Magic Numbers

One Saturday morning this July I’ll hit the Publish button and it will be number three hundred. No matter how you feel about math, numbers are mystical.

If you don’t understand them, there’s the feeling of exclusion, a certain dread that rises from the pit of your stomach whenever things turn numerate—like a shy person who’s called upon to tell a joke in public.

Churchill, who certainly didn’t suffer from public shyness, hated math. In his autobiographical The Young Winston, he complained that in mathematics you couldn’t be ‘almost right’, and as a schoolboy he found this a great disadvantage.

Of course in today’s brave new world you get brownie points for being almost right, which is why you show your working. You did all the preparation properly, but in the end you burnt the stew.

But then these days ‘trial and error’ no longer exists, now it’s ‘trial and improvement.’ In a similar way, KISS no longer means ‘Keep It Simple, Stoopid’, but ‘Keep It Strictly Simple’. When you start shoving adverbs into acronyms you’ve burnt the stew.

But if you do like math, and above all if it likes you, the awe is greater because numbers share their magical properties with you.

When I was a small boy, I thought it remarkable that you could find out if three went into a number by adding the digits. So 84 added to 12, and since 3 goes into that 4 times, you can divide 84 by 3 exactly, which gives you 28.

I didn’t realize then, but I just did now (decades later) that you can use this recursively (i.e. keep doing the same thing) and it’ll work. If you take a large number, say the first two blocks of a credit card, 3276 8433, and add them together, you get 36—most of us can work out that 3 goes into that 12 times, but if you can’t, just add 3 to 6—all of us know 3 goes into 9—that tells you the original number is divisible by three.

Or I thought it wondrous that if you wrote down the numbers zero to nine in order, and then next to them wrote them backward, you came up with the nine times table.

And that if you wanted to multiply 53 by 11, you added the two numbers and dropped them in the middle: 5-8-3, hey presto.

If all this turns you off, or you’re asking ‘so what?’, then you’re in step with most of humanity. And after all we are the only animals that even know about numbers, you don’t get felines worrying about whether 67 is a prime.

Sieve of Eratosthenes animation, an ancient algorithm for finding prime numbers.  (licensed under CC BY-SA 3.0 via Wikimedia Commons).

The Sieve of Eratosthenes, an ancient algorithm for finding prime numbers. (animation licensed under CC BY-SA 3.0 via Wikimedia Commons).

But if numbers are your friends, then you’ll know 67 is in fact prime, and you’ll enjoy this fusion of old and new, where digital animation from the twenty-first century meets the ideas of Eratosthenes of Cyrene, developed twenty-three centuries before.

The eminent Greek mathematician was born in Libya in 276 BC, and rose to become the chief librarian at Alexandria. As often happens, his enemies scorned him with the nickname beta, or second-rate (think alpha male).

Since he died in 194 BC, calculating how many years he lived requires a basic understanding of negative numbers—but a life of 82 years is monumental by the standards of the time, perhaps equivalent to five generations.

The sieve is simplicity itself. Once we establish that 2 is a prime number (not rocket science), the algorithm blocks out all the multiples (4, 6, 8… in red in the diagram). It then does the same for 3, 5, 7, and so forth. Anything left is prime—the ones that fall through the sieve.

But there is no known way to test whether a given number is prime without building up factors, and that finds its way into cryptography.

Mr. E. might have been β to his foes, but due to his eclectic nature he made some amazing astronomical discoveries. He was the first to calculate the circumference of the earth, and he did it with incredible accuracy, considering part of the calculation involved the time it took to ride a camel from Syene to Alexandria.

What is the earth's circumference? Two points on a meridian and a camel ride.

What is the earth’s circumference? Two points on a meridian and a camel ride.

Now even if you don’t give a damn about prime numbers, it’s pretty easy to see the importance of figuring this one out. The astronomers in The India Road certainly did, and if Columbus had paid a little more attention in class he might well have spared himself some embarrassment.

In Clear Eyes, the astronomers from King John’s Mathematical Junta make a cameo appearance as they interview the admiral of the ocean sea and examine his plan for a western passage to India.

Columbus defended the ideas of Marinus of Tyre, the eminent founder of mathematical geography, and mentor of Ptolemy.

“Marinus divided the circumference of the earth into twenty-four hours.”

“And we agree with him,” Abraham said.

“But he considered fifteen of those hours from the end of Western Europe—the city of Olissippo, where we now sit, to the very eastern tip of Asia.”

Abraham looked at him benignly. “And you disagree…”

“Indeed! Marco Polo has shown us how much greater Asia is. Therefore”—he stated with the unshakeable conviction of the self-taught man—“the remaining distance west from Lisbon to Cipango, to Cathay, will be smaller.”

The admiral’s eyes had a febrile glow. “Much smaller, sirs. Do you not see? If it is a nineteen hour interval from Lisbon to Cipango, there remain only five hours unaccounted.”

“Yes, it would place Japan in the vicinity of Spain,” Vizinho said. “Captain, we know the term the Celestials use for Japan is Ji Pen Koue, the Land of the Rising Sun.”

“Polo corrupted the name to Cipango, as you are aware,” Abraham added. The two astronomers subtly let Columbus know they too were familiar with the work of the Venetian merchant.

“Your world, captain, is missing many hours to the west of Lisbon, before the setting sun of the Atlantic meets Ji Pen Koue,” said Vizinho. “Polo’s jorney was arduous, perhaps the time he took somehow colored his judgement. You see, Cristovam, we also disagree with Marinus of Tyre.”

“But then…” Columbus said. “We are of the same view?”

“Sadly no,” said Abraham. “We are of the opposite view—we believe fifteen hours is too great a number. Which increases the distance west to the Indies.”

By the time this conversation takes place, Martin of Bohemia has stormed out of the room, incensed at Columbus and his geographical crimes.

Three hundred articles in as many weeks is a long time in anyone’s life. If you’ve been with me from the start, it’s been almost six years. Up until the end of 2014, the raw texts add to 270,000 words, or three full-length books. I expect we’ll be nearer three hundred thousand by article three hundred—one thousand words per week sounds about right.

At the stage I’ll select the best one hundred articles, and edit those to ninety-thousand words, i.e. a three hundred page book. As usual, the book will be published on Kindle, and since it collects my errant thoughts, it will be called Air and Thought.

I like the way the number 300 navigates across this plan. Maybe it’ll take more than 300 minutes to do, but it certainly won’t take 300 hours.

Atmos Fear and The India Road. Quick links for smartphones and tablets.

Atmos Fear and The India Road. Quick links for smartphones and tablets.


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