Subgroups of (R, +) are Dense or Cyclic

I prove all subgroups of additive group of real numbers are either cyclic or dense. Then, we use that to show the set of rationals of the form 2^m 3^n is dense in [0, infty).
A more elementary proof of the fact that rationals of the form 2^m 3^n is dense in [0, infty).: • A Dense Set in Positiv...
Fractional Part of n alpha is Dense in [0, 1]: • Fractional Part of n a...

[@zeggwaghismail827]

Very well done. Density is on trend everywhere now!

[@DrEbrahimian]

Interesting! I wonder why that is!

[@derkritiker9434]

I think in 9:30 the ineaquality should be a

[@DrEbrahimian]

True! Thanks for catching that!

[@ilemt0923]

I'm interested in this. Could you provide some references or materials about this topic?

[@DrEbrahimian]

The proof is presented in the video. Which topic interests you, especially?

[@acrommclain5233]

I have a question about a slight generalization.
Let $A = \{(2^{n}3^{m},5^{n}7^{m}):\space n,m\in\mathbb{Z}\}\subset\mathbb{R}^{2}$
What is $D(A)$ (the set of the accumulation points of A)?
Is A dense in $\mathbb{R}^{2}?
What if $A = \{(2^{n}3^{m},5^{n}7^{m}):\space n,m\in\mathbb{Z}\spacen\leq0\leqm\}\subset\mathbb{R}^{2}$?

[@DrEbrahimian]

This is a very interesting quetsion. I will give it some thought.

[@acrommclain5233]

@@DrEbrahimian For the big set the answer is not to difficult. I will not spoil it though if you want to come up with it. For the n

[@MathEnthusiast-od8yu]

great video. Love it

[@nicolascamargo8339]

Excelente video

[@srishtinegi9930]

So If it is dense it must not be cyclic, right?

[@DrEbrahimian]

Yes.

[@adfr1806]

I dont understand what happens at 10:40, you just showed the infimum belongs to G not that there is an elements smaller than epsilon... what am I missing

[@DrEbrahimian]

That turns the problem into the previous case where there is the smallest positive element in G.

[@jiaweihuo6951]

Does it mean that for any different numbers $a$ and $b$ of $\mathbb{R}$, then the additive group of $a,b$ is dense in $\mathhb R$?

[@DrEbrahimian]

Good question. Not necessarily. For example the additive group generated by any two rationals is cyclic, and hence not dense.

[@jiaweihuo6951]

​@@DrEbrahimian Thank you! I have another question. Is the additive group generated by two differential irrational (for example, $sqrt 2$ and $\sqrt 3$) dense in $\mathbb R$?

[@DrEbrahimian]

Not necessarily. This is only true if the two numbers are linearly independent over Q. For example the additive group generated by 2sqrt(2) and 3sqrt(2) is cyclic.