1.3333.. does NOT equal 4/3!

like how 0.9999... does NOT equal 1!!!1one

Seriously, I have to sift through one of these 0.999.. posts every week on the warcraft forums. One might ask why I bother with them but it allows me to flamebait and flame people repeatedly for having an inferior understanding of numbers.

Do you think that 0.9999... is a different (smaller) number than 1? If so what is the difference between the two numbers, and how did you find that difference when there should be more 9's?

Remember infinity is not a number.

*shrug*

Somebody help me convince somebody who doesn't understand the difference between a number and a function

Okay. Why do you try so hard?

It's fun to question people's beliefs when they are clearly wrong. I also have the same fun asking religious people about their beliefs and showing that (at the very minimum) they are inconsistent.

Help pls.

Duh, it's called rounding up!

0.9 recurring (0.999999999999999999999999.....)

Is the same as 1.

0.99999 is not.


Hmmm, although Im a little confused by the point of the topic...

I think the topic is based on the fact that 1/3 = 0.3333333333333333333..., implying that 3*1/3 = 0.9999999999999999999... = 1 yet assumes 0.9999999999999999999... is a distinct and seperate number to 1 indicating that 1 =/= 0.9999999999999999999... I, for one, don't know what to do...

If we let 0.9999999999999999999... be an infinitessimally smaller number than 1 (ie, 1-dn where dn is an infinitessimally small number), then let dn tend to 0, 0.9999999999999999999... = 1-0 = 1. If we then say all numbers are seperated by a value dn, we get a result that all numbers equal all other numbers! Does that equal the end of mathematics?

I once read that it has to do with the tens system. That 10 is not to have direct value or some weird :skull::skull::skull::skull:. I was stressed, and taking a 3d mathematics course. Maybe it is something to look at. if 10 equaled 9 + .999999 and not 9 + 1, wouldn't life be grand.

I like how people continue to oversimplify the relationship between ratios (4/3) and approximations (1.34). People also employ rounding, to make numbers nicer, and to obey the grand necessity of significant figures (uncertainty).

Quote:

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Originally Posted by Pureghetto

Somebody help me convince somebody who doesn't understand the difference between a number and a function

---

Thank you for trying to convince someone who does not have the fundamental base nor the knowledge to be able to intelligently argue, nor to discern proper mathematical terminology. Thank you. Lots. Seriously.

As for the topic, whether I believe that .999... is == 1, is irrelevant. Once you hit limit theory, it doesn't matter, since a function can approach a number without end, but never get there, but it may very well be so close, that you may as well, just finish it off and round it.

If you're going to quarrel about how an approximation, which represents an uncertain definition of a ratio, and whether or not multiple of those will equal whatever number you're trying to get to, I suggest you look over some of those yummy number postulates, such as

A/B = ?
as long as B != 0
and A == B
therefore, A/B = 1.

And some others. To me, I'm all about making life easier on yourself, as long as it's within a reasonable range, depending on the question, percentage of error, but if I view .999... == 1, if the .999 stops, then it is its own number, but since .999... does not end, you can't really make a mature decision about it until it reaches its conclusion.

I agree with the title of the thread. :p

0,999... is not 1 imho. I don't really care what the truth is. :smash:

0.999999999999 is not 1.
0.999999999999... may as well smurfing be 1.

If you need to carry the repeating 9 out a billion places to ensure that errors do not occur in equipment or tests, than do so. But for all practical purposes, we ought to view it as simply 1.

This how I view it, and I will be the first to tell you I am not gifted in mathematics. But from what I would call a logical, practical standpoint, I think my statement is true.

.aaaaaaaa.... is b. shut up.

I still don't get it. Why have an infinity? Infinity is just for people to lazy to count out the number. Slackers.

Quote:

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Originally Posted by bipper

.aaaaaaaa.... is b. shut up.

I still don't get it. Why have an infinity? Infinity is just for people to lazy to count out the number. Slackers.

---

Try to divide 1 by 3 using long division. It never ends. That's why.

