Poll: 0.999... = 1

[Addendum_Forthcoming]

All positive numbers have two square roots, one positive and one negative. -6 x -6 = 36, in the same method as 6 x 6 = 36.

No they don't the square root of a positive number is the positive number which squared equals the first.

(-6)^2 = 36, yes.

but

sqrt(36) = 6, and that's that.

The confusion comes from the fact that when the teacher tells you that

if x^2 = 36 then x = 6 or x = -6, and that's right, but the math isn't complete because

x^2 = 36

does not imply

x = sqrt(36),

it implies

|x| = sqrt(36) = 6

which implies

x = 6 or x = -6.

From what I can remember of year 6 maths, all positive numbers have two square roots. Besides, as I said .. example of idiotic math games not unlike .9~ = 1

 

[Maze1125]

Every math major I have talked to and showed that to has described that as "shady".

Well, that is how my dad explains it, and he's a math professor, so I'd rather take his word over that of some math majors who can't come up with anything better than calling it 'shady'.

Well, they were nice enough to supply me with a more satisfying answer using infinite series. I am not arguing against the result; I am arguing against the proof used.

Yes, that proof is a bit shady, it's perfectly correct, but it implies a lot of complicated mathematics that isn't explicitly said. And in that it is a bit shady.
If you were talking to a mathematician. That's a fine proof to use, as they should be able to fill in the gaps themselves. But if you're talking to someone who doesn't know the maths, then you're essentially keeping half of what you're doing "under the table" to avoid confusion which, even though all the stuff you're hiding is perfectly correct, is a bit shady.

 

[Ishnuvalok]

Man, screw maths. It's 0430 and I am tired, lol. I give up. I either don't get this at all, or maths is broken.

It's the former option. It's that you don't understand this.

Never assume because you don't understand something, that everything that has to do with that field is wrong. That's ignorant.

 

[Agayek]

That doesn't make sense. You're saying that 10x-x is a number with an infinite number of decimal places occupied by nines, but the last decimal place is occupied by a one. Which means that you're essentially saying that 10x-x has both an infinite number of decimal places and a finite number of decimal places. That's a contradiction.

You cannot perform mathematical operations on an infinitely repeating number. Therefore, you must at some point terminate the string. At that point, you can then multiply it by 10 and proceed.

However once you do that, 9.9999...999 will have shifted to the left, so 0.999...999 will have one more significant digit. Thus, you get 8.999...991.

Edit:

But it goes on to infinity, so technically there's no 1.

Same answer to you too.

 

[Vanaron]

All positive numbers have two square roots, one positive and one negative. -6 x -6 = 36, in the same method as 6 x 6 = 36.

No they don't the square root of a positive number is the positive number which squared equals the first.

(-6)^2 = 36, yes.

but

sqrt(36) = 6, and that's that.

The confusion comes from the fact that when the teacher tells you that

if x^2 = 36 then x = 6 or x = -6, and that's right, but the math isn't complete because

x^2 = 36

does not imply

x = sqrt(36),

it implies

|x| = sqrt(36) = 6

which implies

x = 6 or x = -6.

From what I can remember of year 6 maths, all positive numbers have two square roots. Besides, as I said .. example of idiotic math games not unlike .9~ = 1

I agree that the 0.999... = 1 maybe useless in actual math (although it may reveal problems on basic understanding of math).

But square roots of positive numbers are by definition positive... Or else:

if sqrt(36) = 6 and sqrt(36) = -6

then

sqrt(36) * sqrt(36) = -36

or worse

sqrt(1) = 1 and sqrt(1) = -1

then 1 = sqrt(1) = -1

which implies 1 = -1 and that's just wrong.

And that's not useless or a idiotic math game.

 

[Maze1125]

From what I can remember of year 6 maths, all positive numbers have two square roots. Besides, as I said .. example of idiotic math games not unlike .9~ = 1

These aren't games. These are facts.

Square roots are factually defined to always be positive.
The equation x[sup]2[/sup] = 36 has two solutions 6 and -6.
But that is not the same thing as saying sqrt(36) = -6.
sqrt(36) = 6, always. This is done in order to ensure that "sqrt(x)" is a valid function.

