It's Johnny Ball's birthday, and if you are not from the UK and getting a little gray, you probably never heard of him. I had not; until one day recently I wandered through the McCracken County Library in Paducah, Ky, and as I usually do, spent my time in searching through the 510-512 section in non-fiction, and I met him. The book was published in 2017, and I had been in England from 2001 until 2011... and never heard of him.
When I looked up his bio in Wikipedia I realized why, he didn't spend a lot of time hanging out at Cambridge or Oxford. In fact, he seems to have no post high school math education at all except for his own curiosity and creativity.
So here is the highlight of his wiki bio:
Graham Thalben Ball, professionally known as Johnny Ball (born 23 May 1938), is an English television personality, children's television presenter, writer and populariser of mathematics. Ball regularly appeared on British television for several decades, predominantly on the BBC, from 1967, featuring in or presenting programmes such as Play School, Think of a Number, Think Again, Think It ... Do It, Think...This Way and Johnny Ball Reveals All. He has also published a number of non-fiction books. *Wik
Well, the one I stumbled across was "Wonders Beyond Numbers", Bloomsberry Sigma, 1917.
I opened the book and found (carefully hidden on the opening page) one of the oldest, and yet persistent items in mathematical folklore.
He uses colorful language to tell about an event in his childhood where he learned about "Russian Peasant Multiplication" in a British Pub which apparently had a Children's room... at least when Dad was getting snockered, the kids could get a bit of education/entertainment.
He present's the problem in this way:
To multiply 13 x 9 you make two columns,
13 9 now divide the left column by two, ignoring fractional parts, and double the other.
6 18
3 36at's right
1 72
Now if you imagine back to a time when there were almost no schools, addition was much simpler than multiplication...so we learn to transfer the labor.
to find the product, ignore all the even numbers on the left and thei corresponding doubled number on the right, so cross out the 6----18 and for the rest, add the numbers that remain in the right column, 9, 36, and 72. I got 117, and checking 9x13 on my calculator, that's right.
but what if we switch the halving and doubling:
9 13
4 26
2 52
1 104.
crossing out the 4,26 row and the 2,52 row and you are left with 13 + 104= 117... well gosh, at least it's commutative in a sense.
He goes on to show that the method dates all the way back to the Rhind Mathematical papyrus where he does a different multiplication, but why not use the same numbers to show both methods are essentially the same.
So we take the powers of two on the left column until they exceed the multiplier (lets say 9) and take the multiplicand, 13 , and double it on the right column.
1 13
2 26
4 52
8 104
Which looks familiar to the second example above. Now we find numbers that add up to nine in the left column (That would be 1 and 8) and add the corresponding numbers in the right column... 13 + 104 (hmmm same as above?) and still get 117.
Take a moment and think about the number nine in binary notation, 1001 and it may all become a little clearer.
Entertaining, and that is not all bad working with children, but for us Big Folk, what's the History.
Well, it really is :
The Rhind Mathematical Papyrus records multiplication by successive doubling (e.g., to multiply
n by 13, double, n repeatedly and select the rows that sum to 13). This is why it is often called “Egyptian multiplication.” The method survived through Greek mathematical texts and commentaries, where it was described as a practical approach to multiplication.
And then, throughout the generations, lots of authors kept showing it worked. Around the 9th–12th centuries) the method was known in the West before the widespread adoption of Hindu–Arabic numerals. It was especially useful when people were calculating on line boards or with Roman numerals, since it avoided having to memorize or apply a large multiplication table.
In some parts of Europe—including Russia and Eastern Europe—the halving-and-doubling method survived as a common practice among peasants and merchants, long after written multiplication tables had become standard in schools. That’s why Western writers in the 19th and 20th centuries often labeled it “Russian peasant multiplication.” In truth, it was not unique to Russia
The Venerable Bede, an Anglo-Saxon monk, describes multiplication by halving and doubling in his treatise on the reckoning of time. This is one of the earliest explicit mentions in Latin Europe.
In Propositiones ad Acuendos Juvenes (Problems to Sharpen the Young), some problems can be solved naturally using halving-and-doubling multiplication.
Gerbert of Aurillac (Pope Sylvester II, c. 950–1003) introduced the abacus with counters to Europe. His methods of multiplication often used halving and doubling, because with counters on a line board, it was simpler than memorizing a multiplication table.Some reconstructions of his abacus schools show this very procedure.
Leonardo of Pisa (Fibonacci), in his famous book, Liber Abaci (1202; 2nd ed. 1228) explicitly describes multiplication by doubling and halving (calling it “duplation and mediation”), alongside other multiplication methods. He notes it as useful when written multiplication is impractical.
Nicholas Chuquet, a French mathematician, includes halving-and-doubling multiplication. He also shows positional decimal multiplication, but presents duplation/mediation as a practical alternative.
Adam Ries (1492–1559) in his widely used German arithmetic textbooks (such as Rechnung auff der linihen und federn, 1522), Ries describes halving and doubling as a method still current among merchants.
His influence helped spread “schoolbook multiplication” across German-speaking lands, but he records the older method as well. It seems that even today in Germany his influence remains in the phrase "all according ton Adam Riese', similar to the US where something that is done right is "According to Hoyle."
No comments:
Post a Comment