What is your field/career?

[Dreylin]

Dreylin, what are all the pretty lines? And I think you have a threadwinning meme at the bottom. 

Geology humour, it Rocks!  

This is a seismic section showing a vertical slice through the earth; the lines are the rock layers. Oil company Geologists use maps like this and information derived from it to determine where to drill for oil.

For comparison, here's the Lulworth Crumple, an outcrop in SW England that contributed to the development of the theory of tectonics:

[scooter]

Fun thread. I was actually just thinking some sort of demographic survey of RB might be interesting. I expected a steady stream of CS/Chem/Math responses, but it looks like there is still some variety.

1) What degrees and diplomas do you have?
2) Is there anything of note you specialized in during your education?
3) What is your current job and/or your career?
4) Is there anything of note you specialize in at your job?
5) Are there any past jobs you've done that are of interest?
6) What is your name, address, and credit card PIN?

1) BS in Computer Science
2) Not really
3) Software Developer
4) Kinda. It's a mid-sized marketing company, so part of that is my work often ends up at large public events. If you've ever gone to a major US auto show (especially Detroit, Chicago, NY, or LA), there's a decent chance you've seen things I've worked on, so that's fun I guess.
5) When I was 17-18ish just starting college and looking for any sort of job prior to starting college, I knew someone who owned a "mess removal" business. Long story short, I spent the better part of two summers doing trash-outs of previously foreclosed homes. So that was good for some bizarre stories.
6) hunter2

[darrelljs]

Fun thread.

+1  .  Mr. Cairo's post made me go back to my bookshelf, I went through a Minoan/Mycenaean phase, had three books and will probably re-read my favorite of the lot...

1) What degrees and diplomas do you have?
2) Is there anything of note you specialized in during your education?
3) What is your current job and/or your career?
4) Is there anything of note you specialize in at your job?
5) Are there any past jobs you've done that are of interest?

1) MS in Electrical Engineering
2) Wireless communications
3) System Architect working in public safety (mostly focused on the edge devices)
4) Well, I did participate in the design and standardization of a few wireless protocols.  I'm at 24 patents as a result...
5) Lived in Malaysia for three years and still get to travel to Asia a lot (to the tune of 600K frequent flyer miles last year).

Darrell

[darrelljs]

Adrien, if -1 + ∞ = ∞, but N - (N - 1) = 1, does 1 = 0?

There are many ways of answering this saying no, and one pointless answer that says yes. Which one do you want ?

Curiosity is going to make me ask for the pointless answer .  I still recall a professor explain how there are degrees of infinity, using the simple example that inf/inf = 2 where the numerator was:

2* and the denomiator was

Darrell

[AdrienIer]

Unfortunately I can't see what's after the "2*" and after "was"...

The pointless answer is that modulo 1 all numbers are equal, therefore 0=1 in this particular circumstance. But taking numbers modulo 1 is entirely pointless, so they're only equal in very limited circumstances.
The actual answer, of course, is that inifinity is not a number, and (N+1) - N = 1 is only true for N a number. *insert gif about infinity saying "I am not a number"*

There definitely are different degrees of infinity. What's funny is that twice an infinity is still the same infinity. n times an infinity as well. But there are different degrees of infinity because while the number of Integers is the same infinity as the number of Rationals (the numbers written a/b with a and b integers), it's not the same as the number of Real numbers. There are infinitely more Real numbers than there are Rational numbers, while there are the same number of Rational numbers as there are Integers.

[suboptimal]

suboptimal, I never thought to meet a food or beverage flavorist. What an unusual and interesting occupation. Have soft drinks lost their fizz for you yet?

No, not yet.  You'd be surprised at how many different tasting orange and grape sodas are out there, though.

[thestick]

Ooh, goody, a census! Let the enumeration begin!

1) What degrees and diplomas do you have?
2) Is there anything of note you specialized in during your education?
3) What is your current job and/or your career?
4) Is there anything of note you specialize in at your job?
5) Are there any past jobs you've done that are of interest?
6) What is your name, address, and credit card PIN?

1. Pursuing a bachelor's in economics. I am a dirty capitalist pig, and way too young to be on this site.
2. Going down the quantitative/Ph.D prep/poor man's statistics track with more math and statistical theory than needed.
3. Student, but have been working as a research assistant in behavioral macrofinance. Most of it's just econometrics with unusual datasets, though.
4. Know how to code by the abysmal standards of economics/statistics. Trying to 'specialize' in data analysis de-spaghettification.
5. Somehow ended up licensed to practice real estate at a barely legal age. Never actually practiced, though.
6. Robert Maynard Marshall, 0 Homo Economicus Drive, 666.

[Coeurva]

There definitely are different degrees of infinity. What's funny is that twice an infinity is still the same infinity. n times an infinity as well. But there are different degrees of infinity because while the number of Integers is the same infinity as the number of Rationals (the numbers written a/b with a and b integers), it's not the same as the number of Real numbers. There are infinitely more Real numbers than there are Rational numbers, while there are the same number of Rational numbers as there are Integers.

If anyone cares besides those who obviously already know it -- here's the proof of that as "classically" taught since Cantor, and whose amusing paraphrase was first devised by Hilbert in his excellent headwear. First off, the definitions we need:

We define two sets A and B as having the same cardinality ("size") when there exists a bijective function f : A -> B.

"Bijective" means both injective and surjective. "Injective" means that f(x) = f(y) => x = y -- that is to say, no two members of A are mapped to the same element of B, i.e. no element of B is mapped to more than once. "Surjective" means that every element of B is mapped to at least once. Bijective, therefore, means that every element of B is mapped to exactly once. That's all we need.

