1=.999999...?

[AdrienIer]

Which brings you to the point of the discussion. It's a thought experiment like the original tortoise and Achilles race. Convergence is not identity. Modern math says it's close enough. Look back on the discussion and see how often limits are used. The definition of limit is just rigorous mathspeak for close enough.

J

What you are missing here is that in this case 0.99999... is just a convention, it doesn't really exist by itself. It is literally the limit when n -> infinity of the sum for k from 1 to n of 9^(-k). Which is 1.

Lim (for x -> 1) of x = 1. It's not "close enough". It is literally 1.

 

[Akka]

Nearly four years later, and you're still wasting time with a self-important clown ?

 

[warpus]

I mean, it's like the thread where people say that evolution didn't happen or the earth is flat. From a mathematician's point of view. Not to say that I am one.. Either way, non-mainstream opinions can be interesting, even if they're obviously wrong, and at least in the context of math, it is tempting to attempt to explain what is happening in a slightly different way, so the person holding the incorrect view can be convinced. With evolution that is not so easy, but math is all black and white, true or false, so it's tempting to try again with a slightly different approach

 

[Lohrenswald]

I mean, it's like the thread where people say that evolution didn't happen or the earth is flat. From a mathematician's point of view. Not to say that I am one.. Either way, non-mainstream opinions can be interesting, even if they're obviously wrong, and at least in the context of math, it is tempting to attempt to explain what is happening in a slightly different way, so the person holding the incorrect view can be convinced. With evolution that is not so easy, but math is all black and white, true or false, so it's tempting to try again with a slightly different approach

Thing is that onejayhawk is just repeating the same thing over and over, even fter he's proven wrong

 

[Akka]

I mean, it's like the thread where people say that evolution didn't happen or the earth is flat. From a mathematician's point of view. Not to say that I am one.. Either way, non-mainstream opinions can be interesting, even if they're obviously wrong, and at least in the context of math, it is tempting to attempt to explain what is happening in a slightly different way, so the person holding the incorrect view can be convinced. With evolution that is not so easy, but math is all black and white, true or false, so it's tempting to try again with a slightly different approach

I'd say it's the opposite, as math are absolute with provably true or false answers, it's even more grating and takes even more stubborn ignorance to refuse facts.

 

[warpus]

Maybe we're just not good enough teachers of math. It should be easy, you're right.

 

[onejayhawk]

What you are missing here is that in this case 0.99999... is just a convention, it doesn't really exist by itself. It is literally the limit when n -> infinity of the sum for k from 1 to n of 9^(-k). Which is 1.

Lim (for x -> 1) of x = 1. It's not "close enough". It is literally 1.

There is no dispute that the limit = 1. Never was. The dispute is whether 0.999... is "literally the limit" is true or not. Common usage says not. Undefined I would give you.

As to the definition of limit,

lim·it [ˈlimit]

• mathematics
  a point or value that a sequence, function, or sum of a series can be made to approach progressively, until it is as close to the point or value as desired.

https://en.oxforddictionaries.com/

How is that not a dressed up form of close enough?

I'd say it's the opposite, as math are absolute with provably true or false answers, it's even more grating and takes even more stubborn ignorance to refuse facts.

Not always. Like life, there are fringes and gray areas. They have to do with definitions. Sometimes things must stay undefined.

Maybe we're just not good enough teachers of math. It should be easy, you're right.

You have a point. I'm a math teacher and you are not getting it.

J

 

[Lexicus]

Did you just use a dictionary to make a math argument

 

[Lohrenswald]

As to the definition of limit,

lim·it [ˈlimit]

• mathematics
  a point or value that a sequence, function, or sum of a series can be made to approach progressively, until it is as close to the point or value as desired.

https://en.oxforddictionaries.com/

oxford dictionary is wrong

 

[W|M]

n/n = 1 for any real number n

1/3 = 0.333...

0.333... * 3 = 0.999...

1/3 * 3 = 3/3

3/3 = 0.999...

3/3 = 1

0.999... = 1

/thread

 

[AdrienIer]

Do you have a mathematical definition of 0.9999... then ? I proposed the limit of the sum, which is the definition accepted by pretty much the totality of mankind that has studied some maths.

