0.99~=1

ljcox · Apr 21, 2007

arod said:
Yes, isnt that the point of the proof? Showing that .999~ = 1...
I don't think you understand. When I type 0.999~, it means that the 9s repeat to infinity.

Instead of me reinvinting the wheel, here is a page showing the convergence of an infinite geometric series proving that 0.999~ = 1

Thank you.

rumiam · Apr 25, 2007

ljcox said:
Your error is in adding the decimal values for 1/3 and 2/3

These have an infinite number of decimal places and so I don't see how you can add them.

As you can see..I just did add them

bloody-orc · Apr 26, 2007

Originally Posted by ljcox
Your error is in adding the decimal values for 1/3 and 2/3

These have an infinite number of decimal places and so I don't see how you can add them.

Why can't you add two totally normal numbers as:

Code:

[U]1[/U]  and  [U]2[/U]  ?
3       3

There are no infinite numbers i can see in them... just 1,2 and 3...

Glyph · Apr 26, 2007

Proof that 0.99~ is not 1

Rewrite 0.99~ as (1-x)

(1) Therefore (1-x) =0.99~

(2) Therefore x = 0.00~1

(3) Assumption (to be proven later) x != 0

(4) Therefore, although x may be infinitely small is still not 0

(5) so if 0.99~ = 1 (as asserted by OP) then substituting into (1) the statement (1-x) =0.99~ becomes (1-x) = 1.

(6) Rearranging gives 1-1 = x

(7) therefore x = 0

(8) VIOLATION of original assumption (3) where X !=0

Proof by "Reductio ad absurdum" that 0.99~ does not equal 1

---------------------------------

Proof that assumption (3) in the above proof is itself correct, where x is not 0

(1) Axiom: 1^infinity = 1

(2) (1+0)^infinity = 1 since (1+0) = 1, therefore from (1) it must be 1

(3) base of natural logaritms "e" is give by (1+1/n)^n as n approaches infinity as given by https://en.wikipedia.org/wiki/E_(mathematical_constant)

(4) let x = 1/n

(5) as n approaches infinity, x will approach 0, therefore x = 0.000~1

(6) substitute (4) into (3) to give (1+x)^n = e since x =1/n and from (3) (1+1/n)^n = e

(7) if x = 0 then (1+x)^n = (1+0)^n = 1^n

(8) as n approaches infinity 1^n = 1 from statement (1)

(9) VIOLATION: if x = 0 then from (3) the base of natural logarithms must be e = 1 from (7) and (8) but this violates (3)

Proof that x =0.000~1 is not zero even though it is very small

proof by "reductio ad absurdum"
----------------

Therefore from the above two proofs it is shown that 0.99~ is NOT 1

if it were true then from the above proofs the base of natural logarithms should be 1.... you can check your calculator and spreadsheets to show that it is not 1.

If this proof doesn't satisfy you i honestly don't think any proof will.

Sceadwian · Apr 26, 2007

I don't care how much of a math geek you are. If you can't come to the realization that .99~ is not 1 without prompting.... There's just no hope for you as a thinking human being

arod · Apr 26, 2007

Sceadwian said:
I don't care how much of a math geek you are. If you can't come to the realization that .99~ is not 1 without prompting.... There's just no hope for you as a thinking human being

But you are wrong
https://en.wikipedia.org/wiki/.999

arod · Apr 26, 2007

Glyph said:
(2) Therefore x = 0.00~1

0.000~1 = 0
Reason:
1/(10^n) represents .000~1 as n approaches infinity

limit as n -> infinity(1/(10^n)) = 0

Therefore, .00~1 = 0.

These are fundamental laws to mathematics. This is not a debate.

Glyph · Apr 26, 2007

Ok i think i'm seeing where the confusion lies.

there are certain unusual aspects in various number systems that appear as inconsistentcies when those systems are compared against each other. An obvious example is that 1/3 while being simple and discrete appears as the rather absurd 0.3333~ when put into the decimal based system. Likewise putting the same number into the base-2 number system gives even stranger constructs.

The thing is, they are all equivalent if properly defined.

The problem arises within the definition of a number system or number set. But if we're going to get into a discussion of the finer points of number system theory we're all going to need to explicitly define what types of number systems we're using and speak in the language of mathematicians. Concepts such as real analysis and the like need to be explored. Limits and the like need different approaches than other equations to be properly expressed.

I think the problem with this thread is one of communication, we're not on the same wavelength so to speak in discussing our numbers.

Realizing this i think i'm going to withdraw from further debate.

As a final note.

It is true: 0.99~ can equal 1. It is also true that 0.99~ does NOT equal 1

the statements are not contradictory, they just apply under different circumstances that are mathematically self-consistent.

ljcox · Apr 27, 2007

Your reasoning is flawed

Glyph said:
Rewrite 0.99~ as (1-x)

(1) Therefore (1-x) =0.99~

(2) Therefore x = 0.00~1 I assume you mean 0.00 followed by an infinite number of zeros and then a 1. Ie. the last digit is 1. This is not true if there are an infinite number of zeros. There is no last digit.