To use a practical example, if all the students of Linear Algebra 101 enter the auditorium and sit down, and everyone finds exactly one seat and not one remains vacant, then we map every student to a unique chair and thus the sets of chairs and students have the same cardinality. With finite sets, this is an intuitive definition; it becomes more amusing if we hold infinite sets to the same standards.

Let's define the cardinality of N, the set of all natural numbers, of which there are infinitely many (precisely because every natural number, by definition, has a successor), as ℵ_0 ("aleph-null") or, for this post and because we lack inline LaTeX, just ℵ -- since the Greek alphabet did not suffice to suggest this craft as magical, or perhaps due to Kabbalistic influence. The name isn't too important.

Let's assume, in boundless optimism, that we have as many attendent students in Linear Algebra 101 as there are natural numbers, and an auditorium that holds just as many seats, i.e. ℵ students on ℵ seats, so that student No. x is mapped to the seat No. f(x) := x. The students enter the auditorium and sit down, and obviously, everyone finds their place.

One student, however, is always late. Let's call him Zero. This guy barges in and demands a seat. But isn't the auditorium full already? -- No issue, as it turns out: we simply shift everyone one place to the right. More formally, we define a map

f(0) := 1,
f(x) := x + 1 for all x > 0.

This is a bijective mapping from N_0 (the set of natural numbers including zero) to N (the set of natural numbers not including zero). These sets, therefore, have the same cardinality.

What's also always late is public transportation. The tramway arrives fifteen minutes later, which coincidentally holds just as many students wanting to attend Linear Algebra 101 as are already in the auditorium. Let's call them -1, -2, -3, etc. ("Obviously", there are as many negative as positive numbers; formally, you'd look at the bijective function f : N -> -N; f(x) := -x.)

Can they all find a seat?

Yes.

f(x) := 2x + 1 for all x >= 0
f(x) := -2x for all x < 0

Conclusion: There are as many positive integers as there are integers total (including zero or not).

Because public transportation is so notoriously unreliable, fifteen minutes after that, there arrive ℵ tramways each holding ℵ students. (When constructing the rational numbers from the integers, you can choose any integer for the numerator and just as many for the denominator.) Can they all find a seat?

This is a considerably more surprising answer: Yes! To help visualize the solution, arrange them in a grid like this:

0/1 0/2 0/3 0/4 0/5 0/6 ...
1/1 1/2 1/3 1/4 1/5 1/6 ...
2/1 2/2 2/3 2/4 2/5 2/6 ...
3/1 3/2 3/3 3/4 3/5 3/6 ...
4/1 4/2 4/3 4/4 4/5 4/6 ...
5/1 5/2 5/3 5/4 5/5 5/6 ...
6/1 6/2 6/3 6/4 6/5 6/6 ...

Consider, for instance,
f(m/n) := (n+m)^2 + (n-m)

(Start in the top-left corner, and go along the diagonals while counting the numbers -- skipping all fractions that could be simplified further, like 6/2 or 3/6.)

Now for why it doesn't work with real numbers.

Assume we could have a countable (this word designates any set of cardinality ℵ 0) list of all real numbers (in any order) between 0 and 1 (non-inclusive), specifically including non-periodical fractions, like this:

Code:

f(1) = 0.1324959636....
f(2) = 0.2493598029....
f(3) = 0.3423892363....
f(4) = 0.2359236923....
[and so forth]

Define a number z := 0.z_1 z_2 z_3 z_4 z_5...., where the z_n are the digits of said number. Now define

z_n := anything you want! -- except the n-th digit of f(n).

Is z on the list? It cannot be. Consider if z were equal to f(n). But z is -- by definition! -- distinct from f(n) in at least one digit past the decimal point, namely, at z_n. However, z is a real number between 0 and 1, so it should be on the list if these were countable. Conclusion: They are not.

The real interval (0, 1) "is larger" (has a larger cardinality) than the set of rational numbers. (Because (0,1) is a mere subset of the real numbers, obviously the same applies to all the real numbers.) We could call this cardinality ℵ 1, and one could naturally ask "how much bigger" this is than ℵ 0 -- or rather, we could pose the question like this: are there any cardinal numbers between ℵ 0 and ℵ 1? This is left as an exercise to the reader.

Hush, Adrien

[Ichabod]

After reading Couerva's post, I'm starting to believe the bathroom inscription.

About the novel I'm writing, it's a coming of age story about a boy that changes schools. It's very realist in nature, and it's written in third person narration, as opposed to first person, which I think most coming of age stories are. I've been working on it for about 5 years, but not in a consecutive or properly organized manner.

I would love to share the bits I have ready, but it's all in portuguese.

[darrelljs]

Unfortunately I can't see what's after the "2*" and after "was"...

The pointless answer is that modulo 1 all numbers are equal, therefore 0=1 in this particular circumstance. But taking numbers modulo 1 is entirely pointless, so they're only equal in very limited circumstances.
The actual answer, of course, is that inifinity is not a number, and (N+1) - N = 1 is only true for N a number. *insert gif about infinity saying "I am not a number"*

There definitely are different degrees of infinity. What's funny is that twice an infinity is still the same infinity. n times an infinity as well. But there are different degrees of infinity because while the number of Integers is the same infinity as the number of Rationals (the numbers written a/b with a and b integers), it's not the same as the number of Real numbers. There are infinitely more Real numbers than there are Rational numbers, while there are the same number of Rational numbers as there are Integers.

Ponders.  Ponders some more.  Decides to ponder more later .

Darrell

P.S. What was after the 2* was SUM(n), n = 1...inf