0.99999... = 1

[Leicester Fox]

So are we on page 11 or page 10 + 0.999.......

 

[Harrypotter_fan]

And there is no end to infinite either, and that's why its called infinite. Isn't it? Then the very theory of infinite 9's getting sometimes equal to 1 is proven wrong!

G.P.

 

[Sureshot]

The difference between it is 2/3.

1/3 is a complete fraction. 0.333... is a recurring decimal.

Joe, shut up!

 

[svijay]

There are 1.999... kinds of people in the world. Those who understand limits, and those who don't.

 

[sohum]

.9999.... is a finite representation of an infinite number. You can pretend like the ... represents your notion of infinity, but you have to understand that that representation is not accurate. Unless you keep account of the fact that there are an infinite number of digits following it (which you cannot, without simplifying it to a nice, round number) you really cannot draw any conclusions from it.

Notice that 1 - 0.999... = 0.00...1. In particular, note that the ellipses are in the middle of the finite representation. Given the whole notion of infinity, you will never reach that last one (or rather, it will take an infinite amount of time to reach that last 1). Since no one can actually experience an infinite amount of time, that is simplified to 0, proving simply that 1 == 0.999.... (the last . was actually a period).

Basically, this thread is silly and is really argument for the sake of argument. The argument that calculations do not use the real values is a matter of precision rather than mathematics. If you choose to round off earlier than infinity, that is your choice in making your calculations more imprecise. If you rounded off everything to infinity, using these numbers, you would probably end up with a round number at the end of your arithmetics.

 

[svijay]

I have a simple question for you, Sohum. What is the smallest positive real number?

 

[Kshitiz_Indian]

I have a simple question for you, Sohum. What is the smallest positive real number?

That's the same if I ask you which is the largest one...

 

[svijay]

That's the same if I ask you which is the largest one...

It's not. The smallest positive integer is 1. There is no largest positive integer.

The first thing to get your head around when you talk about infinity is that there are enumerable infinities (like integers) and non-enumerable infinities (like real numbers).

Also, there are numbers you can define without writing the decimal expansion, e.g: the positive root of the equation x^2 - 2 = 0, or the ratio of the circumference of a circle to its diameter. 0.999... is syntactically defined as the sum of a certain geometric series and semantically equals 1.

 

[Kshitiz_Indian]

It's not. The smallest positive integer is 1. There is no largest positive integer.

Now when on earth did you say Integers? I guess we were talking about real numbers isn't it? Can you give me the smallest real number? :

 

[svijay]

Now when on earth did you say Integers? I guess we were talking about real numbers isn't it? Can you give me the smallest real number? :

We were not talking about real numbers, you were talking about whether the existence of a smallest positive real number is as trivial a question as the existence of a largest positive real number. It is not. There are infinite sets , such as positive integers, which have a smallest element but not a largest element. Even the set of integers, which have neither a smallest element nor a largest element, can be systematically enumerated, e.g: 0, 1, -1, 2, -2, 3, -3,... The big difference with real numbers is that it is impossible to systematically enumerate all of them, as was proved by Cantor in the 1870s.

I brought up the smallest positive real number because Sohum, whose judgement I generally respect, mentioned a quantity known as 0.00...1. Now that is an example of what certain non-standard models of mathematics, such as surreal numbers and hyperreal numbers, call an infinitesimal. Since the standard model of positive real numbers has no smallest element, the aforementioned quantity is ill-defined in our model, or just a proxy for 0 if you prefer.

You guys are ahead of the curve, and are really thinking of hyperreal numbers in most of your discussions. Section 1.6 (see Section 1.5 for definitions) should put this baby to bed.

http://www.math.wisc.edu/~keisler/chapter_1a.pdf

 

[shravi]

Seriously? This has gone on for 11 pages?! Must be the crack.

 

[Magz]

Discussions about this sorta thing are interesting . Especially in the case of Physics and Chemistry.

Yeah, most of the posts are correct, sohummisra's post sums it all up very well.

 

[Kshitiz_Indian]

Reading it again, today. svijay had hit the nail on the head, the concept of limits really helped me understand this.

Any people still wondering?

 

[ZoraxDoom]

Oh God No. Please don't tell me this thread is alive again...