Real Numbers as the set of Dedekind Cuts
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- เผยแพร่เมื่อ 26 ส.ค. 2024
- We define the set of real numbers to be the collection of Dedekind cuts. We give an example of a Dedekind cut that corresponds to an irrational number.
#mikedabkowski, #mikethemathematician, #profdabkowski, #realanalysis
I think Terence Tao , constructed Real Number set perfectly in his book , I really love that , maybe it's purely algebraic but it's SO strong !
@AmirErfanian Yes Terry is one of a kind!
OK...Uncle Doug here...explain to me how you can write so straight across the screen??? I mean as a retired engineer, I like straight lines. However, for the life of me, I can't even sign my name anymore without trailing off the page…
Thank goodness you aren't designing conveyer belts that trail off! Did you know that real numbers are really defined as Dedekind cuts?
No, but when I order a Easter ham I specify a Dedekind cut in stead of spiral cut. @@mikethemathematician
Great stuff mate appreciate the effort!
Thanks @LeonTagleLB
Great video!
What I don’t understand is how dedekind cuts create the real numbers if they are subsets. If you define a set a a collection of subsets, and those subsets only contain rationals, how to the reals arise? Are the reals a set of all supremums of the dedekind cuts?
I was also wondering how we can create a dedekind cut for the cube root of two before having defined what a cube root means in the reals.
@obikenobi3629 Great question. In every construction of the real numbers, the definition is a bit surprising. In the Cauchy sequence definition, real numbers are defined as Cauchy sequence of rationals modulo Cauchy sequences converging to zero. This sequence and all of its representatives can be thought of as a subset of Q as well!
excellent
I’m glad you liked the video!
Great video
Thank you!
Could you make a video with the construction of the dedekind cut correspondent to pi?
Sure thing!
is it possible to construct a unique dedekind cut for an uncomputable number?
Yes. Let's take for example pi. You can define a subset of the rational number such as every element of the set is less than any partial sum of a series that converges to pi. Then, you have defined a Dedekind cut that represents pi.
where are my irrational numbers?
@princez2835 Every irrational number can be realized as a Dedekind cut! I will add more example videos on this, but most of the famous irrational numbers can be represented by a convergent infinite series (convergent needs to be properly defined, but the Least Upper Bound Principle will help us), and this representation will help us build the cut.
@@mikethemathematician thanks! but i have another question. if real numbers are subsets of the rational numbers, than does it mean that all the real numbers are inside the rational numbers? It doesn't make sense since rational numbers do not contain irrational numbers.
and are there any examples of the subset of Q is not bounded above? I think real numbers are not bounded above@@mikethemathematician
@princez2835 Great question! The integers are a subset of Q that is not bounded above.
@princez2835 Good question! If we look at the set of rational numbers {p: p^3 < 2} (this is a Dedekind cut), then we can identify this set with its least upper bound which happens to be 2^{1/3}, which is an irrational number!
Likewise, for any rational number r, we can identify the Dedekind cut {p: p< r} with its least upper bound of r.
There is certainly a lot to unpack here, but you got it @princez2835!