• Choose a category
• All
• Classic-Fiction

By Root 754 0

what has become standard algebraic notation. It is the first book that looks like a modern maths book, full of as, bs and cs and xs, ys and zs. It was Descartes’s decision to use lower-case letters from the beginning of the alphabet for known quantities, and lower-case letters from the end of the alphabet for the unknowns. When the book was being printed, however, the printer started to run out of letters. He enquired if it mattered if x, y or z was used. Descartes replied not, so the printer chose to concentrate on x since it is used less frequently in French than y or z. As a result, x became fixed in maths – and the wider culture – as the symbol for the unknown quantity. That is why paranormal happenings are classified in the X-Files and why Wilhelm Röntgen came up with the term X-ray.

Were it not for issues of limited printing stock, the Y-factor could have become a phrase to describe intangible star quality and the African-American political leader might have gone by the name Malcolm Z.

With Descartes’ symbology, all traces of rhetorical expression had been expunged.

The equation that Luca Pacioli in 1494 would have expressed as: 4 Census p 3 de 5 rebus ae 0

and Viète would have written in 1591 as: 4 in A quad – 5 in A plano + 3 aequatur 0

in 1637 Descartes had nailed as: 4x2 – 5x + 3 = 0

Replacing words with letters and symbols was more than convenient shorthand. The symbol x may have started as an abbreviation for ‘unknown quantity’, but once invented, it became a powerful tool for thought. A word or an abbreviation cannot be subjected to mathematical operations in the way that a symbol such as x can. Numbers made counting possible, but letter symbols took mathematics into a domain far beyond language.

When problems were expressed rhetorically, as in Egypt, mathematicians used ingenious, but rather haphazard, methods to solve them. These early problem-solvers were like explorers stuck in a fog with few tricks to help them move about. When a problem was expressed using symbols, however, it was as though the fog lifted to reveal a precisely defined world.

The marvel of algebra is that once a problem is restated in symbolic terms, often it is almost solved.

For example, let’s re-examine Diaphantus’s epitaph. How old was he when he died? Translating that statement, using the letter D to symbolize his age when he died, the epitaph says that for D—6 years he was a boy, that another D—12 years passed before he sprouted facial hair, and that he wed after another D—7. Five years after that he had a son, who lived for D—2 years, and four years later Diophant us himself breathed his last. The sum of all these time intervals adds up to D, since D is the number of years Diophantus lived. So:

The lowest common denominator of the fractions is 84, so this becomes:

Which can be rearranged as:

Or:

Which is:

Moving the Ds to the same side:

Multiplying out:

The father of algebra died aged 84.

We can now return to the trick at the start of the chapter. I asked you to name a three-digit number for which the first and last digits differed by at least two. I then asked you to reverse that number to give you a second number. After that, I asked you to subtract the smaller number from the larger number. So, if you chose 614, the reverse is 416. Then, 614 – 416 = 198. I then asked you to add this intermediary result to its reverse. In the above case, this is 198 + 891.

As before, the answer is 1089. It always will be, and algebra tells you why. First, though, we need to find a way of describing our protagonist, the three-digit number in which the first and last digits differ by at least two.

Consider the number 614. This is equal to 600 + 10 + 4. In fact, any three-digit number written abc can be written 100a + 10b + c (note: abc in this case is not a×b×c). So, let’s call our initial number abc, where a, b and c are single digits. For the sake of convenience, make a bigger than c.

The reverse of abc is cba which can be expanded as 100c + 10b + a.

We are required to subtract cba from