Equally, 0.999... = 1, that is a fact. We're not playing games, it just an interesting fact. Just like e[sup]i*pi[/sup] = -1. It's a very interesting fact that is very unintuitive the first time you see it. But that doesn't make it wrong.

 

[Addendum_Forthcoming]

No I get what he's saying, it's not like it's hard to envisage .9~ as being '1', but It's still just an idiotic playing around with established ideals concerning the usage of numbers.

It would be like writing an essay in phonetic English rather than with actual words. One could argue it's still English, it's just a stupid thing to do.

Yup, you got that right. There's no purpose to this other than math wizards mocking the living daylight out of mathematically challenged individuals. 0.(9) is not used for anything else, because it is nothing more than an idiotic way to write down 1.

Math has other wondrous things, ones that actually have meaning and purpose. Such as Euler's Identity that you could see in my avatar, for instance.

The only maths I ever had to do since the end of school was pearson's co-efficient for psych experimentations when measuring and testing relationships between psychological concepts such as happiness, optimism and extroversion using three tests and a lie scale test to eliminate untruthful test subjects.

Of course all that came on a CD you put into a computer an write down the results in the boxes ... hey presto ... computer works out RSS values for all of them <.<

 

[Maze1125]

That doesn't make sense. You're saying that 10x-x is a number with an infinite number of decimal places occupied by nines, but the last decimal place is occupied by a one. Which means that you're essentially saying that 10x-x has both an infinite number of decimal places and a finite number of decimal places. That's a contradiction.

You cannot perform mathematical operations on an infinitely repeating number. Therefore, you must at some point terminate the string. At that point, you can then multiply it by 10 and proceed.

However once you do that, 9.9999...999 will have shifted to the left, so 0.999...999 will have one more significant digit. Thus, you get 8.999...991.

Edit:

But it goes on to infinity, so technically there's no 1.

Same answer to you too.

Here's a more rigorous proof for you then:

An infinite decimal is defined to be:
lim(as n->infinity)sum(from k=1 to n) (a[sub]k[/sub] * 1/10[sup]k[/sup])
where a[sub]k[/sub] is the kth digit of the decimal.

Therefore, 0.999... is defined to be:
lim(as n->infinity)sum(from k=1 to n) (9 * 1/10[sup]k[/sup])
So all we need to do is show that that is equal to one.
Which is true iff for all e>0 there exists an N such that for all n>N |1 - sum(from k=1 to n) (9 * 1/10[sup]k[/sup])| < e

Now sum(from k=1 to n) (9 * 1/10[sup]k[/sup]) is a finite sum, and so we can calculate that
|1 - sum(from k=1 to n) (9 * 1/10[sup]k[/sup])| = |1/10[sup]n[/sup]|

So we need to show that for all e>0 there exists an N such that for all n>N |1/10[sup]n[/sup]| =1 then |1/10[sup]n[/sup]| e>0, then let N = 1/e and then |1/10[sup]n[/sup]| N

Hence the claim that, for all e>0 there exists an N such that for all n>N |1 - sum(from k=1 to n) (9 * 1/10[sup]k[/sup])| < e, is true.
So, by the definition of a limit, lim(as n->infinity)sum(from k=1 to n) (9 * 1/10[sup]k[/sup]) = 1
Therefore, by the definition of infinite decimals, 0.999... = 1

QED

 

[Athinira]

From what I can remember of year 6 maths, all positive numbers have two square roots. Besides, as I said .. example of idiotic math games not unlike .9~ = 1

I think his point was that while the square root of a positive number CAN have two solutions, in reality it only has one (aka. it's one or the other). Which one it is isn't necessarily something we can deduce, which is why he wrote x = 6 or x = -6

 

[Addendum_Forthcoming]

From what I can remember of year 6 maths, all positive numbers have two square roots. Besides, as I said .. example of idiotic math games not unlike .9~ = 1

These aren't games. These are facts.

Square roots are factually defined to always be positive.
The equation x[sup]2[/sup] = 36 has two solutions 6 and -6.
But that is not the same thing as saying sqrt(36) = -6.
sqrt(36) = 6, always. This is done in order to ensure that "sqrt(x)" is a valid function.

