The humanist looks at these enterprises with some bewilderment. Paranoia is not unknown among mathematical geniuses. The matter of priority, of who first took the leaps, is important it is of small value to work out Fermat's last theorem if somebody has done it before and there was a classic row between Newton and Leibniz about which of them first identified what Newton called "fluxions" and Leibniz "calculus". They assume they must answer them logically, for the history of mathematics is one of logical development and amendment, sometimes in startling leaps and bounds. And sometimes even geniuses have enormous difficulty in answering what look like childlike questions. Moreover the child's discoveries resemble, if only a little, those of more recent mathematicians, which may often, though by no means always, be of no use. Yet without their work the modern world would be almost inconceivably different. They knew about pi of course, but geometrically, not as a number with endless decimal points, and couldn't get their heads round square roots, let alone infinity. What the Greeks discovered, and the mathematical universe they believed in, might also seem nowadays to be of little practical value. But the calculating child is emulating the Greek mathematicians. I am fairly sure I was not taught these things at school, perhaps because they are not really much use and don't lead anywhere. And incidentally, why is the sum of two odd numbers, like the sum of two even numbers, always an even number? Twos and fives need no help, and the only number that yields to no such tricks is seven. If a number is divisible by 11, the sums of its alternate digits will always be equal (121, 671, 2541.
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If a number is divisible by three, the sum of its digits is also divisible by three, eg 714, 1,002, 108,762.
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INFINITY AND INFINITESIMALS BOOK FULL
Even those of us to whom calculus was a distant peak we had no prospect of climbing can remember a time of innocence when numbers were full of mysterious interest.