0.9 repeating = 1?

[Cory]

When you get to calculus they'll have you prove that .9999=1 using geometric series.

[JewishDan18]

I like this method. between any 2 different real numbers, there is at least one number between them not equal to either of them.

[Ether]

Originally Posted by JewishDan18

I like this method. between any 2 different real numbers, there is at least one number between them not equal to either of them.

between any 2 different real numbers, there exists an uncountably infinite set of different real numbers.

[BornaE]

Lets try it this way.


X=.99999...
Thus X*10=9.99999....
now 9X = X*10 - X = 9.999... - 0.999... = 9.000
9X = 9
X = 1

And proven.

[ExTexan]

Following your logic of :

X=.99999...
Thus X*10=9.99999....
now 9X = X*10 - X = 9.999... - 0.999... = 9.000
9X = 9
X = 1

if X=1 then 10X=10 and
9X=10-.999999......

I'm not sure anything was proven!

[RoboDesigners]

I'm not quite sure if I completely understand this, but here's another look at it:

1/9 = 0.111...

9*1/9 = 9*0.111...

9/9 = 0.999...

1 = 0.999...

[Ether]

Originally Posted by ExTexan

I'm not sure anything was proven! :confused

The mathematical meaning of the repeating decimal .999... is a limit.

It is the limit of the sequence of partial sums of the infinite series 9/10 + 9/100 + 9/1000 + ...

The sequence of partial sums of the above series is equal to (1-1/10), (1-1/100), (1-1/1000), ... (1-1/10^n)


Using the definition of limit of a sequence:

A real number L is said to be the limit of the sequence Xn if and only if for every real number ε > 0, there exists a natural number N such that for every n > N we have | Xn−L | < ε.

It can be shown that the limit of the above sequence is "1".

Therefore, "1" and ".999..." mean exactly the same thing. They are two different ways of writing the same real number.

[Molten]

Originally Posted by Ether

The mathematical meaning of the repeating decimal .999... is a limit.

It is the limit of the sequence of partial sums of the infinite series 9/10 + 9/100 + 9/1000 + ...

The sequence of partial sums of the above series is equal to (1-1/10), (1-1/100), (1-1/1000), ... (1-1/10^n)


Using the definition of limit of a sequence:



It can be shown that the limit of the above sequence is "1".

Therefore, "1" and ".999..." mean exactly the same thing. They are two different ways of writing the same real number.

The limit of a series of partial sums is not equal to the actual sum at all. This is by definition of a limit. A limit is what the function must approach ever closer without ever reaching it. If it ever actual reaches it at any point, then it is not truly its limit.

[Ether]

Originally Posted by Molten

The limit of a series of partial sums is not equal to the actual sum at all. This is by definition of a limit. A limit is what the function must approach ever closer without ever reaching it. If it ever actual reaches it at any point, then it is not truly its limit.

The meaning of the expression .999... is the sum of the infinite series.

And the sum of the series is the limit of the sequence of partial sums of the series (assuming the limit exists).

The limit in this case exists and is 1, so .999... means 1. They are two different ways of writing the same real number.

[Ether]

Originally Posted by Chris is me

The limit is a point that it will never pass.

This is not a good definition of limit.

For example, consider the function f(x)=sin(x)/x. As x approaches infinity, this function "passes" zero infinitely many times. But the limit exists and is zero.


~

[Chris is me]

Originally Posted by Ether

This is not a good definition of limit.

For example, consider the function f(x)=sin(x)/x. As x approaches infinity, this function "passes" zero infinitely many times. But the limit exists and is zero.


~

Yeah, what you said. I'll get rid of my post.