1 = .999..... Right?
 

So I had an argument with a couple friends about this. They can't be convinced that 1 = .999 repeated

I showed them the Wikipedia proof:

1/9 = 0.11111....

9 * 1/9 = 9 * 0.1111....

1 = 0.99999....

They say that Wiki isn't a reliable source :banghead:

Anyone want to back me up on this? Just for ****s and giggles.

They kept trying to argue that .9999 doesn't equal one because if you have .99999... it's not the same as having 1 of something. What they don't seem to grasp is that .9999 is a limit.

Any math wizards here?

And crap, I wanted a Yes/No poll on this!

They're confusing math with intuitive reasoning.

Yeah they get hung up on the fact that it's .9999.... so that it must end on a 9 right? They obviously can't grasp the concept of infinity.

I hate stubborn people. Oh, and Wikipedia is a great source 95% of the time.

Yes, 0.9 recurring (0.9...) is equivalent to 1. It's related to the idea that every number has a non-truncating decimal equivalent (i.e. a decimal which doesn't end in heaps of zeros). Here's what I remember from an informal 0.9... = 1 proof I saw a few years back:


Consider an infinite set S of numbers {0.9, 0.99, 0.999, ...}. Each element of S has a finite number of 9s and is marginally smaller than 1. 0.9... doesn't belong to S as it has an infinite number of 9s (and hence is bigger than every element of S).

Now imagine a number which is just smaller than 1 - let's call it 1-ϵ, where ϵ is an infinitesimally tiny number. Since S is an infinite set, there inevitably exists a number in S which is bigger than 1-ϵ. Hence 0.9... is also bigger than 1-ϵ. This leads us to the corollary that 0.9... is larger than every number smaller than 1. Now obviously every number bigger than 1 is also bigger than 0.9...

So if every number smaller than 1 is smaller than 0.9... and every number bigger than 1 is bigger than 0.9..., then 0.9... must equal 1.

Technically .999... is not 1, but for all practical purposes it is = 1. But the fact remains, .999 technically Isn't 1.

It is actually. See my proof, and Seltzer's (who's knowledge far surpasses my copy-paste skills).

Technically, .9999... isn't a number. It's a limit, and the limit equals 1.

Explain.

I'm not sure I agree with this, because it is TECHNICALLY a different number. But the differences are so minuscule that it might as well be the same number.

But it isn't. 1.999999.... = 2, TECHNICALLY.

You're not realizing what 0.9999 means. It means 1. It's a mathematical representation of a quantity defined by a limit of not logically being able to be anything but 1.

It's not a matter of numerical semantic or intuitive disbelief. If you use math and logic, as is shown in the thread, it is unequivocally 1.
Why? Because it actually works. TECHNICALLY.

What I don't get is why you would say the difference is miniscule? What exactly is the difference between 2 and 1.999....


People seem to have trouble grasping the concept of infinity.

I understand what Infinity means. Never-ending and all of that jazz.

The difference between "1.999..." and "2" is "0.000...001"
And I also understand the whole "not logically being able to be anything but 1" also, but there's still a difference in 1.999... and 2.

That's wrong, because you then imply that 1.999... ends with a 9, which obviously it doesn't seeing as how it never ends.

1.9999..... is a limit. It can be written as 2 interchangeably.

But, it's never completely two. If it never ends, there's a very small difference, but it never quite reaches two.

Dude, I don't know how I can explain it any better. There's no difference when it never ends.

I'd rather you offer up some sort of proof (in the form of mathematical equations) rather than stubbornly sticking with what you've always believed. If you see fault in the logic I originally posted, feel free to critique it.

I also have no idea how I can explain my side any better.
Agree to disagree, my good friend.

Fair enough, as long as you understand that in the mathematical community it IS accepted as one and the same.

But I respect your thinking.

I do understand that it's accepted as one and the same. Of course.

But I also feel like, if it continues forever (1.99999999999...) it's never REALLY 2. Just that if it continues infinitely, the difference is so miniature that it could be replaced with two, and be pretty much the same (but not quite, at the same time).

Yes, the difference between the two is 0.000000... infinity .... 0001, which is the same as 0.
0.000...001 is infinitely small, as is 0.

I, too, had problems grasping this concept when it was first introduced to me.

Mathematicians knew you would try this, so they made a ton of......FALSE MATH? to keep you occupied with intuition and uneducated "common sense" that got kicked around in grade school for the general betterment of the average guy. Please do not rule this life with that small a bubble, oh king!

