We know that: 1)How integers are constructed from the natural numbers (Just by extending the discrete line of numbers with zero and negatives). 2)How rationals are constructed from integers{p/q: p, qεZ and q≠0}. I think you should make a detailed video on the construction of real numbers from the rationals (I know it happens by filling the holes but still as a student of math I want to see rigor). By the way liked your explaination and name of your channel.
Thank you! I made a short video titled, Handwaving Dedekin cuts, which more or less explains the idea behind Dedekind cuts. There are two (usual) routes to build the reals from the rationals: Dedekind cuts, or Cauchy sequences. Both routes have their pros and cons, and in any case, math students benefit from learning both of them... Note that the importance of building the real numbers lies in proving that there exists an ordered field that satisfies the supremum property (or any other equivalent property that would allow you to do calculus). Once you have that property, you can do calculus... I might explain the construction of the reals from the rationals in a stream, and if i get a good idea for a video, I will make one! Thanks for your suggestion!
Thank you! I've been thinking about making that video for a while. It wouldn't be so much about presenting a proof but how one could think about that problem and perhaps get to prove it... If your comment gather enough traction, I'll make that video next! (I was planning to go on doing more on this series in analysis first but talking about square root of two would be a fun parenthesis)
Don't worry about it then! I wish you luck on your channel. I personally like how you present things. Side note: I think going into the direction of "big picture thinking" of mathematics is very good, only a few channels do that. People usually don't focus on the declarative component of mathematics and think of it as stepping stones to an end solution but its not like that at all.@@academyofuselessideas
@jeevacation I took on your suggestion and I made a video on proving the irrationality of two! I also made one on the infinitude of primes... If you happen to see those videos, let us know what you think of those proofs! Thanks again for the suggestions!
We know that: 1)How integers are constructed from the natural numbers (Just by extending the discrete line of numbers with zero and negatives).
2)How rationals are constructed from integers{p/q: p, qεZ and q≠0}.
I think you should make a detailed video on the construction of real numbers from the rationals (I know it happens by filling the holes but still as a student of math I want to see rigor).
By the way liked your explaination and name of your channel.
Thank you! I made a short video titled, Handwaving Dedekin cuts, which more or less explains the idea behind Dedekind cuts. There are two (usual) routes to build the reals from the rationals: Dedekind cuts, or Cauchy sequences. Both routes have their pros and cons, and in any case, math students benefit from learning both of them... Note that the importance of building the real numbers lies in proving that there exists an ordered field that satisfies the supremum property (or any other equivalent property that would allow you to do calculus). Once you have that property, you can do calculus...
I might explain the construction of the reals from the rationals in a stream, and if i get a good idea for a video, I will make one! Thanks for your suggestion!
Nice video, you should make one on the proof of irrational numbers
Thank you! I've been thinking about making that video for a while. It wouldn't be so much about presenting a proof but how one could think about that problem and perhaps get to prove it... If your comment gather enough traction, I'll make that video next! (I was planning to go on doing more on this series in analysis first but talking about square root of two would be a fun parenthesis)
Don't worry about it then!
I wish you luck on your channel. I personally like how you present things.
Side note: I think going into the direction of "big picture thinking" of mathematics is very good, only a few channels do that. People usually don't focus on the declarative component of mathematics and think of it as stepping stones to an end solution but its not like that at all.@@academyofuselessideas
@@jeevacation Thank you! That's one of your goals!
@jeevacation I took on your suggestion and I made a video on proving the irrationality of two! I also made one on the infinitude of primes... If you happen to see those videos, let us know what you think of those proofs! Thanks again for the suggestions!