The Language of Mathematics 1

[CajunCenturion]

Multiplying by 10 increases by 1 order of magnitude, not 1 order of magnitude plus .000...1.

That is correct. There is no addition of .000...1. But remember that 0.9999... is repeating infinitely before the multiplication and it is repeating just the same after the multiplication. The multiplication by 10 has no effect on the repeating non-terminating decimal portion of the value.
Additional reading
wikipedia
Ask Dr. Math Also, see references at bottom of page

You can have infinites within infinities. You an infinty +1, etc. You can have infinity x 2, which is an infinite series followed by another infinite series.

Yes, you can have infinities within infitinities. They are know as degrees of infinity. However, infinity, infinity + 1, and infinity * 2, are the same infinity, or to be more precise, all these infinities are of the same degree, or those sets of numbers have the same cardinality. If a 1-to-1 mapping can be established between two infinities, then the two infinities have the same cardinality. The cardinalilty is know by the Hebrew Letter Aleph ([ℵ])

The smallest infinity is the set of integers, or counting numbers, and has cardinality of [ℵ]₀ (Aleph-Null). The set of integers, odd integers, even integers, integers + 1, and integers * 2 all have the same cardinality because a 1-to-1 mapping can be established with the integers. The values on either side of the mapping do not have to be, and usually aren't the same, but as long as the 1-to-1 mapping exists, the size of the infinity is the same.

There are just as many even integers as there as integers. The mapping is E_n = I_n * 2 or conversely I_n = E_n [÷] 2
[tt]
Integer Even Integer
1 2
2 4
3 6
4 8[/tt]
For every integer there is exactly one corresponding even integer, and vice versa; for every even integer, there is exactly one corresponding integer. The values of corresponding entries are not equal, in this mapping, the even value is twice that of the integer value, but both sets still have the same number of terms. There is no integer that does not have corresponding even integer, nor is there an even integer that does not have a corresponding integer. That means that a 1-to-1 mapping exists, therefore the numbers of integers is same as the number of even integers, or mathematically speaking, both sets have the same cardinality, which is [ℵ]₀. When dealing with the cardinality of infinities, it is not the values of the terms that matter, it is the number of terms that matter.

Now, can we establish a 1-to-1 mapping between the integers and the real numbers. Let's start with the integer 1 which maps to the real number 1. What is the next highest real number, i.e., the real number that will map to the integer 2? It is unknown. No matter what answer is submitted, there is a real number halfway between that value and 1. Therefore, you cannot map the real numbers to the integers and since the real numbers is clearly the larger set, the real numbers have a higher degree of infinity than the integers. The set of real numbers has a cardinality of [ℵ]₁ (Aleph-One).

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[Stella740pl]

Tim, let's say, your question #2 would be off-topic in this forum.

 

[craigsboyd]

Well, I'm ready to leave just as soon as majority of Americans agree that the 1 == .999999...

Then I'll be adrift, in search of a nation where the majority of the population finds that notion as irrational as I do. Oh, and they have to have TVs with parental lock features, newspapers, and a good classic novel.

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[Stella740pl]

Oh, and they have to have TVs with parental lock features, newspapers, and a good classic novel.

You probably won't belive me, but there are many non-English speaking countries that have all this! ;-)

Then I'll be adrift, in search of a nation where the majority of the population finds that notion as irrational as I do.

And those, too, are aplenty! (But do they all have to be proficient in math to resist all the "proofs" of the opposite notion? Oh, you might be adrift for long...)

I guess, in many places people will agree that those values are equal for all practical purposes, as for computing, statistics, accounting, etc. - but not exactly equal, mathematically speaking.

But I don't want to get started on this.

 

[craigsboyd]

I guess, in many places people will agree that those values are equal for all practical purposes, as for computing, statistics, accounting, etc. - but not exactly equal, mathematically speaking.

That is precisely the camp I sit in. Mathematically I've heard that it is possible to turn a sphere inside out without breaking or creasing the surface...

http://en.wikipedia.org/wiki/Bernard_Morin

...and in a previous post I showed a link that clearly shows that 1 = 2, but that doesn't make either any more probable or the reality. It merely shows IMHO, that mathematics cannot be applied like this and still be logical.

Now, I am probably speaking from ignorance, because I must admit that a few of the arguments (not all of them mind you) that were given as proof in the posts above went right over my head. No matter how much I slanted my head to one side and squinted my eyes, I was unable to fully comprehend the implications of that which was being shown. And, I believe those examples to all be in good faith by the poster.

For me, I guess it's the difference between what can be manipulated, and what can be rationally conceived. Oh well, not earth shattering either way I guess.

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[BobRodes]

To all the "Bobs" out there, please don't take my foolishness seriously.

I won't if you won't.

 

[CajunCenturion]

...and in a previous post I showed a link that clearly shows that 1 = 2, but that doesn't make either any more probable or the reality. It merely shows IMHO, that mathematics cannot be applied like this and still be logical.