Look, I know it seems counter-intuitive to believe that 0.9 recurring = 1, but you have to trust me on this one ;) Your reasoning follows on from this:

2 - 1.9 = 0.1 = 10^-1 (which is more than 0)
2 - 1.99 = 0.01 = 10^-2 (ditto)
2 - 1.999 = 0.001 = 10^-3 (ditto)

General Form:
2 - 1.9{n} = 10^-n where 9 is repeated n times (which is more than 0)

That reasoning is fine for a finite n. But as n tends towards infinity, limit(10^-n) = 0. In other words, that 0.000...001 number you quoted is basically 1/∞ which is considered to be zero, meaning that 2 - 1.999... = 0 (which makes sense since 2 is 1.999...)


Here's another simple proof that 0.9... = 1. This is a method commonly used to convert recurring decimals into fractions and it demonstrates that 0.9... is 1/1 as a fraction.

x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 9/9 = 1, therefore 0.999... = 1


The important thing to understand is that 1.000... and 0.999... are both decimal representations of them same number (1). This applies to other numbers too (i.e. 0.8324 = 0.8323999... ) but not to all numbers (i.e. 1/3 = 0.3... and there is no other decimal representation).

Did we really bring the nerd-dom of MMO's to MB?

MMOers wish they could be this nerdy ;)

Oh really? Then feast your virgin eyes upon the godless math skills of the Theorycrafters at Elitist Jerks...Behold!

http://elitistjerks.com/f47/t82625-shaman_elemental/

Don't even read the page, just scroll down and watch the formulas and graphs scroll by. This is for one spec, of one class, in one video game - nerds.

WoW nerds do put an impressive amount of effort into theorycrafting. I also came across this recently - it's an academic paper written about the probabilities of getting various skills when leveling up in Heroes III :laughing:

it's good news that .9 repeated is the same as 1, otherwise i'd never make it home (i'd just get closer, and closer, and closer... but never quite there!)

of course, if you break my journey down into an infinite number of steps, each step is zero meters... so how will i ever get home?

You can find the sum of geometric series that converge. Thus you can find the sum or rational form of a repeating decimal. By using n/(1-r)

.9/(1-.1)=.9/.9=1

I'm a math major
this is fact
close thread

I'm kinda turned on right now.

Yes, 0.9999 recurring can be mathematically defined as 1. In fact, in many applications, including aerospace engineering, the specificity of a number is usually used to the 5th decimal place. Anymore than this would usually be supercilious. For example when using pi, 3.14159 is the generally accepted figure to be used.

Here's a fun fact for math nerds, the closest fraction for pi was found to be 355/113. Not 22/7 which was probably used when you were about 9years old to calculate circumference.

So at any rate, 0.9999 can be substituted as 1 in any mathematical circumstance. It can be proved as a proof as Seltzer did or simply by common sense as I have just shown here.

is this like a cross between "super silly" and "super serious"?

It's a word :p:

But I meant to say superfluous.

I'm flattered.

Where's your proof? Wikipedia (amongst many other sources) would say otherwise.

This is EXACTLY what I had wondered about... I had put together everything but the conclusion, then I just dropped the subject.

take a distance, A to B.
move 90% of that distance.
then 90% of the remaining distance.
and so on and so on, to infinity.
how will you ever get to B?

reminds me of zeno's dichotomy paradox...

This isn't a matter of limits. It's matter of the rationalizing repeating decimals.

So let's look at 1/3 it divides out to .3 repeating.
These two quantities are equal.

so if we multiply both quantities by three we will 1 and .9 repeating.
These two quantities are equal

there's no discussion to be had

.3 repeating is very close to 1/3, but even if you carry it out to the end of the universe, they're never actually equal, are they?

Um, yes, they are.

If 1/3 doesn't equal .3 repeating, then enlighten all us math noobs as to what the decimal equivalent of that fraction is...

You seem to stuck on the concept of limits.

.3 repeating is a rational number so you can rewrite as a fraction if you wish and the are the same. It all goes back to geometric series really. And 3rd grade division.

Wait... so if you take 1 and break it up into smaller and smaller pieces... it's ultimately MADE OF ZEROES?!?! HOLY SHIT!!! EVERYTHING IS NOTHING!!

... I guess that's why it's so easy to get from A to B. Ain't really goin' anywhere, are ya now?