Does 1.0 equal 0.9999999999999999...?

[Robert James Liguori]

Does 1.0 equal 0.9999999999999999...?
http://robertjliguori.blogspot.com/2017/02/does-10-equal-09999999999999999.html

[Campbell Ritchie]

No, they are not equal. What you are seeing is the inability of the floating‑point datatypes to distinguish such small differences.

[Campbell Ritchie]

Surely you shou‍ld have written
if (bigDecimal1.compareTo(bigDecimal2) != 0) ...
rather than using <. It still shows not equal.

[Jesper de Jong]

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.

[Jeanne Boyarsky]

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

[fred rosenberger]

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.

[Campbell Ritchie]

Jesper de Jong wrote:. . . (infinitely repeating 9's) is equal to 1. . . . an approximation with a finite number of 9's. . . .

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.

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

[fred rosenberger]

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]

Yes, that looks right. If x is ∞ however, 0.0...01 degenerates to 0.0 and the two numbers become mutually equal.

[Randall Twede]

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

[Stephan van Hulst]

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.

[Henry Wong]

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