I beg to differ, as your link does nothing of the sort because it contains an undefined operation. Division by 0 is undefined and never permissible; therefore, any sequence that contains such operation is complete garbage. They can be fun puzzles – both in hiding and in finding the division by 0 – but there is absolutely no mathematical legitimacy to any sequence of steps which contain a division by 0.

That is completely different from dealing with infinities. I can understand the reluctance to accept some of the statements presented in this thread, because by their very nature, and to varying degrees, infinities can be difficult to understand. A repeating decimal, such as .999…, is an infinity, and as such, it is not as straightforward to wrap our minds around the concept as it is with standard finite mathematics.

Please don’t confuse, nor in any way equate, the serious mathematics of infinities with parlor room games and puzzles built on undefined operations.

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[craigsboyd]

OK, that makes sense. I may be back when I've got more knowledge on this subject. I'm quite sure that I am not up to speed, and perhaps if I was then 1 = .9999... would make more sense to me. Thanks CajunCenturion.

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[simmiekastner]

I found this thread by Googling my name, Simmie Kastner. I was looking at my cyberspace trail reletative to my work as Managing Director of Burning Coal Theatre Company, although my core work is managing the HRIS Operations for Nortel. I must reply to the thread below that mentions me, and it DOES mention ME.

I did not marry anyone named Colin. I married Jerome Davis, and we have been happily married for 10 years. MaryAlice Lilieholm married Tim Colin and became MaryAlice Colin. Though considerably less amusing, it is also true.
I have not worked for Oracle, in management or otherwise, for about 8 years.

I do think it is fitting that I should show up in this thread, though, because my colleagues here at Nortel sometimes call me the Grammar Nazi.

 

[CajunCenturion]

simmiekastner - I belive the thread you you're referring to is thread1256-1080099.

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[KenCunningham]

I was confused and then I Googled and came up with Thread1256-1080099 as a possible source for Simmie's interjection. Welcome Simmie.

 

[BJCooperIT]

Welcome Simmie,
I took several classes taught by you and I look forward to your participation here.
Barbara

^{Beware of false knowledge; it is more dangerous than ignorance. ~George Bernard Shaw}
Consultant Developer/Analyst Oracle, Forms, Reports & PL/SQL (Windows)
My website: Emu Products Plus

 

[grande]

This is how I look at it. Take a circle, and divide it into three thirds. You have 1/3 + 1/3 + 1/3, which is 1 whole.

Therefore, 0.33333333333... + 0.33333333333... + 0.33333333333... = 1

BUT, through basic math, we see that 0.33333333333... + 0.33333333333... + 0.33333333333... actually equals 0.999999999...

Therefore, 0.999999999... = 1.

QED?

-------------------------
Just call me Captain Awesome.

 

[craigsboyd]

grande,

That is best explanation I have encountered. Thank you for helping my be able to get my head around it.

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[AngelB]

I don't wish to untwist your head again, but if you had trouble understanding why 0.999999... = 1, don't you have the same problem with 1/3 = 0.3333333333...?

Gez



Let's think the unthinkable, let's do the undoable, let's prepare to grapple with the ineffable itself, and see if we may not eff it after all

 

[medic]

tgreer,

No one has proven that the human brain has any limits, nor that the universe is infinite...

The fact that no one can prove it just proves how limited human brain is.

sbudzynski,

... Money is the root of evil ...

Where did you get that saying? The written saying is:
[blue]“For the [red]love of[/red] money is the root of all evil.”
(1 Timothy, 6:10)[/blue]

 

[SamBones]

Heh, yeah. You can have all the money you can get, you just can't love it!

 

[Turkbear]

Hi,
Carl Sagan once noted the the distance between the largest number you name ( say a googleplex, for instance) and infinity is essentially the same as the distance from the smallest number to infinity...Infinity is just that big...






To Paraphrase:"The Help you get is proportional to the Help you give.."

 

[SilentAiche]

er, is it just y'all, or is anyone else completely avoiding the concept of [blue]"approximation.?"[/blue]

1/3 = .333333333333333333.... until the 3's run out of curves, the 8's lose their apetite, or Frosty loses his head.

Therefore, 0.33333333333... + 0.33333333333... + 0.33333333333... = 1

Um, what the heck does [red]Kwed[/red] mean?


1st Timothy in my family (but only 6'4") [X] (there's supposed to be yellow smiling thingamajig where the X is).

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[CajunCenturion]

QED is stands for "quod erat demonstrandum" which is Latin for 'thus it has been demonstrated'. It is conventionally used to signify the end of a proof.

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[SilentAiche]

i wore kweds as a kwid, when I was kwite a kwute twoddler.

I'll twie to kwit being phunny....

Twim (er, hardly) (ask a Geekfester...)

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