Math is fun n.n

I'm only in geometry right now, though. I do believe that .333... is equal to 1/3 and .6666... is equal to 2/3; etc.

While we're on the subject of math, what does any number with an exponent of zero equal one instead of zero or that same number?

Quote:

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Originally Posted by Ouch!

Quote:

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Originally Posted by bipper

.aaaaaaaa.... is b. shut up.

I still don't get it. Why have an infinity? Infinity is just for people to lazy to count out the number. Slackers.

---

Try to divide 1 by 3 using long division. It never ends. That's why.

---

Or does it? I accuse you of lazyness :hot:

This is what 3/10 looks like on paper.

Quote:

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Originally Posted by Jessweeee♪

This is what 3/10 looks like on paper.

---

Thanks for the approximation and using the rules of significant figures.

Continuing on...

Quote:

---

Originally Posted by Jessweeee♪

While we're on the subject of math, what does any number with an exponent of zero equal one instead of zero or that same number?

---

An exponent is really division/multiplication, except in this case, an exponent is a multiplied number multiplied many times. So 4² is 16 (4*4), or 6³ is 216 (6*6*6), or can be expressed as 6¹⁺¹⁺¹. Simply put, exponents represent how many times a number is multiplied by itself. Is it still not itself when it is alone?

Exponential numbers can be modified by adding and subtracting, a shortcut for signifying that a number can be multiplied that many number of times. There are such things as identities, numbers you use to keep a number that you have. For exponents, an identity is anything to the X¹ == X. Exponents already use to the first power as equalling itself. Addition uses N+0 == N, N*1=N, N¹ == N.

Let's take a number, a variable: (A)
It can equal anything besides zero.

Second variable: (n)
This is our exponent.

Aⁿ/Aⁿ, is the same as saying A^{n - n}, since we can manipulate exponents however we want.

What happens when we subtract any number from itself? We get zero, the additive identity. After all, A + 0 is still A. Applied to division, A^{n - n} is the same as A⁰. When you divide anything by itself, it equals one. Therefore, A⁰ == 1, as long as A != 0.

There's a multiplicative proof using algebra, but the divison proof is easier to remember, even if it does use more mathematical theory than the other.

Quote:

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Originally Posted by Jessweeee♪

Math is fun...

---

Oh, yeah!

Quote:

---

Originally Posted by Jessweeee♪

While we're on the subject of math, what does any number with an exponent of zero equal one instead of zero or that same number?

---

x to the n-th power equals x times (x to the n-th minus one power).
So, 5 cubed is equal to 5 times (5 squared) and 5 squared is equal to 5 times (5 to the first power).
Therefore, 5 is equal to 5 times (5 to the zero power).
5 = 5 x 1
:D

Edit: Tav. :)

Quote:

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Originally Posted by Odd Eye

Edit: Tav. :)

---

I prefer the division, since it's more familiar to me, although I guess I had elements of both.

I saw a proof of why .9999... = 1 before in calc but I totally forget it now :D?

Why wouldn't it be true in other number systems? I just tried it in base 2 and it seems to be pretty much the same.

Decimal, the ".999. . . ." http://www.snowy-day.net/stuff/decimals.png

Fractions, the "3/3"
http://www.snowy-day.net/stuff/fractions.png

Why don’t you just ask God?

Oh not this again! Maths is the epitome of boring.

Thanks, Tavrobel n.n

My teacher just did the table thing and said it had to be one otherwise the pattern wouldn't fit, and offered no other explanation. I thought that was silly, but it gave me a headache thinking of a broken pattern like that :\

Quote:

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Originally Posted by rubah

Why wouldn't it be true in other number systems? I just tried it in base 2 and it seems to be pretty much the same.

---

Most odd number systems (such as x/13), certain even numbers with an odd number prime root, and prime numbers have never-ending approximations for a fraction value. Not that I needed to tell you that, but it does hold true.

To answer the title, it does. I don't remember the proof exactly, but I think it was this.

1/9 = 0.1111...
=> 10/9 = 1.111...
=> 90/9 = 9.999...
=> 9/9 = 0.9999...
=> 1 = 0.9999...

