Cliff Pickover@pickover is one of many people who pointed out that (111,111,111)2 = 12,345,678,987,654,321 the first nine multiples of 1 rising and falling. If you try that with twos, it won’t work because of the carries, 2222^2 = 4937284.

I had thought, from reading Dickson’s History of the theory of Numbers that this little tidbit first appeared in The Gentleman’s Diary in 1810 by a man named Peter Barlow who was born in 1776 in Norfolk,England,just down the road from my old teaching location on RAF Lakenheath. He was a pretty good math sci guy. Check him out on Wikipedia …. Turns out, Peter B wasn’t the first. Not by about 700 years, and I don't know that he was the first.

I recently read an old (1966) journal article about the earliest known Arabic arithmetic in its original language by a scholar whose very long name is usually shortened to al-Uqlidisi, who wrote in the tenth century. In it, he includes the above written out in nine steps beginning with 1^2=1, then 11^2 = 121 , and continuing up to the same nine digit repunit squared above. He also points out that you can also do the same with (222222222)^2, and get the sequence of multiples of 2 from 4 up to 36 and then back to 4, but he has an advantage, he wrote numerals in sexigesimal notation.

The ancient scholar also used decimals and pointed out that you can avoid the problem of the carry in base ten by inserting one or more zeros between each non-zero digit and get 20202020202020202^2 = 408121620242832363228242016120804. Try it yourself with other repdigits of nine digits, and if you get a problem with carries, just add more zeros.

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