The Black Vault

Frrosty your logic is thinking of the fractional part of the whole number, which is not growing it is stuck in a cycle of greater than or lessor than, with each progressive decimal point. That cycle may continue forever, right of the decimal point, but the fact the next higher whole number( pi forinstance) is known to be 4 left of the decimal point, it is not infinity where no whole number is greater.

frrostedman wrote:You're the expert so I will bow out on the point, but, technically speaking if a number increases in value and never stops, that means it grows infinitely and that can equal nothing but infinity.

Ok I'll stop. You win. I begrudgingly hand you the trophy.

My agenda is not to win but to inform. I may be shot for saying this but your questions and challenges about this are better than I ever had the pleasure of witnessing while teaching calculus.

As for the point you raised, what if a variable increases in value but at an ever slowing and smaller rate? (I prefer to say a variable increases to a number increasing because it sounds like a number is changing when numbers don't change exactly.)

For example, this sequence of numbers is always increasing:
{1, 1.3, 1.33, 1.333, 1.3333, 1.33333, ....}

Yet all terms are bound by 1.4.

So that sequence does not diverge to infinity.

There's a theorem in calculus that says that if a sequence is bounded (in this case, it is bounded by 1.4) and always increasing (ok, technically non-decreasing), then it must converge to some real number.

In this case, the "limit" of the sequence is what's known as the least upper bound of the sequence. 1.4 is an upper bound but not the least upper bound. 1.3333333.....(repeating) is the least upper bound for the provided sequence.

Just because I'm stubborn doesn't alone make me right. I always tried to encourage skepticism because it is when skepticism is addressed and satisfied (one way or the other) that genuine learning can occur. I think that's a general principle with skepticism, even in matters of faith. If the skepticism is not addressed then genuine learning will not occur, in my experience.

I think math is worthy of study for, among other things, the new avenues of thought that can develop. But when it is learned, I think one should be able to abandon mathematical thinking when one needs to.

greeney2 wrote:Frrosty your logic is thinking of the fractional part of the whole number, which is not growing it is stuck in a cycle of greater than or lessor than, with each progressive decimal point. That cycle may continue forever, right of the decimal point, but the fact the next higher whole number( pi forinstance) is known to be 4 left of the decimal point, it is not infinity where no whole number is greater.

I should drop it because I am not confused and I do understand that because logic says (in my example) the number can never be greater than 1.34, or even reach that value, it cannot be equal to infinity.

However, my point was, infinity exists between any 2 rational numbers because, like in my example, the fact that the decimal continues to move to the right, that means the number is increasing in value. And can increase in value until the end of time. Infinity.

at1with0 wrote:My agenda is not to win but to inform. I may be shot for saying this but your questions and challenges about this are better than I ever had the pleasure of witnessing while teaching calculus.

I don't know if I should be embarassed to admit this or not, but, that's one of the best compliments I've received in a very, very long time. Thanks man.

[quote]As for the point you raised, what if a variable increases in value but at an ever slowing and smaller rate? (I prefer to say a variable increases to a number increasing because it sounds like a number is changing when numbers don't change exactly.)

True, the increase exponentially smaller and smaller but... it still qualifies as an increase. In fact, it's a HUGE increase each time, considering the fact that an infinite number of smaller numbers than that of the increase value itself, exist.

You're welcome dude.
And it's true.

How can it increase to even be above 1.4 though, let alone go all the way to infinity?

at1with0 wrote:You're welcome dude.
And it's true.

How can it increase to even be above 1.4 though, let alone go all the way to infinity?

It can't. But infinity doesn't have to be above 1.4 ... infinity, we must admit, somehow exists between any 2 numbers you can think of. Just like, as I said before, an infinite number of universes can exist in a grain of sand.

Infinity doesn't have to be above 1.4, just like infinity is not above (infinity + 1), which in and of itself is a tiny number when we consider that there is an infinite number of numbers higher than (infinity + 1), i.e. (infinity + infinity)

One way to phrase that and make it true is that there are infinitely many numbers between any two distinct numbers.


You might find cardinal arithmetic amusing.
http://en.wikipedia.org/wiki/Cardinal_n ... arithmetic
The article isn't the greatest though...

That mathematical traveler book is much better.

There are infinitely many fractions between 0 and 1

Pi/4 is a fraction heehee

at1with0 wrote:Pi/4 is a fraction heehee

http://www.numberworld.org/misc_runs/pi-5t/details.html

5 Trillion Digits of Pi

That was interesting how 39 digits of Pi are theoretically all you'd ever need...

They use Pi to test the accuracy and speed of computers as well as quality of heat dissipation as it is brutal on the cpu.