Or something like that. This exact wording actually doesn't hold up because the first line assumes that my conclusion is already correct...

Quote:

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Originally Posted by Firo Volondé

1/9 = 0.1111...

---

Quote:

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Originally Posted by Tavrobel

I like how people continue to oversimplify the relationship between ratios (4/3) and approximations (1.34).

---

alright, let's let x = .999999...
x = .999999...

I trust everyone agrees that if we times it by 10, we're gonna get 9.999999...
10x = 9.999999...

Now we'll find 9x. To do that, we just simply take 10x, and minus 1x.
10x - x = 9.999999 - .999999
9x = 9.

Divide both sides by 9 to find x by itself...
9x/9 = 9/9
x = 1

If we remember back to the start, we let .999999... equal x. So lets substitute that back in. x = .999999...

.999999... = 1.

And there you have it. .999999... DOES equal 1. Solid mathematical proof.

Quote:

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Originally Posted by Anaisa

Oh not this again! Maths is the epitome of boring.

---

This. If you're going to make fun of someone's opinions, can't you at least make them interesting ones? :p

Quote:

---

Originally Posted by Mittopotahis

alright, let's let x = .999999...
x = .999999...

I trust everyone agrees that if we times it by 10, we're gonna get 9.999999...
10x = 9.999999...

Now we'll find 9x. To do that, we just simply take 10x, and minus 1x.
10x - x = 9.999999 - .999999
9x = 9.

Divide both sides by 9 to find x by itself...
9x/9 = 9/9
x = 1

If we remember back to the start, we let .999999... equal x. So lets substitute that back in. x = .999999...

.999999... = 1.

And there you have it. .999999... DOES equal 1. Solid mathematical proof.

---

YES! that is correct *sniff* that is so beautiful...i learned that in maths c (the hardest and optional maths at my school) a while ago and it brought a tear to my eye when i saw it...thank you...you are exactly right, btw: i did have a WHOLE webpage proving it but i lost it :(

EDIT: here's a page, not the one i was thinking of, but very close to it: Point nine recurring equals one @ Things Of Interest

I believe it is 1

no 0.9999999... in thier right mind would think otherwise

Quote:

---

Originally Posted by Mittopotahis

alright, let's let x = .999999...
x = .999999...

I trust everyone agrees that if we times it by 10, we're gonna get 9.999999...
10x = 9.999999...

Now we'll find 9x. To do that, we just simply take 10x, and minus 1x.
10x - x = 9.999999 - .999999
9x = 9.

Divide both sides by 9 to find x by itself...
9x/9 = 9/9
x = 1

If we remember back to the start, we let .999999... equal x. So lets substitute that back in. x = .999999...

.999999... = 1.

And there you have it. .999999... DOES equal 1. Solid mathematical proof.

---

Hey, thanx for that proof. That just made my day.

lmao.

You don't need math in the real world. Just like you don't need CPR certification.

Quote:

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Originally Posted by Harmless

You don't need math in the real world. Just like you don't need CPR certification.

---

So, I guess you work out stuff like how much stuff costs with magic then? Cool. We could throw away checkout tills, just have a little moneybox to collect the cash.

It only gets problematic once you launch a space-station in the orbit and find out that somebody approximated one decimal too many.

http://www.hoax-slayer.com/images/Columbia05.jpg

Convinced?

math is to be treated like a multidimensional array, not a continuation of numbers.
0 0123456789
1 0123456789
2 0123456789
3 0123456789
4 0123456789
--
aka

00 01 02 03 04 05 06 07 08 09
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49

therefore taking one third of a cluster, leaves you will an inequality, and I would think, leave a remainder instead of simply being .999.... equaling 1. So to me, in my head .9999 != 1 outside of mathematical error and/or laziness. I mean, you have ten digits, it would be impossible to cut them in three ways. Where as a pie, you can. Therefore, pies are by far the most mathematically boggling, and awesome thing on this planet.