Equally, 0.999... = 1, that is a fact. We're not playing games, it just an interesting fact. Just like e[sup]i*pi[/sup] = -1. It's a very interesting fact that is very unintuitive the first time you see it. But that doesn't make it wrong.

As far as I remember, all positive numbers have 2 sq roots. A square root is just that. I hardly see how you're able to debate semantics after committing yourself to idiotic number games.

An analogous argument if I were to use similar semantics would be to simply say that .9~ isn't 1 because it doesn't use the same figure. A faultless argument on the basics of semantics (and what I still think is a logical one to make regardless in response to such foolish games)

 

[Maze1125]

From what I can remember of year 6 maths, all positive numbers have two square roots. Besides, as I said .. example of idiotic math games not unlike .9~ = 1

I think his point was that while the square root of a positive number CAN have two solutions, in reality it only has one (aka. it's one or the other). Which one it is isn't necessarily something we can deduce, which is why he wrote x = 6 or x = -6

No, he wasn't.

If you have the equation x[sup]2[/sup] = 36 then it has two solutions x = 6 or x = -6.
Equally, if you have the equation sqrt(36) = |x| then x = 6 or x = -6.

But if all you have is sqrt(36) = x, then that equation only has one solution, x = 6.
x = -6 would be wrong as the sqrt function always give a positive answer.

 

[Maze1125]

As far as I remember, all positive numbers have 2 sq roots. A square root is just that.

No, a square root is absolutely defined as the positive solution and only the positive solution. Some school teachers will say otherwise, but that pretty much because explaining the full truth would be overly complex and confusing, and so hinder their students more than help.

I hardly see how you're able to debate semantics after committing yourself to idiotic number games.

That's because, as I explained in my previous post, I'm not playing games, I'm defending known facts.

An analogous argument if I were to use similar semantics would be to simply say that .9~ isn't 1 because it doesn't use the same figure. A faultless argument on the basics of semantics (and what I still think is a logical one to make regardless in response to such foolish games)

If that was a valid augment, then 0.5 =/= 1/2, as they're both written differently. Or, in fact, 1 =/= 1.00

 

[Addendum_Forthcoming]

No, a square root is absolutely defined as the positive solution and only the positive solution. Some school teachers will say otherwise, but that pretty much because explaining the full truth would be overly complex and confusing, and so hinder their students more than help.

Whilst this is from wikipedia, it makes sense.

Every positive number x has two square roots. One of them is \scriptstyle \sqrt{x}, which is positive, and the other \scriptstyle -\sqrt{x}, which is negative. Together, these two roots are denoted \scriptstyle \pm\sqrt{x} (see ± shorthand).

If that was a valid augment, then 0.5 =/= 1/2, as they're both written differently. Or, in fact, 1 =/= 1.00

Anyways .. I'm bored <.<

 

[Maze1125]

No, a square root is absolutely defined as the positive solution and only the positive solution. Some school teachers will say otherwise, but that pretty much because explaining the full truth would be overly complex and confusing, and so hinder their students more than help.

Whilst this is from wikipedia, it makes sense.

Every positive number x has two square roots. One of them is \scriptstyle \sqrt{x}, which is positive, and the other \scriptstyle -\sqrt{x}, which is negative. Together, these two roots are denoted \scriptstyle \pm\sqrt{x} (see ± shorthand).

Exactly sqrt(x) is positive and if you want the negative, you take -sqrt(x). And if you want to solve the equation y[sup]2[/sup] = x then you take y = ±sqrt(x)

 

[Addendum_Forthcoming]

No, a square root is absolutely defined as the positive solution and only the positive solution. Some school teachers will say otherwise, but that pretty much because explaining the full truth would be overly complex and confusing, and so hinder their students more than help.

Whilst this is from wikipedia, it makes sense.

Every positive number x has two square roots. One of them is \scriptstyle \sqrt{x}, which is positive, and the other \scriptstyle -\sqrt{x}, which is negative. Together, these two roots are denoted \scriptstyle \pm\sqrt{x} (see ± shorthand).

Exactly sqrt(x) is positive and if you want the negative, you take -sqrt(x). And if you want to solve the equation y[sup]2[/sup] = x then you take y = ±sqrt(x)

So therefore there are two square roots for every positive number... I fail to see the argument.

