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[Reelya]

How could you possibly have a cycle of infinite length?

Doesn't that mean it comes back around?

Well the literature about the Collatz problem refers to such non-settling patterns as a cycle so what I really meant was a non-repeating pattern that never settles, so it is a "cycle" in that it constantly cycles through new numbers, in the sense of chaos theory - cyclic patterns that never quite repeat. So even if you construct an inductive proof that give "n" you can prove the shortest cycle must be n+1, then you can prove that it must be n+2 and so on, you can't rule out chaotic attractors, that have an infinite number of unique states.

And you can in fact have infinite sequences of numbers that don't hit all numbers. e.g. the doubling pattern 1,2,4,8,16,32...and so on ... What percentage of all natural numbers does this pattern contain? 0% How many numbers does it contain? infinity. How many natural numbers are there? Infinity. Can you map the pattern one-to-one with all natural numbers? Yes. Therefore infinity is 0% of infinity.

[MagmaMcFry]

You can have a cycle of numbers of infinite length that hits any percentage of real numbers that exist, including 0%. Yes, an inifinite list of numbers can in fact take up 0% of all the numbers.

Oh, that statement greatly depends on the nonzero finite measure you use to define how a set contains a certain 'percentage' of real numbers. There isn't any canonical such measure over the real numbers, there isn't even a translation-invariant measure that is nonzero and finite. And what's worse, in every measure of the real numbers dominated by the usual Lebesgue measure, any countable set has measure zero, which means that for a cycle of infinite length to hit any "percentage" of real numbers instead of just always 0%, you need a truly contrived definition of "percentage".

Well the literature about the Collatz problem refers to such non-settling patterns as a cycle so what I really meant was a non-repeating pattern that never settles, so it is a "cycle" in that it constantly cycles through new numbers, in the sense of chaos theory - cyclic patterns that never quite repeat. So even if you construct an inductive proof that give "n" you can prove the shortest cycle must be n+1, then you can prove that it must be n+2 and so on, you can't rule out chaotic attractors, that have an infinite number of unique states.

The word "cyclic" really doesn't make sense in the case of nonterminating aperiodic Collatz sequences. Collatz sequences either go to 1, or they become periodic, or they simply go off to infinity, nothing else.

[Reelya]

i was really pointing out that having infinite things doesn't mean you have all the things. you can fit an infinite number of infinite patterns in the real numbers without any of them overlapping.

[Parsely]

Spoiler (click to show/hide)

Uhhhhh...

1. y = a(b)^t
2. 50,000 = a(2)^90/20
3. log(50,000) = 90/20 * log(2a)
4. log(50,000)/(90/20) = log(2a)
  4.1. log(50,000)/(90/20) = log(2) + log(a)
5. [log(50,000)/(90/20)] - log(2) = log(a)
6. 10^{[log(50,000)/(90/20)] - log(2)} = 10^a
7. 5.535865894 = 10^a

Okay this isn't working. I'm clueless. How do I solve this?

[frostshotgg]

First of all, notation for logarithms is usually log_base(stuff). Just log(stuff) can either mean log_10 or log_e/ln depending on the context, so it's generally good form to just always write the base.

So there's a bunch of things that seem wrong. I'm not totally up to snuff on my log rules, but the math error in what you did looks to be between 5 and 6.  Why did you make 10^(log_10(a)) = 10^a? The 10^log_10 cancel out and you should just be left with a.

The other thing is... Why the hell are you even using logs? Your equation 2 is {constant = x * constant}. Just evaluate 2^(90/20) and be done with it.

Spoiler (click to show/hide)

The answer should be 2209.71.

[Culise]

While I strongly recommend you follow the above advice for being simpler (especially the advice on logarithm notation; I had some brief mental confusion before I realized you weren't actually trying to take base-10 logarithms of a base-2 exponentiation ), another minor note that sticks out to me and may come up again in another problem is how you moved from step 2 to step 3.  It looks like you took the logarithm of both sides in an attempt to simplify.  However, here's something to note: while
   log(B^C) = C * log(B),
it is absolutely not true in the general case that
   log(A * B^C) = C * log(A * B).

The reason is because A * B^C is not the same thing as (A*B)^C.  You need to work from the outside in.  The former in its simplest form is simply two things multiplied together, so you need to break up that multiplication first.  First, you need to split up the multiplication using the product rule:
   log(A*B) = log(A) + log(B)

Thus, what you would find is that...
   log(A * B^C)
   = log(A) + log(B^C) Product rule
   = log(A) + C * log(B) Power rule

Now, let's show the difference with the other equation.  If you were trying to handle (A*B)^C, the exponention is the outermost operation, so it needs to be broken first to get to the multiplicative nougat inside.
   log (A*B)^C
   = C * log(A*B) Power rule
   = C * (log A + log B) Product rule
   = C * log A + C * log B Multiplicative distribution

[Parsely]

First of all, notation for logarithms is usually log_base(stuff). Just log(stuff) can either mean log_10 or log_e/ln depending on the context, so it's generally good form to just always write the base.

So there's a bunch of things that seem wrong. I'm not totally up to snuff on my log rules, but the math error in what you did looks to be between 5 and 6.  Why did you make 10^(log_10(a)) = 10^a? The 10^log_10 cancel out and you should just be left with a.

