Decimal Expansions

Construction of Real Numbers using Decimal Expansions

In this video, I construct the real numbers using decimal expansions instead of Dedekind Cuts. This way is much more direct, but of course makes proving the least upper bound property a bit harder. Enjoy!

[jyotingoel]

Man you Make MATHS so interesting
Keep going

[DrWeselcouch]

Today I learned that I know more digits of pi than Dr. PI-yam! Nice video as always :)

[iabervon]

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.

[plaustrarius]

@0:50 Diving right into that Cauchy sequence, love it haha

[dp]

11:19 – But in the end it's "Doch! Doch! Doch! ...", 0.999... is in fact equal to 1 😊

[moshadj]

This is essentially the Cauchy sequence construction of the Reals

[plaustrarius]

@3:14 should 10^n be 10^(n-1) though? coincidental timestamp as well very nice

[dstigant]

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<0 or x^2 < 2 } ) and pi ( pi = { x€ Q | x<0 or there exists an integer N such that x^2 < 6(1 + 1/4 + 1/9+...+1/N^2}). I think I can see how to get -cuberoot(2), but I’m stuck on cuberoot(2). The difficulty is in the step where you have to show that if x € C then there exists a y > x which is also in C. In the case of sqrt(2) and other roots of quadratic equations, I used y=f(x) where f(x) is derived from the continued fraction for the number. For -cuberoot(2), I think you can find y=f(x) by using a newton’s method approach to finding the real root of x^3 +2=0. However, neither of these approaches will work with cuberoot(2) because there isn’t a nice CF for cuberoot(2) and because the NM approaches cuberoot(2) from above rather than below. Could you do a video about this?

[aviralsood]

Professor what do you think of Prof NJ Wildberger's videos on YouTube (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.

[jabahalder]

Please make a video on ramanujans pi formula

[Kdd]

4:18 BAM!! very nice video it was very interesting :)

[toaj]

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?

[Apollorion]

Does "a digit between zero and nine" not exclude zero and nine themselves?

[lowlifeuk]

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.

[kirangkumar]

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