Powers of Ten in Base Two and Five

Make two lists; one consists of all powers of 10 in base 2, and the other all powers of 10 in base 5. (Powers of 10 are 10, 100, 1000, etc.)

Fun Fact: for any integer N > 1, there is exactly one number in exactly one of the lists that is exactly N digits long!

Presentation Suggestions:
Show students the first terms in each list: they are 
(base 2) 1010, 1100100, 1111101000, … 
(base 5) 20, 400, 13000, 310000, …

The Math Behind the Fact:
This fact is quite startling when you first see it. Proving it makes a great exercise for a Putnam problem-solving seminar…

How to Cite this Page: 
Su, Francis E., et al. “Powers of Ten in Base Two and Five.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
R. Vakil, A Mathematical Mosaic, 1996. p.131.

Fun Fact suggested by:
Ravi Vakil

Did you like this Fun Fact?

Click to rate it.

Average rating 2.7 / 5. Vote count: 10

No votes so far! Be the first to rate this Fun Fact

Share: