Uploaded February 2010 | Updated September 2026, 3 weeks ago
This follow-up video to "LOGARITHMS: Brief History and Brief Math" goes a little bit further with the mathematics of these scary sounding creatures. This is Part II of two Parts.
This follow-up video to "LOGARITHMS: Brief History and Brief Math" goes a little bit further with the mathematics of these scary sounding creatures. This is Part II of two Parts.

![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)








![Parentheses Numbers / Catalan Numbers: Part I (Tanton Mathematics)
Here is a classic puzzle: In how many ways can one arrange parentheses around a sum of N terms so that one is only ever adding two things at a time? [This is a great puzzle for students to do when they are first learning about parentheses!]
It turns out to be tricky, but in this video we come up with a good way of computing these numbers. In part II of this video, we go further and find an actual explicit formula for them! Parentheses Numbers / Catalan Numbers: Part I (Tanton Mathematics)](https://i.ytimg.com/vi/m9Khxn2g-6w/mqdefault.jpg)