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