posted 10 months ago

Does 1.0 equal 0.9999999999999999...?

http://robertjliguori.blogspot.com/2017/02/does-10-equal-09999999999999999.html

http://robertjliguori.blogspot.com/2017/02/does-10-equal-09999999999999999.html

Campbell Ritchie

Marshal

Posts: 56529

172

Campbell Ritchie

Marshal

Posts: 56529

172

posted 10 months ago

Actually, 0.9999...... (infinitely repeating 9's)

But the program in the blog is not really showing that, because it isn't working with the number 0.9999...... (infinitely repeating 9's), but with an approximation with a finite number of 9's.

- 1

Campbell Ritchie wrote:No, they are not equal.

Actually, 0.9999...... (infinitely repeating 9's)

*is*equal to 1.

But the program in the blog is not really showing that, because it isn't working with the number 0.9999...... (infinitely repeating 9's), but with an approximation with a finite number of 9's.

posted 10 months ago

Fun! That's a lot of 9's in your blog post!

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posted 10 months ago

regardless of how many 9's there are in that post, that's not even CLOSE to an infinite number of 9's.

I haven't watched the video, but there are a couple of simple ways to show that 0.999... is exactly 1. The simplest relise on the fact that multiplication and division are inverse operations. If I divide by 2, then multiply by 2, i get back my original number. That is a fundamental property of multiplcation.

so, i can take a number...Say 1, and divide it by 3, then multiply that result by three, and i'm back to my original number.

1 / 3 = 0.33333....and on and on forever.

0.3333...and on and on forever * 3 is 0.9999....and on and on forever. And so it must equal 1.

I haven't watched the video, but there are a couple of simple ways to show that 0.999... is exactly 1. The simplest relise on the fact that multiplication and division are inverse operations. If I divide by 2, then multiply by 2, i get back my original number. That is a fundamental property of multiplcation.

so, i can take a number...Say 1, and divide it by 3, then multiply that result by three, and i'm back to my original number.

1 / 3 = 0.33333....and on and on forever.

0.3333...and on and on forever * 3 is 0.9999....and on and on forever. And so it must equal 1.

There are only two hard things in computer science: cache invalidation, naming things, and off-by-one errors

Campbell Ritchie

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posted 10 months ago

1: 0.999999999999999999 = 1.0 (Assumption) 2: 0.9999999999999999999 is closer to 1.0 ∴ 0.9999999999999999999 = 1.0 3: 0.999999999999999999 ≠ 0.9999999999999999999 (Arithmetic) 4: 1.0 = 1.0 (Arithmetic) 5: 1.0 ≠ 1.0 (Deduction from lines 1, 2, and 3) 6: 0.999999999999999999 ≠ 1.0 (

Yes, that is the problem. It should be possible to show that all finite sequences of multiple 9s after the decimal point do not equal 1.0.Jesper de Jong wrote:. . . (infinitely repeating 9's)isequal to 1. . . . an approximation with a finite number of 9's. . . .

*Reductio ad absurdum*of assumption 1 under 4 and 5)

posted 10 months ago

it should be simpler than that.

1 - 0.<x nines> = 0.<(x-1) zeros>1

I think i have that right...

so if there is a non-zero difference between two number, they are not the same.
There are only two hard things in computer science: cache invalidation, naming things, and off-by-one errors

- 1

Campbell Ritchie wrote:Yes, that is the problem. It should be possible to show that all finite sequences of multiple 9s after the decimal point do not equal 1.0

it should be simpler than that.

1 - 0.<x nines> = 0.<(x-1) zeros>1

I think i have that right...

so if there is a non-zero difference between two number, they are not the same.

Campbell Ritchie

Marshal

Posts: 56529

172

posted 3 months ago

i was just gonna mention that infinity problem. i dont want to take sides in this argument

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Stephan van Hulst

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posted 3 months ago

There is no argument here Randall :)

There is a mathematical proof that 0.‾9 is equal to 1, so there is no side to pick. It's just a fact.

There is a mathematical proof that 0.‾9 is equal to 1, so there is no side to pick. It's just a fact.

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posted 3 months ago

Yeah, there are lots of proofs out there -- including one using 9th grade algebra. So, most 14 and 15 year old children (in the U.S. that is, other countries may be younger) knows this fact...

Henry

Stephan van Hulst wrote:

There is a mathematical proof that 0.‾9 is equal to 1, so there is no side to pick. It's just a fact.

Yeah, there are lots of proofs out there -- including one using 9th grade algebra. So, most 14 and 15 year old children (in the U.S. that is, other countries may be younger) knows this fact...

Henry