Poll: Does 0.999.. equal 1 ?

[Eclectic Dreck]

1. There are many reasons why the study of limits is NOT pointless. They have enormous utility in many applied fields.

Calculus did indeed change the world and it is based entirely upon the concept of a limit. Differentiation? Just a limit. Integration? Limit of a Riemann Sum. Think they're useless? I suppose technical advancements of the last 200 years were just lucky breaks.

2. The "if you do understand that .999~ will never quite get that last little bit..." statement is wrong, because there is no "last little bit". There is no "bit" or "number" etc. that separates the two real numbers, therefore they are the same. It's very important to realize that this is NOT a matter of opinion. This is a factual, direct conclusion from the basic axioms which define what a real number is.

And this is probably as explicitly as it could be stated.

Now, if you DO decide to change what a real number is, then some ambiguity may arise. But this is equivalent to saying "I won this track meet because I redefined what "running" is, and showed up in a jet plane. Doesn't my argument make sense?"

And even though I already posted I think this is a /thread worthy remark.

 

[Volkov]

Here is another proof actually.

1/9 = 0.1111... (1 repeating infinitely)

Multiplying both sides by 9.

Left: 9*1/9 = 1. (Since for any nonzero n, n*(1/n) = 1).

Right: 9*0.1111.... = 0.9999... (Again, obviously).

Therefore, 1 = 0.999...

 

[veloper]

So many elegant proofs and wasted because the question is flawed. It should say recurring, not just "0,999" which is 0,9990.

the ellipsis indicates recurring

0.(9) is both shorter and clearer.
It would also have ensured fewer bad answers. I can hardly believe the poll would have otherwised turn out that discouraging and with wikipedia only a click away.

 

[Volkov]

0.(9) is both shorter and clearer.
It would also have ensured fewer bad answers. I can hardly believe the poll would have otherwised turn out that discouraging and with wikipedia only a click away.

While it's a well-closed problem in mathematics, it's actually a very well-known and important problem in mathematics education. Many students don't find the answer intuitive, and reject it. So it's actually not at all unexpected that the answer turned out the way it did; it basically just says that most of the people voted have the mathematical understanding of ~5th-grader (which is probably the case).

 

[MrCollins]

yes 10/3=0.333333333333333333333333
and 9.999999999999999/3= 0.3333333333333333333

 

[Hawgh]

No. Approximates are not equalities

Also 1/3 \neq 0.333...

0.333...*3 \neq 1

There are Real numbers(or is 1/3 rational?) that cannot be represented by a finite string of integers.

 

[Regiment]

Yes it is.

Another simple proof:

1/3 = 0.333...
(1/3) * 3 = 1
0.333.. * 3 = 1
(1/3) * 3 = 0.333... * 3
Therefore 0.999... = 1

 

[Coldie]

No. Approximates are not equalities

Also 1/3 \neq 0.333...

0.333...*3 \neq 1

There are Real numbers(or is 1/3 rational?) that cannot be represented by a finite string of integers.

Yeah, that's exactly why 0.999... = 0.(9) and 0.333... = 0.(3) are infinite strings of integers. There are no approximations involved here at all.

 

[mattsipple4000]

So many elegant proofs and wasted because the question is flawed. It should say recurring, not just "0,999" which is 0,9990.

the ellipsis indicates recurring

0.(9) is both shorter and clearer.
It would also have ensured fewer bad answers. I can hardly believe the poll would have otherwised turn out that discouraging and with wikipedia only a click away.

wikipedia is not FACT! and I don't think any one was confused by m y question! everybody knew I was talking about reoccurring!
you are the only one that didn't get the question here

 

[Volkov]

No. Approximates are not equalities

Also 1/3 \neq 0.333...

0.333...*3 \neq 1

There are Real numbers(or is 1/3 rational?) that cannot be represented by a finite string of integers.

0.999... stands for an INFINITE string of integers. So does 0.333.... If the strings are finite, you are exactly correct. But the notation used in this thread (at least before your post) was such that 0.999... is infinitely many 9s (equal to 0.(9)) and 0.333... is infinitely many 3s (equal to 0.(3)). So, since the strings are INFINITE, your last statement, while mostly correct, is inapplicable here.

Also, 1/3 is rational, and, as all other rational numbers, is real. Not every rational number can be represented by an infinite string of repeating integers, but both 1 and 1/3 can be. In fact, such string does not even have to be unique. Case in point:

1 = 0.999... (infinitely many 9s).
but also
1 = 1.000... (infinitely many 0s).

 

[chieften]

Essentially, yes.

1/9 = 0.111..
2/9 = 0.222..
etc
so 9/9 = 0.999..

but 9/9's is a whole, so it is also 1.

Yeah my maths teacher told us this once

but .999 times 9 does not equal nine

 

[Bon_Clay]

No. Approximates are not equalities

Also 1/3 \neq 0.333...

0.333...*3 \neq 1

There are Real numbers(or is 1/3 rational?) that cannot be represented by a finite string of integers.

Its not an approximation at all though. The two are perfectly equal because there is theoretically no difference between them of even the smallest amount. (Bit of a tautology I guess)

I think most people should be able to understand it if I phrase is this way. The is no real value difference between the two. This infinitesimally small difference a lot of people think there is doesn't exist. You can not write the 1 after zero point infinite zeros. The one will NEVER come up, because the zeros preceding it will NEVER end.

The two will no longer be equal when the 1 after infinite zeros comes up. So if you want to prove they are unequal you can sit down and write what 1 - 0.999... is, in proper mathematical notation, that isn't 0.

 

[Volkov]

wikipedia is not FACT! and I don't think any one was confused by m y question! everybody knew I was talking about reoccurring!
you are the only one that didn't get the question here

Wikipedia is a website, whether or not the information it provides is correct, it cannot be fact, so you are right.

But, it's worth noting that vast, vast majority of mathematics on Wikipedia is actually extremely good. Really, of all fields, mathematics is probably the best represented on Wikipedia of all of them. I am not sure why, but that is the case.

 

[Volkov]

but .999 times 9 does not equal nine

Assuming you mean 0.999... here - yes it does.

 

[kebab4you]

No, it does not ,never had never will. You need to round it off for it to be equal 1. Else it will be equal 0.99999......

 

[veloper]

So many elegant proofs and wasted because the question is flawed. It should say recurring, not just "0,999" which is 0,9990.

the ellipsis indicates recurring

0.(9) is both shorter and clearer.
It would also have ensured fewer bad answers. I can hardly believe the poll would have otherwised turn out that discouraging and with wikipedia only a click away.

wikipedia is not FACT! and I don't think any one was confused by m y question! everybody knew I was talking about reoccurring!
you are the only one that didn't get the question here

Don't worry I got the question and wikipedia is a much, much better source for your math needs than a random forum, trust me.

 

[Volkov]

No, it does not ,never had never will. You need to round it off for it to be equal 1. Else it will be equal 0.99999......

This is incorrect.

 

[mattsipple4000]

wikipedia is not FACT! and I don't think any one was confused by m y question! everybody knew I was talking about reoccurring!
you are the only one that didn't get the question here

Wikipedia is a website, whether or not the information it provides is correct, it cannot be fact, so you are right.

But, it's worth noting that vast, vast majority of mathematics on Wikipedia is actually extremely good. Really, of all fields, mathematics is probably the best represented on Wikipedia of all of them. I am not sure why, but that is the case.

thanks ! you can never count on it to be 100% correct tho !

 

[doggie135]

General Knowledge tidbit: The answer is yes. Ask any professor of mathematics...or a standard High School student for that matter.