0.9_ = 1?

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you haven't actually demonstrated the statement in bold.

Wrong.



Unfortunately true.



Wrong. Repeating it doesn't make it right.



Only technically true. Exactly equal is close enough for any purpose, practical or not.



Wrong again.



This is the root of your misunderstanding. There is not such a number and there cannot be such a number. If there were, it would mean there are a finite number of numbers between 0 and 1, and there is not.

In other words, proof by contradiction:

Let x = 1 - 0.999_
You have stated that 0.999_ < 1, therefore it follows that x > 0
Let y = x/2.
Since x > 0, that means that y > 0 and y < x
Add 0.999_ to both sides and you find that:
0.999_ + y < 0.999_ + x
But from rearrangement of the initial equation, 0.999_ + x = 1
Therefore 0.999_ + y < 1

Which means that (0.999_ + y) is a number that is less than 1, but greater than 0.999_, contradicting your assertion that 0.999_ was the closest number you could get to 1, UNLESS y = 0, but if y = 0 then x = 0, which contradicts your assertion that 0.999_ is not equal to 1.

We conclude that if 0.999_ is less than 1, then there are an infinite number of numbers greater than 0.999_ which are still less than 1.



Back to this again. It is one.

Why can you not understand that there is more than one way of representing a number? 1 = 1/1 = 2/2 = 3-2 = 0.333_ * 3 = Pi/Pi = -e^(i*Pi) = sin(Pi/2) = 0.999_ = 0b1010 / 0xa = 0b.111_ = 1.



There's no error. Enlighten us where you think you see errors.





If I'm taking an offensive tone here, and I admit that I probably am, it's because I'm offended by the incredible arrogance of this sentence. You are wrong and this has been established for hundreds of years, the proofs and demonstrations have been linked here from multiple sources and it is easy for you to go and verify it. Instead you've chosen to say there are errors in a demonstration of 0.999_ = 1 without giving any hint as to what you're talking about.



Keep thinking.



Well yes, if it were easy people would have stopped arguing about it. If you're truly convinced you've out-thought the mathematicians of the ages on this issue, I suggest you take up a problem in the millennium prizes (I can't post the link since I'm new here) because it would advance humanity if you could solve one of those.



Unfortunately but evidently true.

No, he was wrong. They are theoretically the same and a perfectionist must consider them the same, because 0.0_1 doesn't mean anything (well -- it looks like a cool emoticon for somebody flexing their muscles, but other than that ). 0.0_ = 0, there's no "after".

Maybe it will help to think about this: what's 0.0_1 * 10? What about 0.0_1 / 10?

I think people get hung up on the pencil & paper method of subtracting decimal numbers. There is no "last" 9 in 0.9_ so there is no "last" 0 in 0.0_ to put a 1 after -- that's what infinity means.

(sorry, I double-posted, so I'm editing the second to new content because I don't see a delete button)

Last edited by YourMaster on 31 Aug 2009, 1:29 am, edited 1 time in total.

When I went to my first engineering conference I was at the dinner table with a bunch of young engineers. An older engineer gave us a simple mechanics problem and asked which of two conditions created more stress. They were almost the same. There were a couple of answers (hey, we were calculating in our heads), and then I said "they're the same". The engineer beamed and said "right. you're the first young engineer I've heard answer that correctly in years".

In engineering, all numbers have error associated with them, and it's a big mistake to allow your precision to exceed your accuracy. These days, if the computer analysis says something is over-stressed by 0% (rounded), people will say the design is inadequate.

Yes, this is a story about life, not engineering.

True, which is why this sort of problem is rarely relevant to actual engineering (I am an engineer myself). Exact numbers such as 0.9_ have arbitrary precision and therefore do not affect the margin of error, but you are limited to your other measurements.

However, from a pure mathematical perspective, 0.9_ is exactly equal to 1, which is fun for pure mathematics (and the few times in practical engineering when you do care about this infinitely precise equality).

It has the ordinal structure omega + 1.

Look up theory of transfinite ordinals.

ruveyn

Probably what robo37 is thinking when he writes 0.0_1 is the limit of the sequence (0.1, 0.01, 0.001, 0.0001, ...) which actually is zero. So 1-0.9_=0_1 is really just the statement that 1-1=0.