Quote:

---

Originally Posted by bipper

math is to be treated like a multidimensional array, not a continuation of numbers.
0 0123456789
1 0123456789
2 0123456789
3 0123456789
4 0123456789
--
aka

00 01 02 03 04 05 06 07 08 09
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49

therefore taking one third of a cluster, leaves you will an inequality, and I would think, leave a remainder instead of simply being .999.... equaling 1. So to me, in my head .9999 != 1 outside of mathematical error and/or laziness. I mean, you have ten digits, it would be impossible to cut them in three ways. Where as a pie, you can. Therefore, pies are by far the most mathematically boggling, and awesome thing on this planet.

---

You couldn't ever initialize the array because you'd need another dimension for each integer exponent of 10 across (infinity,-infinity), and therefore could never use it. :p
And that would seriously muck up exponents that aren't in the integers. :p

So there's the proof that .999... = 1.

Code:

---

1/9 = .111...
=> 9*1/9 = 9*.111...
=> 9/9 = .999...
=> 1 = .999...

---

To assume the validity of this proof is to assume that the theorems and sets it is based upon are not fundamentally flawed. If you don't want to read a lot, skip the next paragraph. :p

For example:
The most fundamental number set in use is the set of all positive integers, P. P was used until it was realised that there is a fundamental need for a zero-value digit.
From this, the natural numbers were constructed, which is N = { x | x in P or x in {0}} or in other words, x is a postive whole number or zero.
Then they realised that negative numbers were necessary, so the integers or Z were made. Z includes all positive integers, 0 and all negative integers.
But what if you don't want an integer? You can have one cake or two cakes or nothing, but not half a cake. So they made Q, the rational numbers, which are defined as { m/n | m,n in Z }, or any number that can be obtained by dividing two integers. This gives some numbers in between two whole numbers, but not all.
They realised that some numbers couldn't be represented by a decimal, numbers like the square root or 2, or Pi. There needed to be a continuous scales, so they made the real numbers, R, which includes all of Z and everything in between.
Then they realised that the Real numbers only operated in one dimension, positive and negative, so they created the complex numbers, C, which is a set of ordered pairs, extending in two dimensions. The real numbers can be visualised as a line in C, while C itself as a plane.
Each set is a superset of the preceeding one, or in other words each set contains all of the previous one and more.

Each of these number sets were created for one reason: the previous one was flawed.
You can imagine this theorem in the real numbers:

Code:

---

For any numbers m, n in R:
There exists some k such that |k| < |m - n|

---

What I'm saying there is that no matter how close m and n are, there is always a k that is closer to 0 than the difference between m and n.
For that reason, I can only conclude that there is a number between 0.999... and 1, and therefore that they are indeed different.
But wait a minute, 0.999... is infinite in length, so I can only conclude that it is that k.
But wait a minute, that's like saying |k| is the least positive number in R, even though R is infinite. If that's the case, what's half of |k|?

What I'm really trying to say is that the real numbers are flawed. This is just one example of how they break down under certain circumstances. With conversion between decimals and fractions rounding and approximation is inevitable, especially with irrationals, infinites and infinitesimals.
As far as I'm concerned, the two numbers are different, but they represent the same quantity.

This thread still going on? Can somebody tell me the smallest real # > 0?

I express it as 1/∞ but we all know '∞' is not a number and so it doesn't work.

1/∞ = 0 so it wouldn't satisfy your criteria of # > 0 even if ∞ was a number.

Quote:

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Originally Posted by Little Blue

1/∞ = 0 so it wouldn't satisfy your criteria of # > 0 even if ∞ was a number.

---

By using limits it works

1/ x when x -> ∞ is a teeny bit > 0

Well of course that works. Tending to ∞ just means the numbers are getting bigger and bigger without actually reaching ∞. The fact remains that 1/∞ = 0, a fact proven by the limit you just posed.

Fact remains I answered the question properly :tongue:

No you didn't. The question was what's the smallest number > 0. I challenged the statement that 1/∞ would give a valid answer that fit the criteria.

lim 1/x = 0⁺
x->+∞

Is a mathematical statement that as x approaches +∞, 1/x approaches 0 from the positive side. This limit is simple enough to state that if x = ∞, 1/x = 0 (unlike some limits where direct substitution lead to issues, and treating ∞ as a number, even though such a concept is false).

