Tuesday, 30 June 2009

One More "Times"



The guys (and girls??) over at 360 have been going through a number of ways to multiply, and with some contributions from several others are approaching 25 (more or less) different ways to multiply two numbers.
I am on my road trip across America's Midwest and southwest (today from Roswell NM, where the Aliens abide) but finally found a moment to write.

And as it seems to happen with things like this, I was going through some old journal articles that Dave Renfro had sent me, and came across a posting in the September 1921 Philosophical Magazine with the catch title, The Mental Multiplication and Division of Large Numbers, by V. A. Bailey, M.A.; Queen's College, Oxford.

It was a method I had never seen before, and hazard to guess that the folks at 360 haven't either. The author, who states that he arrived at it independently, declares it to have the following advantages:
1) It is speedier.
2) Less liable to error.
3) Less Fatiguing for Large Numbers.
4) By reversing the process we can perform long division mentally.

I will ignore the last , and illustrate the method, then amplify a little about its history (which goes well before 1921).

for ease of operation, I will use a smaller pair of numbers than the author, who multiplied 24968 by 4352. I will instead use 257 x 62 to shorten the process.

Write the first number on a piece of paper, and then the second number should be written backwards on a seperate piece of paper.
Align the second sheet under the first as shown below, and multiply only the numbers aligned vertically (the 7 and 2) to get 14. Record the four, and keep the one that is to carry in the mind (this is the mental part referred to in the title):
_______257
_________26
_______ 4

Now move the bottom sheet one digit to the left aligning the 2 under the 5...


Now multiply and add the vertical digits, (7x6+5x2=52) and then add the one that was carried from before to get 53. Record the 3 and carry the five mentally
_______257
________26
_______ 34

Once more we move the bottom number one to the left, and again multiply the vertically aligned digits and add the carry to get 2x2+5x6+5=39 and record the units and carry the tens

_______257
_______26
_______934

Continue to move and shift until no numbers are, aligned, so the last multiplication is 2x6+3=15.. record the five, and since there are no more digits to shift under, append the one
_______257
______26
_____15934

And sure enough, 257 x 62 = 15934. This is really not much more difficult with longer numbers since only the single carry must be memorized from digit to digit...

But of course, nothing is new under the mathematical sun, and the method had been previously described in Nature in the Dec 31st issue of 1891 by a Dr. K. Haas. But Dr. Hass mentioned that he had not invented it, and that it had been mentioned by Pappus, who credited to Appolonius of Pergia (circe 250 BC); and had also been known to the Hindus under the name of Vajrabhyasa.
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