(3) Assumption (to be proven later) x != 0

(4) Therefore, although x may be infinitely small is still not 0

(5) so if 0.99~ = 1 (as asserted by OP) then substituting into (1) the statement (1-x) =0.99~ becomes (1-x) = 1.

(6) Rearranging gives 1-1 = x

(7) therefore x = 0

(8) VIOLATION of original assumption (3) where X !=0

Proof by "Reductio ad absurdum" that 0.99~ does not equal 1

---------------------------------

Proof that assumption (3) in the above proof is itself correct, where x is not 0

(1) Axiom: 1^infinity = 1

(2) (1+0)^infinity = 1 since (1+0) = 1, therefore from (1) it must be 1

(3) base of natural logaritms "e" is give by (1+1/n)^n as n approaches infinity as given by https://en.wikipedia.org/wiki/E_(mathematical_constant)

(4) let x = 1/n This is not valid. You can't isolate the 1/n from the whole expression. I showed how this = e in an attachment above

(5) as n approaches infinity, x will approach 0, therefore x = 0.000~1

(6) substitute (4) into (3) to give (1+x)^n = e since x =1/n and from (3) (1+1/n)^n = e

(7) if x = 0 then (1+x)^n = (1+0)^n = 1^n

(8) as n approaches infinity 1^n = 1 from statement (1)

(9) VIOLATION: if x = 0 then from (3) the base of natural logarithms must be e = 1 from (7) and (8) but this violates (3)

Proof that x =0.000~1 is not zero even though it is very small

proof by "reductio ad absurdum"
----------------

Therefore from the above two proofs it is shown that 0.99~ is NOT 1

if it were true then from the above proofs the base of natural logarithms should be 1.... you can check your calculator and spreadsheets to show that it is not 1.

If this proof doesn't satisfy you i honestly don't think any proof will.

ljcox · Apr 27, 2007

bloody-orc said:
Originally Posted by ljcox
Your error is in adding the decimal values for 1/3 and 2/3

These have an infinite number of decimal places and so I don't see how you can add them.

Why can't you add two totally normal numbers as:

Code:

[U]1[/U] and [U]2[/U] ? 3 3

There are no infinite numbers i can see in them... just 1,2 and 3...

Of course you can add 1/3 + 2/3. My problem is in adding or subrtacting 2 numbers that have an infinite number of decimal places.

The proof that convinced me was the sum of an infinite geometric series, ie. ar/(1-r)

For 0.9999~, a = 9 and r = 0.1. Thus the sum is = 1.

Sceadwian · Apr 27, 2007

It makes a bit more sense logically when I read

For example, 28.3287 is the same number as 28.3286999…, 28.3287000, or many other representations

It just doesn't make sense to me intuativly, one of the reasons I don't like math I guess =>

Roff · Apr 30, 2007

One of you guys in the ".999... not equal to 1" gang should rewrite the Wikipedia entry. Thousands, yea, millions, of people are being misled by it.
You might want to read it first.

3v0 · Apr 30, 2007

Ron H said:
One of you guys in the ".999... not equal to 1" gang should rewrite the Wikipedia entry. Thousands, yea, millions, of people are being misled by it.
You might want to read it first.

Perhaps they should just use BCD.

phalanx · May 2, 2007

tkbits said:

How about an alternate proof?

Code:

    0.9999....
   -----------
9 ) 9.0000....
    8 1
    ---
      90
      81
      --
       90
       81
       --
        90
        81
        --
         9....

That's not a proof. That's math a 3rd grader would laugh at. When your first remainder is the same size or larger than your divisor it means you used the wrong value in the quotient. 9 goes into 9 one time, not zero.

tkbits · May 4, 2007

Which means you didn't notice the disturbing part.

The division pattern follows one of the patterns of valid repeating decimals. The remainder is the same at each step. In the more obvious invalid divisions, the remainder will grow at every step.

ljcox · May 5, 2007

tkbits said:
Which means you didn't notice the disturbing part.

The division pattern follows one of the patterns of valid repeating decimals. The remainder is the same at each step. In the more obvious invalid divisions, the remainder will grow at every step.

I don't understand why it is a problem if the remainer is the same at each step since that is what happens with any division where the result is a recurring figure, eg. divide 1 by 3.

tkbits · May 7, 2007

It's not a problem for me.

The objection is that I didn't reduce the first significant digit fully, as is normally done. Does that make the division invalid? Or, because of the repeating remainders, is it a valid division?

Electronics4you · May 21, 2007

1-0.9999999... will NEVER EVER be zero
That's wrong due to the laws of math
The right way to do it is to take the limit of the function
That's zero but only when the x-value is undefined

Roff · May 21, 2007

Electronics4you said:
1-0.9999999... will NEVER EVER be zero
That's wrong due to the laws of math
The right way to do it is to take the limit of the function
That's zero but only when the x-value is undefined

I wrote this in a previous post:

One of you guys in the ".999... not equal to 1" gang should rewrite the Wikipedia entry. Thousands, yea, millions, of people are being misled by it.
You might want to read it first.

Perhaps you would volunteer for this task?

3iMaJ · May 21, 2007

For those of you too lazy to read the wikipedia article, here is your proof. Now lets stop arguing about something that is clearly TRUE.

0.99~=1

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