Math Puzzle of the Week #1: Decimal Expansions

1 replies · 1,567 views · Started 31 December 2009

Revfan9 31 Dec 2009, 20:40

The number 13717421/1111111111 is quite curious: Its decimal representation is all the digits in order, repeated over and over again. In other words, 0.0123456789012345678901234567890....

The number 1/3 doesn't look like anything special at first, until you look at its base 2 decimal expansion, which is starkingly similar to the example above. 0.0101010101... For the purposes of this exercise, we shall call numbers which repeat all the digits in a given base in ascending order a Nigerian Number.

Other examples of Nigerian numbers include 5/26, 9/85, 97/1562, which are the Nigerian numbers of bases 3, 4, and 5 respectively. Base 16 (Hexadecimal's) Nigerian number is 5465701947765793/1229782938247303441.

What is the fraction form of the nigerian number of base 42?

Saffith 01 Jan 2010, 06:46

It's not obvious with the numbers written that way, but there's a simple pattern.

1/3 in base 2 is 1/11
5/26 in base 3 is 12/222
9/85 in base 4 is 123/3333
97/1562 in base 5 is 1234/44444
The decimal number, 13717421/1111111111, can be rewritten as 123456789/9999999999
And 5465701947765793/1229782938247303441 in base 16 is 123456789ABCDEF/FFFFFFFFFFFFFFFF

If I did this correctly (it's 1:30 in the morning, so I could be wrong...), the corresponding number for base 42 is 89310492293454236976068989984684388943514557149755370158175807861 / 150130937545296572356771972164254457814047970568738777235893533016063. I'm not about to try and reduce that, though.