By Ian Stewart

At its center, arithmetic is ready numbers, our basic instruments for knowing the realm. In Professor Stewart’s outstanding Numbers, Ian Stewart deals a pleasant creation to the numbers that encompass us, from the typical (Pi and a couple of) to the unusual yet no much less consequential (1.059463 and 43,252,003,274,489,856,000). alongside the best way, Stewart takes us via major numbers, cubic equations, the idea that of 0, the potential positions at the Rubik’s dice, the position of numbers in human heritage, and past! An unfailingly genial advisor, Stewart brings his attribute wit and erudition to undergo on those outstanding numbers, supplying a fascinating primer at the ideas and gear of math.

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At that stage we get: Square Fig 26 If we keep going using three colours we get stuck. Arizona now borders states that we have coloured medium grey, dark grey, and black. Since all colours up to this point are forced by the way states adjoin, three colours won’t work for the whole map. Therefore we need a fourth—light grey, say—to keep going: Fig 27 A fourth colour comes to the rescue. 51 52 Incredible numbers With 38 more states to go, excluding Alaska and Hawaii, it seems possible that maybe we’d need a fifth colour at some point, or a sixth .

However, we can continue, making sensible guesses and changing colours if things go wrong. One resulting colouring has only three light grey states: Arizona, West Virginia, and New York. Even though there are 50 states, we’ve coloured the entire map with just four colours. Fig 28 A fifth colour isn’t needed. (Another technical point: Michigan occurs as two disconnected regions, with lake Michigan in between. Here both have been coloured dark grey, but disconnected regions sometimes lead to more colours.

Most numbers, but not all, can be written as a sum of three squares. The list of those that can’t begins: 7 60 15 63 23 71 28 79 31 87 39 92 47 95 55 103 Again there is a pattern to the numbers, and again it’s difficult to spot. It was found in 1798 by Adrien-Marie Legendre. He stated that the sums of three squares are precisely those numbers that are not of the form 4k ð8n þ 7Þ. The list of exceptions, above, comprises all numbers that are of this form. Thus if n ¼ 0 and k ¼ 0 we get 7, if n ¼ 1 and k ¼ 0 we get 28, and so on.

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