I think the definition of the real numbers that is both beautiful and works with decimal expansions is that real numbers are equivalence classes of sequences of rational numbers with finite limits, where two sequences are equivalent if they get and stay arbitrarily close together. (And they have finite limits, without using real numbers, if for any epsilon greater than 0, there's an N and a rational approximation q such that all terms after N are within epsilon of q.) I haven't actually done it, but I feel like proving the least upper bound property on these equivalence classes would be not too hard, but also not trivial like it is for Dedekind Cuts.
Professor what do you think of Prof NJ Wildberger's videos on UA-cam (from the channel 'Insights into Mathematics') where he claims that there is no rigorous basis for the definition of the real numbers and limits, and that we should focus on working with mathematics that does not involve infinitely long and complex sequences (like the expansion of pi) like that of the rational numbers? I am very curious.
Nice video! After watching your video about Dedekind cuts, I’ve spent the summer playing with them. I’ve managed to find and prove DCs for any rational number, any real solution to a quadratic equation ( ex: sqrt(2) = { x € Q | x
Thank you very much for this video Professor. Please, I know I am a nuisance as I have already asked you, could you tell us something on the hyperreal numbers ? Sorry to bother.
Man you Make MATHS so interesting
Keep going
Today I learned that I know more digits of pi than Dr. PI-yam! Nice video as always :)
@0:50 Diving right into that Cauchy sequence, love it haha
I think the definition of the real numbers that is both beautiful and works with decimal expansions is that real numbers are equivalence classes of sequences of rational numbers with finite limits, where two sequences are equivalent if they get and stay arbitrarily close together. (And they have finite limits, without using real numbers, if for any epsilon greater than 0, there's an N and a rational approximation q such that all terms after N are within epsilon of q.)
I haven't actually done it, but I feel like proving the least upper bound property on these equivalence classes would be not too hard, but also not trivial like it is for Dedekind Cuts.
There is a video on it on my playlist :)
Great video!
This is essentially the Cauchy sequence construction of the Reals
No that’s something else
@3:14 should 10^n be 10^(n-1) though? coincidental timestamp as well very nice
4:18 BAM!! very nice video it was very interesting :)
11:19 - But in the end it's "Doch! Doch! Doch! ...", 0.999... is in fact equal to 1 😊
Please make a video on ramanujans pi formula
Professor what do you think of Prof NJ Wildberger's videos on UA-cam (from the channel 'Insights into Mathematics') where he claims that there is no rigorous basis for the definition of the real numbers and limits, and that we should focus on working with mathematics that does not involve infinitely long and complex sequences (like the expansion of pi) like that of the rational numbers? I am very curious.
Nonsense lol
Does "a digit between zero and nine" not exclude zero and nine themselves?
Inclusive
When you said that we do not know right now if all real numbers have a decimal expansion, what did you mean by real numbers?
What I meant was real numbers as we know them, like pi or e or zeta(3). But of course after this video every real number has a decimal expansion
Nice video! After watching your video about Dedekind cuts, I’ve spent the summer playing with them. I’ve managed to find and prove DCs for any rational number, any real solution to a quadratic equation ( ex: sqrt(2) = { x € Q | x
we can also say if there is no real number between 2 real numbers eg(0.9999.. &1) ,then they both are same ,paradox solved?reply
Thank you very much for this video Professor. Please, I know I am a nuisance as I have already asked you, could you tell us something on the hyperreal numbers ? Sorry to bother.
I'm Indian hii