talking about the binary expansion of base ten numbers that are made up of repeating nines...

"I discovered a cool property of positive integers of the form 10

^{n}-1, that is, integers made up of n digits of 9s: they have binary representations that have exactly n digits of trailing 1s. For example, 9,999,999 in decimal is 100110001001011001111111 in binary. "

Rick went on to show that this must always be the case with some pretty simple algebra. He noticed that the part of the number preceding the repeating ones is the binary for 5

^{n}-1... ok here is a simple example... 10

^{2}-1 =99 in base ten and in base two that is 1100011,.... the two repeating 1's at the end have a value of 2

^{2}-1, the 11000 in front is the binary expression for 24 or 5

^{2}-1.

Rick showed that 10

^{n}-1 can always be expressed as

(5

^{n}– 1) 2

^{n}+ (2

^{n}– 1)...

It occurred to me as I read his blog, that we could also expres 10

^{n}-1 as

2

^{n}(5

^{n}) - 5

^{n}+ 5

^{n}-1 by adding and subtracting 5

^{n}instead of 2

^{n}

Regrouping that as (2

^{n}-1 )5

^{n}+( 5

^{n}-1) shows that when expressed in base five, 10

^{n}-1 will begin with 2

^{n}-1 base five followed by a string of n fours,

9 [5] = 1 4

99[5]= 344

999[5]= 12444

9999[5]=1114444

A similar event happens (I think, but do not prove) for any (pq)

^{n}-1 if p and q are (relatively?) Prime when expressed in base p or base q

... for example instead of 10

^{n}-1, if we used 6

^{n}-1 we would get

2

^{n}(3

^{n}) - 2

^{n}+2

^{n}-1 In base two that would show up as the base two number for 3

^{n}followed by a string of n ones.

For example 6

^{3}-1 in base two is 11010111. The front part, 11010, is 26 or 3

^{3}-1 in base ten, and then followed by three ones..

We could do the same switch to express it in base three.

The same number (6

^{3}-1) in base three would give 21222, with 21 in front being the base three for 7 or 2

^{3}-1 and then followed by a string of three twos..

If you want to explore binary problems, he Rick also has a convertor on his web page to go between decimal and binary. Of course you can always use Wolfram-alpha [I have a little "alpha" gadget on my I-google page to keep it handy]... By writing "convert 315 base 8 to base 3" it gave me 21121

_{3}.

As a side note, I find that using alternate bases helps lots of kids really get a feel for polynomials they never had before (and perhaps a deeper understanding of the decimal system they

*thought*they had fully mastered. Quickly students, can you see why 11010111 in base two would be 3113 in base four?