 

[onejayhawk]

n/n = 1 for any real number n

1/3 = 0.333...

0.333... * 3 = 0.999...

1/3 * 3 = 3/3

3/3 = 0.999...

3/3 = 1

0.999... = 1

/thread

You assume the conclusion on the first line. Poor logic butt good debating skills. Are you in politics?

Do you have a mathematical definition of 0.9999... then ? I proposed the limit of the sum, which is the definition accepted by pretty much the totality of mankind that has studied some maths.

When I reopened the thread P commented that there does not seemto be a rigorous definition.

J

 

[warpus]

You really want to see a proof that an integer divided by itself is equal to 1?

 

[AdrienIer]

When I reopened the thread P commented that there does not seemto be a rigorous definition.

Then 0.9999... doesn't exist as a number. Either it is a rigorously defined thing or it is not. I proposed a rigorous definition (that is the definition commonly accepted by, well, everyone). Do you disagree with it ?

 

[Perfection]

There is no dispute that the limit = 1. Never was. The dispute is whether 0.999... is "literally the limit" is true or not. Common usage says not. Undefined I would give you.

The point of repeating decimals is to represent a number. If you declare .999... to not be a number but some undefined thing then you lose all usefulness of repeating decimal notation. This strikes me as super lame.

 

[Kyriakos]

I wish that cfc was the only place where some would brag for just mentioning a definition. It is rather obvious that OJ was talking about the idea, not if current/standard/main/whatever math notation identifies 0.999........ as 1 or as something approaching 1 and reaching it only in infinity. Most people don't really care either. Should be clear that only the ideas (distinct in each case, 1 or not 1 etc) would draw interest. And ideas, mr Cready, are notated in specific logic which itself produces the result based on context; definitions are not something anyone should brag about.

Otherwise, keep thinking you are Euler when you just read a book under a whale oil lamp or something /oil /euler /eu iller

 

[onejayhawk]

Then 0.9999... doesn't exist as a number. Either it is a rigorously defined thing or it is not. I proposed a rigorous definition (that is the definition commonly accepted by, well, everyone). Do you disagree with it ?

The point of repeating decimals is to represent a number. If you declare .999... to not be a number but some undefined thing then you lose all usefulness of repeating decimal notation. This strikes me as super lame.

We don't even try to define the number 'one' or the operation 'plus'. Undefineds are not just permitted, they are inescapable. Refer to Gödel, Escher, Bach: An Eternal Golden Braid, by Douglas Hofstadter.

This sort of problem is ancient. Euclid had five axioms. The sixth would have been that there is only one line through a point parallel to any line not containing the point. He tried and failed to prove it. The problem is the definition of plane. We now use two major forms of non-Euclidean geometry because we accept three types of planes.

Otherwise, keep thinking you are Euler when you just read a book under a whale oil lamp or something /oil /euler /eu iller

Oy.

J

 

[warpus]

You can read a bit about the mathematical definition of addition here

I assume you mean addition. The + operator can be used in different ways depending on the situation. One of those situations is addition.

Anyway, the page I link describes how to mathematically and logically construct a number system, including addition. And before you define addition you must first define the things you are adding, meaning all the numbers, including 1, which the top answer (with 21 votes) does, proving everything along the way.

You can do a search for "Once we have these definitions and theorem, we can start defining addition" on the page (without the quotes), if you want to skip the intro which defines all that is required for you to be able to define addition

There is a lot of interesting related stuff on this page actually, but most of it is pretty involved. But there do seem to exist several ways to define number systems and operators like addition, and not just one.

 

[Lohrenswald]

When I reopened the thread P commented that there does not seemto be a rigorous definition.

It's a number
you don't define individual numbers

like, what's the definition of 5?

 

[onejayhawk]

You glimpse some of the difficulties. Generally, 'one' and 'plus' are left undefined to ease other definitions. In geometry the equivalent are point, line and plane. Here is a tutorial video. https://www.brightstorm.com/math/ge...s/three-undefined-terms-point-line-and-plane/

It's a number you don't define individual numbers like, what's the definition of 5?

5 is defined as 1+1+1+1+1. All the integers and rational numbers (and others) are defined from '1' and '+'.

J