 

[Soraryuu]

I stand by the opinion that there's no such thing as infinity. Maybe for time, maybe for the multiverse, but not for matter/energy. Therefore, any mathematic equation that uses infinity is not valid in my eyes.

1) Time as the 4th dimention in higher dimensional physics is not necessarily infinite.
2) Can you show that there is a final decimal place to 0.999...? If not you are going to have to accept that some things involve the concept of infinity.

1: Yes, I'm not an expert or anything, so therefore I said "might". Just an FYI.
2: Thinking about my counter-argument, I've come up with a new term for myself: impossible numbers. To explain, I'll have to get a bit more practical instead of theoretical.

In my mind, an impossible number is a number that's impossible to create. There is only a finite amount of matter/energy in the universe, and as such, there isn't enough resources to create an infinite number, even if you count the smallest, tiniest pieces of matter in a unary(base 1) system. Therefore, infinitelarge numbers are impossible, and sensible mathematical equations are impossible with them. As for infinitesmall numbers, that's an unknown area. We still don't "know" what the smallest amount of matter is, or even if there is such a thing. In other words, discussing this would be like discussing about god's existence, and that rarely works out, does it? But anyway. To answer your original question, there can be two answers...
1: If matter can be split to infinity: No; it has infinite decimals.
2: If there's no such thing as infinitesmall matter: The number doesn't exist, ergo your question is invalid.

Now, BV, if you can prove that there's such a thing as infinitesmall matter, then by all means, you've won the argument, and more. Until then, this issue is far from over.

Oh, and back to the OP's question, if 0.999... = 1: No. Also, that "9x = 10x-x" argument doesn't work, because x's decimals is n, and 10x's decimals is n-1. 10x can never reach that last decimal it needs to keep it's 9, so 9x = 8.999....

 

[zoulza]

Now, BV, if you can prove that there's such a thing as infinitesmall matter, then by all means, you've won the argument, and more. Until then, this issue is far from over.

What in the world does matter have to do with a simple math question?

Oh, and back to the OP's question, if 0.999... = 1: No. Also, that "9x = 10x-x" argument doesn't work, because x's decimals is n, and 10x's decimals is n-1. 10x can never reach that last decimal it needs to keep it's 9, so 9x = 8.999....

Like it's been pointed out a million times on this thread already, there is no last decimal!

 

[Soraryuu]

Now, BV, if you can prove that there's such a thing as infinitesmall matter, then by all means, you've won the argument, and more. Until then, this issue is far from over.

What in the world does matter have to do with a simple math question?

Oh, and back to the OP's question, if 0.999... = 1: No. Also, that "9x = 10x-x" argument doesn't work, because x's decimals is n, and 10x's decimals is n-1. 10x can never reach that last decimal it needs to keep it's 9, so 9x = 8.999....

Like it's been pointed out a million times on this thread already, there is no last decimal!

1: Numbers represent reality. Math is based on our perception of it. So, when we don't know something about reality, we can't know about the same thing in math. That's why it's relevant.
2: That works too. The last decimal it needs to keep it's 9 doesn't exist. There's your answer.

And please don't pull a "if I can't understand it it's not true" argument.

 

[zoulza]

That works too. The last decimal it needs to keep it's 9 doesn't exist. There's your answer.

Uhm, no. Neither decimal ends, so you always have something to subtract.

Here's a question for you. If 1 and .999... are two different numbers, then there must be some other number between them. What is it?

 

[zoulza]

1: Numbers represent reality. Math is based on our perception of it. So, when we don't know something about reality, we can't know about the same thing in math. That's why it's relevant.

You seem to have a pretty strange idea of what constitutes reality, i.e. that a number doesn't exist unless you can string together that many particles. Does pi not exist then? Pi is irrational, so like .9999..., it never ends, but unlike .99999..., is also never repeats itself. So, by your logic, pi can't exist unless you can divide matter into infinitesimals! Congratulations, you've just disproven circles!

And please don't pull a "if I can't understand it it's not true" argument.

Pretty ironic coming from someone who, over the course of this thread, has been given at least five different mathematical proofs for why this is true and still refuses to accept it.