Differential Equations: Geometric Theory, 2nd ed by Solomon Lefschetz

By Solomon Lefschetz

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Extra resources for Differential Equations: Geometric Theory, 2nd ed

Example text

Mathematicians consider two knots to be ‘the same’ – the jargon is topologically equivalent – if one can be continuously transformed into the other. ‘Continuously’ means you have to keep the string in one piece – no cutting – and it can’t pass through itself. Knot theory becomes interesting when you discover that a really complicated knot, such as Haken’s Gordian knot, is in fact just the unknot in disguise. Haken’s Gordian knot. The trefoil knot is genuine – it can’t be unknotted. The first proof of this apparently obvious fact was found in the 1920s.

Answer on page 260 * Strictly speaking, ‘dice’ is the plural, and I should have used ‘die’ – but I’ve given up fighting that particular battle. I mention this to stop people writing in to tell me I’ve got it wrong. Anyway, the proverb tells us ‘never say die’. Why Does Minus Times Minus Make Plus? // 37 An Age-Old Old-Age Problem The Emperor Scrumptius was born in 35 BC, and died on his birthday in AD 35. What was his age when he died? Answer on page 262 Why Does Minus Times Minus Make Plus? When we first meet negative numbers, we are told that multiplying two negative numbers together makes a positive number, so that, for example, ðÀ2Þ6ðÀ3Þ ¼ þ6.

Try it with other three-digit numbers – you’ll find that exactly the same trick works. Now, mathematics isn’t just about noticing curious things – it’s also important to find out why they happen. Here we can do that by reversing the entire calculation. The reverse of division is multiplication, so – as you can check – the reverse procedure starts with the three-digit result 471, and gives 471613 ¼ 6;123 6;123611 ¼ 67;353 67;35367 ¼ 471;471 Not terribly helpful as it stands . . but what this is telling us is that 47161361167 ¼ 471;471 So it could be a good idea to see what 1361167 is.

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