Mathematics Thread

[pmborg]

9 minutes ago, Lisias said:

In which Set? In Natural, Integers and Real Sets, there's no such thing as 0,99999(9)!!

However, on the Irrational Domain, given f(x) = x , Limit(f(x)) when x approaches 1 is considered 1.

Almost there...

Assuming X= 0.999(9)   (The repeating decimal continues infinitely.)

If we calculate:

10x - x = 9.999(9) - 0.999(9)

9x = 9

x = 9/9

x = 1

 

Edited February 11, 2020 by pmborg

[Lisias]

1 hour ago, pmborg said:

Almost there...

Assuming X= 0.999(9)   (The repeating decimal continues infinitely.)

If we calculate:

10x - x = 9.999(9) - 0.999(9)

9x = 9

x = 9/9

x = 1

I think there's something fishy... 0.9999(9) is a Rational number that we got by 9 * 0.1111(1) , being this last equivalent to 1/9.

So 0.99999(9) is 9 * 1/9, or 9/9, or 1.

0.9999(9) is already one since the beggining!  So it's not considered 1, it is 1.

source: https://en.wikipedia.org/wiki/0.999...

 

Edited February 12, 2020 by Lisias
better choose of words

[K^2]

9 hours ago, pmborg said:

Almost there...

Assuming X= 0.999(9)   (The repeating decimal continues infinitely.)

If we calculate:

10x - x = 9.999(9) - 0.999(9)

9x = 9

x = 9/9

x = 1

 

Fun trick. Lets see what else we can do with this. Lets say we write 9s all the way out the other way.

x = ...9999999.0

10x = ...9999990.0

x - 10x = 9

x = -1

Uh......

 

There's a reason why you have to talk about limit of a series here. 0.9999(9) represents a series with sequence of partial sums {0.9, 0.99, 0.999, 0.9999, ...}. This sequence has a limit and converges to 1. It's fairly easy to prove. On the other hand, the sequence {9, 99, 999, 9999, ...} does not have a limit. Or otherwise is sometimes said to have an infinite limit. And the moment I trick you into doing algebra with something that doesn't have a limit, bad things happen and nonsensical results are obtained. So in order to do 10x - x in the first place, at a minimum, you must prove that limit exists and that it is finite. And the easiest way to do that is just to prove that 1 is that limit, in which case, you don't need to do the mathematical trick in the first place.

Don't get me wrong, it's still a useful trick, but only for cases where you can tell right away that the series converges and just want to express it as a nice rational number. For example, if I asked you to express 0.123123(123) as a fraction, you could make a quick deduction that this is, indeed, a finite number, but you probably wouldn't be able to just guess what that number is without applying the above method.

P.S. I don't mean to be too negative here. What I'm trying to say is that any time you have infinite amount of anything, you have to be careful with it. Infinities are weird.

Edited February 12, 2020 by K^2

[Lisias]

4 hours ago, K^2 said:

P.S. I don't mean to be too negative here. What I'm trying to say is that any time you have infinite amount of anything, you have to be careful with it. Infinities are weird.

When I as a kid, I used to think that Rational and Irrational numbers have this name exactly by this!

Only at college is that I understand that Rational are Real numbers that you can get by Ratios (divisions), and not because they are reasonable.

[kerbiloid]

... by it's still a puzzle, what does the "Rational Rose" mean.

[Nikolai]

6 hours ago, Lisias said:

Only at college is that I understand that Rational are Real numbers that you can get by Ratios (divisions), and not because they are reasonable.

Yup.  And there's a reason "rational" (able-to-be-expressed-as-an-integer-ratio) and "reasonable" are synonymous in conversational speech.

Back in ancient Greece, the Pythagorean cult held to some interesting ideas.  They refrained from eating beans, for example, because they thought that humans were made of the same stuff.  And they held that any number -- any number -- could be expressed as the ratio of two integers.

It's fairly simple to prove that the square root of two cannot be.  This could well have been scandalous.  Imagine giving the impression that one could look forever and never find an "irrational" number, only to find that there was an irrational number with an extremely simple depiction -- the diagonal of a unit square.

The reaction of the Pythagoreans was to attempt to keep the proof secret.  This kind of behavior was -- well -- irrational.

[K^2]

8 hours ago, Nikolai said:

And they held that any number -- any number -- could be expressed as the ratio of two integers.

Αὐτὸς ἔφα.

Edit: To be fair, complete and splitting fields weren't remotely a concept back then. There is no specific reason in mathematics that square root of two should be a number at all. If we don't require that limit of any sequence of numbers is a number itself, then irrational numbers can simply not be a thing. Of course, I doubt the Pythagoreans would feel any better about someone telling them that there is simply no such number that marks the measure of a diagonal of a unit square.

Edited February 13, 2020 by K^2

[Nikolai]

13 hours ago, K^2 said:

I doubt the Pythagoreans would feel any better about someone telling them that there is simply no such number that marks the measure of a diagonal of a unit square.

Indeed not.  I doubt many people would feel any better about that now, even though our mathematical concepts are more refined.

"Every number is rational."

"What about this length?"

"That length can't have a number assigned to it."

"But I can see it!"

... et cetera, et cetera.  Lay folk seem to have a concept that mathematics should be useful, not merely consistent, and as such have little patience for logical abstraction that remains solid even outside of things their senses and intuition tell them that math should be able to handle.  (Witness those who insist that there's no point to so-called "imaginary" numbers.)

[K^2]

11 hours ago, Nikolai said:

Lay folk seem to have a concept that mathematics should be useful, not merely consistent, and as such have little patience for logical abstraction that remains solid even outside of things their senses and intuition tell them that math should be able to handle.

In contrast, I've met pure mathematicians who got genuinely, though not greatly, upset that there is a practical application to something they're working on.  

[έķ νίĻĻάίή]

A squared+ B squared = C squared.

Sin 45 = 0.85

[kerbiloid]

And D squared.

Spoiler