Over-egging the pud... pizza?

 It was with sadness (but not to a soul-withering degree, just a slight world-weary irritation) that I saw this tweet (with celebrity endorsement):

I won't bore you with the whole triptych. I think what evoked La Riley's "best ever" was the final words of the last of these: 

Take Maths seriously!

But I'm afraid @cretiredroy is using a sledgehammer to crack a nut; I blame calculators, which encourage users to let machines do the hack-work and not use their brains. Three whole mega-tweets (by which I mean the Twitter '2.0' version – when Twitter made me a very occasional user by welcoming verbosity).

In 2/3 he gives the formula for the area of a circle, πr², and then laboriously goes through a detailed calculation of the areas of the two pizzas, including a value for π that specifies seven decimal places; (he's clearly an adherent of the my-sum's-got-more-decimal-places-than-yours-so-there school.) I assume the value he specified was near-enough correct; but why calculate it at all? He wants to show that the area of two 5" pizzas < the area of a 9" pizza; so π will appear on both sides of the comparison; it will cancel out. So all he has to do is say 2 x 5²  < 1 x 9. From this it's obvious that even 3 x 5² < 9² (which he gets around to explain about halfway through his mega-tweet).

This later response sums up my feelings:

As Wordsworth might  so easily have put it,

Occam! Thou shouldst be living at this hour

The world hath need of thee...

No, really

 

No wonder kids in school get turned off maths. This sort of Calculomania (as usual, don't bother looking that up: it's an excessive tendency to get involved in complicated/detailed calculation at the expense of  mathematical elegance) contrasts acutely with the ancient Babylonians, who I've been reading about...

<shibboleth-busting>
(and if you think I "should" have written "about whom I've been reading",

1. Get a life
   
2. See this blog, passim [ie all over the place and I can't be bothered to check]; OK, here's one: in short, whether or not you invert, you're going to break somebody's pet "rule" 
   

)
</shibboleth-busting>

...in a Christmas present (yes, I know): Marcus du Sautoy's The Number Mysteries: A Mathematical Odyssey through Everyday Life:

Here's a typical problem. If a rectangular field has an area of 55 square units and one side is 6 units shorter than the other, how long is the longer side? If we call the longer side x, then the problem tells us that x x (x-6)=55 or, simplifying things,

x²-6x-55=0 

...

The Babylonians came up with a neat method: they dissected the rectangle and rearranged the pieces to make a square, which is an easier shape to deal with. We can divide up the pieces of our field just as Babylonian scribes would have done thousands of years ago.

<picture possibility="but not today"> 

The book has a diagram here^ps, which is worth – of course –  N words (where N is a large number), but I installed a brand new version of Linux today, and I hope to put it through its paces (graphics-wise) by trying to draw a copy. I must also ask the good doctor's (apologies, professor's) permission. I was tempted to take the coward's way out, and just photograph it, but the ghost of Doc Lewis (my maths master [who never forgave me for taking the 'soft option' {languages}]) would haunt me forever if I didn't remove the vinculum from the √ sign: "You don't need a vinculum, boy, if there's nothing to vinc" 

</picture>

Start by cutting a small rectangle measuring 3x(x-6) units off the end of the rectangle and move this round to the bottom of the rectangle. The total area hasn't changed, just the shape. The new shape is almost a square with each side x-3 units long, but missing a small 3×3 square in the corner. If we add in this small square we increased the area of the shape by 9 units. The area of this large square is therefore 55+9=64. Now we have the simple task of taking the square root of 64 to discover the length of the side, which must be 8. But the side had length x-3, and so x-3=8 i.e. x=11. Although we've only been shuffling around imaginary parcels of land, behind what we've been doing lies a method for unlocking those cryptic quadratics. 

When maths is elegant like this it's a thing of beauty. But people who use big numbers to show off their ability to use a calculator really get my goat.

But I'm missing the tennis.

b

2022.06.20:35: Update – Made correction. (Incidentally, in British English, "professor" is not the same as  in American English.)

 

 2022.07.17:35: Update – Added footnote

PS (I haven't mastered the curve, and the layout doesn't follow the original precisely, but the text is the same.)

2022.07.20:10: Update – Added PPS

PPS Incidentally. BorExit this morning spiked my guns. I'd just come up with a bon mot about the stability of the Johnson government being measured in Resignations Per Minute when I heard the news: Sic semper satiristibus.

 

2022.08.16:10: Update –  Improved picture STOP PRESS (20:00):... up to a point; I'll fix the obvious glurch when I'm back from holiday.

2022.08.18.11:15: Update –  Picture fixed (after a fashion; for uninteresting system reasons I couldn't get at the source, and had to hack a bitmap, so it's sub-optimal).

 

2022.08.22.12:30: Update –  Picture fixed (properly)