0.999.... != 1 in the sense that 0.999... and 1 differ by 1.0 x 10^-infinity. Two things wrong with that however, 1) infinity cannot be used in arithmetic because it is not a number, and 2) 1.0 x 10^-infinity for all practical and most impractical purposes does not exist. 1.0 x 10^-infinity is 0.000.... . You think that the 1 is coming sometime, but it never is because of the infinite number of times it is being multiplied by 1/10. The 1 would normally appear at the end of the decimal, but this decimal has no end, so the 1 will never appear, so all that is there is 0.000.... which is the same as 0. Although 1.0 x 10^-infinity does exist on some scale, even for mathematical precision, it's existence is insignificant. And therefore, 0.999... = 1.

Now how is it a question that 1.333... != 4/3? That just dumb.

It started with 0.333 x 4 which is the same as 1/3 x 4 which would yield 4/3.....

Problem is in the infinite nature of 1/3 so any multiple of it will have the same running decimals.

Quote:

---

Originally Posted by Pureghetto

This thread still going on? Can somebody tell me the smallest real # > 0?

I express it as 1/∞ but we all know '∞' is not a number and so it doesn't work.

---

It doesn't work because that poses the question:
Let x = 1/∞
What is x/2?

There is always a less positive number for any k in R.

Ok, what number can you fit in between 0.999... and 1?

Quote:

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Originally Posted by ZeZipster

Ok, what number can you fit in between 0.999... and 1?

---

Easy .999.... + .999... Er, at least, I might as well say so! :)

It is like asking what whole number can you fit between 4 and 5. None, they are equal. There is something, as has been mathematically backed by brains far more simple than my own (:jess: ) which prove their is another finite layer that numbers do not correctly represent

Quote:

---

Originally Posted by bipper

Quote:

---

Originally Posted by ZeZipster

Ok, what number can you fit in between 0.999... and 1?

---

Easy .999.... + .999... Er, at least, I might as well say so! :)

It is like asking what whole number can you fit between 4 and 5. None, they are equal. There is something, as has been mathematically backed by brains far more simple than my own (:jess: ) which prove their is another finite layer that numbers do not correctly represent

---

I didn't say whole numbers... I suck at math but the average of two different numbers fits in between those numbers. For example, 4 and 5's average is 4.5 and that fits in between 4 and 5. I'm fairly sure any value you can scrummage up to fit there would in fact have 21 chromosomes.

There are no integers between two integers whose difference is 1. However there are always a real number between two different real numbers. That's how you tell numbers apart.

If 0.999... and 1 were different numbers I could subtract 0.999... from 1 and I would get a number. What is this number? It must exist if the two numbers were different.

Interesting how there's no such thing as a smallest number. I guess that Planck's constant and Planck-time and other Planck stuff do not apply to number theory :)

Uhm... Yes, Planck's.. whatever you just said!

I agree. Yep. Numbers.

Quote:

---

Originally Posted by Pureghetto

There are no integers between two integers whose difference is 1. However there are always a real number between two different real numbers. That's how you tell numbers apart.

If 0.999... and 1 were different numbers I could subtract 0.999... from 1 and I would get a number. What is this number? It must exist if the two numbers were different.

---

The answer to that would be:

Code:

---

1 - 0.999... = 0.000...1

---

except that 0.000...1 is inherently impossible because the zeros go on forever. That's why the real numbers are flawed.

Get me a quantum computer and I might be able to give you the actual answer to that question. :p

Get a life and use a calculator. .

It's actually kind of simple I mean it's like if you had a dog, if you plucked one of his it's hairs that would no longer be a dog, merely a value that gets arbitrarily close to being a dog.

Quote:

---

Originally Posted by Lady Rikku

Get a life and use a calculator. .

---

Show me a calculator that can calculate the difference between 0.999... and 1 and I'll gladly use it.