The other thing is... Why the hell are you even using logs? Your equation 2 is {constant = x * constant}. Just evaluate 2^(90/20) and be done with it. 

It's always log₁₀ unless otherwise stated. On my TI-84 if I punch the log button, it shows "log(", and that stands for a base 10 log. If log₁₀ = log_e/ln then it doesn't really matter which one I write unless I need one over the other. I can see the benefit of being explicit though.

I didn't do it, because I didn't know that.

Like, do you think I'm just pretending that I don't know how to solve this equation, or what? Is it really necessary to put me down during your explanation? I'm not sure what you mean by "equation 2 is {constant = x * constant}". I'm interested in how you simplified things to "2^(90/20)". Could I trouble you to show the steps?

This is why I get so apprehensive about asking for help with math. I don't want to be demeaned just because I made a mistake... And then things just spiral down and get worse because I'm terrified of asking for help. Why is it NOT OKAY for me to need more explanation?

-snip-

Thank you so much for this! I see now what I messed up with the log rules. This is going in my notes.

Well, we know that 50000=a*2^90/20; ergo, a=50000*2^-90/20

You overthought this.

I'm sorry, could you explain this a little more? I see you inverted t, but I'm not sure how that allows you to switch a and y.

[Parsely]

I just took step 2 that you presented and divided both sides by 2^90/20, since it is a constant.

...

Oh.

Wow that IS dumb! I'm super dumb. Thanks for humoring me, everyone. :V

[Arx]

It's always log₁₀ unless otherwise stated.

At the level you're currently working at, yep. As you get further into academia, it increasingly means log_e(n). For instance, WolframAlpha I think will take log(n) as being log base e. It's wonderful, isn't it? Who needs standardisation, amirite.

[Parsely]

It's always log₁₀ unless otherwise stated.

At the level you're currently working at, yep. As you get further into academia, it increasingly means log_e(n). For instance, WolframAlpha I think will take log(n) as being log base e. It's wonderful, isn't it? Who needs standardisation, amirite.

Good to know!


New subject:
I'm taking Calculus 1 soon and I want to get a head start on it. Any recommendations on online resources that could help with that?

[frostshotgg]

It's always log₁₀ unless otherwise stated. On my TI-84 if I punch the log button, it shows "log(", and that stands for a base 10 log. If log₁₀ = log_e/ln then it doesn't really matter which one I write unless I need one over the other. I can see the benefit of being explicit though.

I didn't do it, because I didn't know that.

Like, do you think I'm just pretending that I don't know how to solve this equation, or what? Is it really necessary to put me down during your explanation? I'm not sure what you mean by "equation 2 is {constant = x * constant}". I'm interested in how you simplified things to "2^(90/20)". Could I trouble you to show the steps?

This is why I get so apprehensive about asking for help with math. I don't want to be demeaned just because I made a mistake... And then things just spiral down and get worse because I'm terrified of asking for help. Why is it NOT OKAY for me to need more explanation?

Like I said, and as Arx said. Log(stuff) might mean Log_10 at a high school level, but past that point it's not clear, so it's better to just always write it. And log_e IS ln, I didn't mean for it to look like log_e divided by ln.

As far as tone goes, I've found that asking questions about why people do specific stuff is a lot more effective at getting them to realize mistakes. Sorry if it came off as demeaning.

[da_nang]

As far as standards go, I've been taught that

base e - ln(x) (natural logarithm)
base 10 - lg(x) (Briggsian logarithm)
base 2 - lb(x) (binary logarithm)
specified base a - log_a(x)


log(x) is usually only used as a generic logarithm.

[TheDarkStar]

New subject:
I'm taking Calculus 1 soon and I want to get a head start on it. Any recommendations on online resources that could help with that?

This website is basically some university math teacher's notes for his classes that he posted online. It has notes/practice problems for four classes - Calc 1 all the way through differential equations (although it doesn't have linear algebra) and I've used it as a second resource if my actual teacher wasn't clear about something.

[Parsely]

Like I said, and as Arx said. Log(stuff) might mean Log_10 at a high school level, but past that point it's not clear, so it's better to just always write it. And log_e IS ln, I didn't mean for it to look like log_e divided by ln.

As far as tone goes, I've found that asking questions about why people do specific stuff is a lot more effective at getting them to realize mistakes. Sorry if it came off as demeaning.

College algebra, for me, actually. But yeah, I take your and Arx's point. Explicit is always better than implicit.

It's okay. I'm kind of prickly on the subject because of lots of awful prior experiences.

New subject:
I'm taking Calculus 1 soon and I want to get a head start on it. Any recommendations on online resources that could help with that?

This website is basically some university math teacher's notes for his classes that he posted online. It has notes/practice problems for four classes - Calc 1 all the way through differential equations (although it doesn't have linear algebra) and I've used it as a second resource if my actual teacher wasn't clear about something.

Very cool, thank you!

To add onto this, I found an open source calculus textbook (!!!). http://scholarworks.gvsu.edu/books/10/

[prefuzek]

So I just finished the first year of my mathematics degree, and I'm trying to figure out which major I want to do. I'm mostly looking at either pure math or combinatorics and optimization, and leaning towards the latter. I looked through the notes for an intro to combinatorics course and it looks super interesting.

My questions are: Why might someone prefer one over the other? What are the differences in learning styles/types of problems? Are there any other reasons to choose one or the other?