Real Numbers as the set of Dedekind Cuts

Рет қаралды 3,504

10 ай бұрын

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

Пікірлер: 22

@obikenobi3629 Ай бұрын

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.

@mikethemathematician Ай бұрын

@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!

@LeonTagleLB 4 ай бұрын

Great stuff mate appreciate the effort!

@mikethemathematician 4 ай бұрын

Thanks @LeonTagleLB

@thisguyisyummy 6 ай бұрын

is it possible to construct a unique dedekind cut for an uncomputable number?

@AmirErfanian 3 ай бұрын

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 !

@mikethemathematician 3 ай бұрын

@AmirErfanian Yes Terry is one of a kind!

@youtubepooppismo5284 10 ай бұрын

Great video

@mikethemathematician 9 ай бұрын

Thank you!

@Alkis05 10 ай бұрын

Could you make a video with the construction of the dedekind cut correspondent to pi?

@mikethemathematician 10 ай бұрын

Sure thing!

@aoy3142 7 ай бұрын

excellent

@mikethemathematician 7 ай бұрын

I’m glad you liked the video!

@user-bd2bn4wj2g 5 ай бұрын

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…

@mikethemathematician 5 ай бұрын

Thank goodness you aren't designing conveyer belts that trail off! Did you know that real numbers are really defined as Dedekind cuts?

@user-bd2bn4wj2g 5 ай бұрын

No, but when I order a Easter ham I specify a Dedekind cut in stead of spiral cut. @@mikethemathematician

@princez2835 7 ай бұрын

where are my irrational numbers?

@mikethemathematician 7 ай бұрын

@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.

@princez2835 7 ай бұрын

@@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.

@princez2835 7 ай бұрын

and are there any examples of the subset of Q is not bounded above? I think real numbers are not bounded above@@mikethemathematician

@mikethemathematician 7 ай бұрын

@princez2835 Great question! The integers are a subset of Q that is not bounded above.

@mikethemathematician 7 ай бұрын

@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!