Quote:

---

Originally Posted by o_O

Quote:

---

Originally Posted by Pureghetto

There are no integers between two integers whose difference is 1. However there are always a real number between two different real numbers. That's how you tell numbers apart.

If 0.999... and 1 were different numbers I could subtract 0.999... from 1 and I would get a number. What is this number? It must exist if the two numbers were different.

---

The answer to that would be:

Code:

---

1 - 0.999... = 0.000...1

---

except that 0.000...1 is inherently impossible because the zeros go on forever. That's why the real numbers are flawed.

Get me a quantum computer and I might be able to give you the actual answer to that question. :p

---

Um, or you realise that since 0.00000....1 doesn't make sense, you acknowledge that 0.999.... == 1 ?

ok, just to put an end to this: 0.999... and 1 are the SAME NUMBER, there is no doubting it, ask the smartest person in the world and he or she will tell you that. scroll back up to my previous post and there is a WHOLE page there that explains everything...bottom line is: there is no use arguing, its a fact...so get over it: they are the same number...ok?

The best part of the ignorant masses who scream that .99999-... does not equal 1 are the ones who took one course in Calculus and try to prove it using limits, despite that the proof that .9999... = 1 is limits in Calculus. They say "limits get closer and closer to a number...." No, "x" in limit equations gets closer asnd closer to a number, but the limit is one number.

I don't remember the limit proof of this problem (I saw Dr Unne post it once - bug him), but a simpler example is lim x->infinity (1/x). This reads, for people who don't know calculus, as "the limit, as x approaches infinity, of 1/x." The answer is 0. As x gets bigger and bigger, closer and closer to infinity, the limit actually becomes 0. Not "closer and closer, but never reaching" 0 as the pseudo-intellectuals in this case want you to believe of .9999... and 1.

The easiest way to demonstrate the fact that .999... = 1 is the fact that there is no number between the two. Not only does any two distinct, real numbers have a number between them, but any two distinct, real numbers have an infinite amount of numbers between them. You can't even name one between .999... and 1, though, because there isn't any. The two numbers are equal.

Didn't Blizzard make an announcement on their main page about .999... = 1 a while ago? I seem to remember reading that.

Quote:

---

Originally Posted by Pureghetto

Um, or you realise that since 0.00000....1 doesn't make sense, you acknowledge that 0.999.... == 1 ?

---

Yes. :p

As I said a couple of posts back, they're different representations of the same number under the equality operator in the real numbers.

Note, however that this holds under operations defined in the real numbers and complex numbers. If there exists a set where infinity or the concept of infinity can be treated as a normal entity then 0.999... could be shown to be inequal to 1.

Hmm, this doesn't have to be looked at through vector calculus.

0.999... = 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + 9(1/10)^4 + ...

That sum is an infinite geometric series, and since the absolute value of it's common ratio is less than 1, it converges. So, you can calculate it's sum. It's involves a lot of Reimann sum symbols and other things that would involve annoying use of the character map, but it boils down to this:

ar + ar^2 + ar^3 + ... = ar/(1-r)

9(1/10) + 9(1/10)^2 + 9(1/10)^3 + ... = 9(1/10)/(1-(1/10)) = 1

So, 0.999... = 1. People can argue someone proving that with limits and say something about the nature of limits to infinity and their thoughts on infinitesimals, but the convergence don't lie.

Quote:

---

Originally Posted by ZeZipster

Quote:

---

Originally Posted by bipper

Quote:

---

Originally Posted by ZeZipster

Ok, what number can you fit in between 0.999... and 1?

---

Easy .999.... + .999... Er, at least, I might as well say so! :)

It is like asking what whole number can you fit between 4 and 5. None, they are equal. There is something, as has been mathematically backed by brains far more simple than my own (:jess: ) which prove their is another finite layer that numbers do not correctly represent

---

I didn't say whole numbers... I suck at math but the average of two different numbers fits in between those numbers. For example, 4 and 5's average is 4.5 and that fits in between 4 and 5. I'm fairly sure any value you can scrummage up to fit there would in fact have 21 chromosomes.

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WTFFER? I never said anything about a rabid T-Rex with bipolar tendencies.