reported pollution readings, results of suspect elections. Applies even when you convert the data, such as from dollars to pounds, or miles to kilometers. The second digit in a number also follows the rule, but with 0 being the highest frequency.

Math game using Benford's Law: you pick a number, I pick a number. If the answer starts with 1, 2 or 3, I win; if it starts with 4, 5, 6, 7, 8 or 9 then you win. Looks like odds weighted heavily in favour of the higher numbers but actually favours the lower ones.

Creationist group heard that the percentages of different minerals in the sea and the earth's crust also follow the law. This was so surprising and amazing that it could only be the work of an intelligent designer.

Eratosthenes was the Greek who calculated the circumference of the Earth. He knew that the Earth was round, bc people had observed the way ships sank below the horizon, and had seen the round shadow of an eclipse of the Moon. He knew of a famous well in Syrene (now Aswan) where the sun lit up the bottom at noon on midsummer's day, which meant the sun was directly overhead then. On the same day he measured the angle of the sun at Alexandria, and found it to be 1/50th of a full circle. That meant that the distance from Syrene to Alexandria was about 1/50th of the way round the whole world. They knew the distance was 5000 stades, calculated by bematists or step-people, whose job it was to walk for days on end, counting their steps.

Eratosthenes was blessed by three bits of geographic serendipity, without which he could not have made his calculation. That the Egyptians had settled as far south as Syrene (on the Tropic of Cancer, the furtherest north that the sun ever directly overhead), that Syrene was due south of Alexandria, and that the road between the two was flat and straight, so that an accurate distance was possible.

Eratosthenes calculation of Earth circumference came out at 41,500 km, which is only 1500 over the real figure.

In C11, Arab scholar Al-Biruni was at a fort in the Punjab when he came across the perfect geographic setup to measure a mountain - a high peak facing a flat plain. He used trigonometry to calculate the height of the mountain. Then, he climbed the mountain and measured the angle of his view to the horizon. He used that to make a right angle triangle to work out the radius of the Earth as a ratio of the height of the mountain. Using trig again he got a figure of 39,800 km for the circumference, which is just 0.5% short.

The Great Trigonometrical Survey of India cost more money and lives than most Indian wars. It was a British military project, commanded by Col George Everest (pronounced EVE-rest). At the time, Chimborazo in Ecuador, was regarded as the highest mt in the world. But the highest mt in Himalayas had no name, so it was eventually named in honour of the colonel. And forever pronounced incorrectly.

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These surveys all relied on triangles, and the principle that same angles means same sides. Even today's GPS, works off satellite signals and calculates location based on lookup tables of sines and cosines stored in it's memory.

The power of exponential growth. If a bottle contains a bacteria that doubles in size every minute, and fills the bottle in an hour, then at 59 minutes the bottle must have only been half full. Even at 55 minutes it is only 3% full, and seems to have plenty of room for expansion.

The Rule of 72 - if a population is growing at 2% a year, it will double in size in 722 = 36 years. Helps you understand loan interest - if you borrow at 10%, the Rule of 72 tells you the debt will almost double in 7 years.

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