0.999999 . . . . = 1

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Miles
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0.999999 . . . . = 1

Post #1

Post by Miles »

.


(Because I'm bored)


Consider a zero followed a decimal point and an infinitely long string of nines: 0.999999 . . . .

What is the value of that unending series? The answer, strangely enough, is that it equals 1.

Here's why.

1. define X as 0.999999 . . . .

2. multiply X by 10: ( 10X )

3. We know that 10 times 0.999999 = 9.999999 . . . So this means 10X = 9.999999. . .

4. Now subtract X from 10X : 9.999999 . . . -- 0.999999. . . . = 9. (the parts to the right of the decimal point disappear because they are the same)

5. So 9X = 9

6. and if 9X = 9, then X must = 1

7. Substituting for X from 1. above, 0.99999 . . . . . = 1


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Re: 0.999999 . . . . = 1

Post #2

Post by Tcg »

Miles wrote: Sat Apr 24, 2021 11:14 pm .


(Because I'm bored)


Consider a zero followed a decimal point and an infinitely long string of nines: 0.999999 . . . .

What is the value of that unending series? The answer, strangely enough, is that it equals 1.

Here's why.

1. define X as 0.999999 . . . .

2. multiply X by 10: ( 10X )

3. We know that 10 times 0.999999 = 9.999999 . . . So this means 10X = 9.999999. . .

4. Now subtract X from 10X : 9.999999 . . . -- 0.999999. . . . = 9. (the parts to the right of the decimal point disappear because they are the same)

5. So 9X = 9

6. and if 9X = 9, then X must = 1

7. Substituting for X from 1. above, 0.99999 . . . . . = 1


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If X is .9999999999..... as you've claimed in step 1.

Then 9X = 8.99999999999....

There is no reason to consider step 4 as a valid part of this calculation.

I thought you were going to discuss the sometimes inaccuracy of using floating point numbers in computer code. This is simply a slight of hand trick that fails to attract attention away from step 4. You've got to point to something that distracts attention for this to work.

Look over there...

Image

...therefore 9X = 9.

Unless I've missed something which is highly likely.


Tcg
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Re: 0.999999 . . . . = 1

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Post by Miles »

Tcg wrote: Sun Apr 25, 2021 2:17 am
If X is .9999999999..... as you've claimed in step 1.

Then 9X = 8.99999999999....


There is no reason to consider step 4 as a valid part of this calculation.
But that's the whole point of this exercise: to consider steps 4, 5, 6, and 7

Now, if you don't want to, fine, but for those of us who do it presents a mathematical conundrum. At least to my brain that never got past calculus I in college many years ago.

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Re: 0.999999 . . . . = 1

Post #4

Post by Tcg »

Miles wrote: Sun Apr 25, 2021 3:05 pm
Tcg wrote: Sun Apr 25, 2021 2:17 am
If X is .9999999999..... as you've claimed in step 1.

Then 9X = 8.99999999999....


There is no reason to consider step 4 as a valid part of this calculation.
But that's the whole point of this exercise: to consider steps 4, 5, 6, and 7

Now, if you don't want to, fine, but for those of us who do it presents a mathematical conundrum. At least to my brain that never got past calculus I in college many years ago.

.
Like I said, I might have missed something and apparently I did.


Tcg
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Re: 0.999999 . . . . = 1

Post #5

Post by Difflugia »

Tcg wrote: Sun Apr 25, 2021 2:17 amI thought you were going to discuss the sometimes inaccuracy of using floating point numbers in computer code.
In a sense, he is.

A base system can only represent a fraction as a terminating decimal if the denominator and the base share all of their prime factors. Base 10 can only represent a fraction accurately if its denominator's only prime factors are 2 and 5. It can represent 1/2, 1/5, 1/10, 1/4, and 1/25, but can't represent 1/3, 1/6, or 1/15. Base 2 can represent fractions whose denominators only have a prime factor of 2, so it can accurately represent 1/2 or 1/4, but can't represent 1/5 or 1/10.

Fractions that can't be represented exactly in a particular base, end up as repeating decimals if one does the long division. The deal with the .9 repeating is that no long division problem will ever end up with that as an answer. That's where the "sleight of hand" is.

Any math problem that could end up as .9 repeating is actually 1:
  • 1/3 = .3 repeating, 2/3 = .6 repeating, 1/3 + 2/3 = ?
  • 1/9 = .1 repeating, 2/9 = .2 repeating, 3/9 = .3 repeating, ..., 8/9 = .8 repeating, 9/9 = ?
  • 1 - .7 repeating = .2 repeating, 1 - .8 repeating = .1 repeating, 1 - .9 repeating = ?
Repeating decimals themselves are a sort of representational sleight of hand trick. A repeating sequence of "base - 1" is the point where the trick breaks down.
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