Finding Dense Sets: Solving the ±m/2100 Conundrum in Elementary Analysis

[ThatOneGuy45]

Hi, I am taking an intro. to elementary analysis class and so far our class has gone over sups/infs, the axiom stuff, archimedean property and now we are on dense sets. I've been stuck on this problem for a really long time trying to find clues on how to do it online. It goes as:

Are the numbers of the form
±m/2¹⁰⁰

for m \in N dense? What is the length of the largest interval that contains no such number?

The book our class is using is Elementary Real Analysis 2nd ed. by bruckner/thomson. The book is online somewhere if needed. But anyways, I am really lost and a really helpful hint would be most appreciated. Thank you!

 

[voko]

What is the definition of "dense"?

 

[ThatOneGuy45]

In my book, a set of E of R is said to be dense(or dense in R) if every interval (a,b) contains a point of E.

 

[voko]

Consider this interval: (0, 1/2¹⁰⁰⁰). Does it contain a point from your set?

 

[ThatOneGuy45]

I am still fairly new with all this analysis stuff. For the interval, what made you choose it? I don't see how it would be in the set entirely. I feel like an idiot in this class.

 

[voko]

The interval that I chose or the interval (a, b) from the definition does not need to be in the set. Rather, it is the set must be such that whatever interval (a, b) is selected, there will be at least one point in that set that will also be in that interval. Your set consists of points that are separated by finite, albeit very small, distances from one another, so it is possible to select an interval - infinitely many intervals in fact - that will be smaller than the distance between any two points in your set, so they will not contain any point of the set. The set is not dense.

A dense set in R is, for example, that of rational numbers, because in any given interval there are infinitely many of them.

 

[HallsofIvy]

I am still fairly new with all this analysis stuff. For the interval, what made you choose it? I don't see how it would be in the set entirely. I feel like an idiot in this class.

You said "numbers of the form ±m/2¹⁰⁰" vokD's number, 1/2¹⁰⁰ is of that form with m= 1 so it certainly is in that set. In fact, it is the smallest positive number in that set. The next smaller number in the set is -1/2¹⁰⁰ and the distance between them 1/2¹⁰⁰- (-1/2¹⁰⁰)= 2/2¹⁰⁰= 1/2⁹⁹ and is the length of the largest interval of real numbers containing no member of the set. That much is not "analysis"- it's basic arithmetic.

 

[voko]

vokD's number, 1/2¹⁰⁰ is of that form with m= 1 so it certainly is in that set. In fact, it is the smallest positive number in that set.

My number was 1/2¹⁰⁰⁰