Uploaded March 2010 | Updated September 2026, 3 weeks ago
The number e arises as the number that makes calculus "easy," but it is often presented to students first in a pre-caclulus class as a number arising from the study of compound interest. Very few folk seem to ask whether or not these two e's really are the same. They are! This video - which assumes familiarity with calculus - proves this.
The number e arises as the number that makes calculus "easy," but it is often presented to students first in a pre-caclulus class as a number arising from the study of compound interest. Very few folk seem to ask whether or not these two e's really are the same. They are! This video - which assumes familiarity with calculus - proves this.


![Fractions take up No Space: Rationals Have Measure Zero (TANTON Mathematics)
How much space do the fractions take up on the number line? NONE!
In this video we add up pieces of ribbon to prove that the fractions have measure zero. It is all very easy and very perturbing because it is easy!
[This video does assume familiarity with the geometric series formula - so you might want to watch a video on that too. I do that with paper tearing!] Fractions take up No Space: Rationals Have Measure Zero (TANTON Mathematics)](https://i.ytimg.com/vi/j2QAhOCdm_o/mqdefault.jpg)







