What is .99 repeating as a fraction?

Quote: Wizard

It shows several explanations why 0.9...=1. For what it is worth, I've never known anyone with a strong mathematical background to claim otherwise.

Mathematics as we know it for well over a thousand years says that 0.999 ... = 1 . But the philosophical debate goes back almost 3000 years. Zeno of Elea was trying to prove mathematically that motion of two bodies always involved the sum of an infinite series. By his argument the sum of an infinite series could never equal a finite number. The point of the discussion was that the motion of two bodies was an illusion, and we were left with a universe that was actually "all is one". He was defending a school of thought championed by Parmenides that we associate nowadays with Zen Buddhism.

But most of the dissenting posts on this thread are not sounding so rational as to be defending a philosophical train of thought.

Somewhere in high school the problem is usually 'solved' as letting S be the sum..
S=.99999 ....
S= 9/10+ 9/10^2+ 9/10^3+ 9/10^4.... (equation #1) definition of S
1/10*S= 9/10^2+ 9/10^3+ 9/10^4.... (equation #2) multiplying (equation #1) by 1/10
S can be replaced by 10/10*S in equation #1
9/10*S = 9/10 (equation #3) gotten by subtracting (equation #2) from (equation #1)
S=1 (equation #4) gotten by multiplying both sides by 10/9

Quote: Wizard

I didn't read all of it, but agree it is a good mathematical treatment of the concept. It shows several explanations why 0.9...=1. For what it is worth, I've never known anyone with a strong mathematical background to claim otherwise.

Here is the link again.

My favorite line?

Fascinating thread. I sure am glad I didn't bet mustangsally that 9/9 is a fractional expression of .999... I think I would have lost :)

This statement that .999... is equal to zero, though, has a little wording problem. You see, they didn't prove that .999... is equal to 1. That would mean, of course, that .999... is always equal to 1. That's not correct. What they proved is that certain fractions are equal to both .999... and also to 1. For instance, 9/9 is a fractional expression for .999... and also 1.

You see, other fractions, such as 2/2, for instance, only equal 1, they never equal .999... The proof, from the website above, would fail for this fraction.

So, to just say .999... is equal to 1 is misleading. It should be stated that .999... is equal to 1 for certain fractions. ex 9/9. See, 1/1 =1, 2/2=1, 3/3 = .999... or 1 There's 2 valid expressions for this particular fraction. However, it can't be both. You can interpret 9/9 to be either .999... or 1. If you have 2 different people one might interpret 9/9 as being 1, another might interpret it as .999..., both would be valid.

Fascinating Stuff.

The website from above did inspire me to try to take a stab at proving that 1 is not equal to .999... mathematically instead of logically.

.999... - .999... = 0

1 - .999... = a number >0

or

solve for x

.999... + x = 1

1 + x = 1.x

The interest thing is of course these fractions with two possible interpretations.

1 - 1 = 0

1 - 2/2 = 0

1 - 9/9 = a number >0 , or just 0 It has two valid answers.

This might make for a nice prop bet out here in Vegas:)

Quote: Doc

Thanks for not exposing us to that drivel twice.

Wow. Dude, chill out.

It's an artifact of the notation. There is nothing "natural" about expressing numbers in relation to 10 (or any other way, really). The numbers themselves are concepts. When you express the number in relation to 10, .9999999..... is the same as 1. Otherwise, the concept is the same, which is shown when you express the number as a ratio (fraction).

Quote: YoDiceRoll11

Wow. Dude, chill out.

YDR11: I think that you and I have had polite exchanges on this topic, even though we disagree. I think that you believe in your position, even though I contend that your sources have errors in logic: I have pointed out the circular reasoning and the attempts to discredit the .999... = 1 position by disregarding/refuting the standard definitions that are used by those holding that position. Even though we disagree, I have found no reason not to maintain a polite dialog.

In contrast, JyBrd0403 has made numerous posts in this thread that make no sense at all. On several occasions yesterday, he noted that he was drunk, and I mostly disregarded his posts, allowing that these may have just been the typical results of intoxication. If that was the case, it's time either to sober up or to spend more time away from this forum. Earlier, I made a comment about sometimes forming the opinion that someone either is an idiot or is putting me on. I suspect (don't really know) that JyBrd0403 is putting us on. I am not convinced that he has considered this topic sufficiently to have actually formulated an opinion. I think he just likes to jerk folks around by posting stuff that I think I complimented by calling it drivel. He may even be another one of the trolls who show up here far too often just trying to provoke arguments by posting inane comments/opinions/claims. He has been a member of the forum for 10 days and has made 54 posts in 3 threads. I don't think I have noticed a post yet that really made much sense.

Quote: JyBrd0403

mathematically instead of logically.

The interest thing is of course these fractions with two possible interpretations.

It has two valid answers

This statement that .999... is equal to zero, though, has a little wording problem.

It's like a piano doesn't hit exact notes; it is tuned to get really, really close. Likewise, it isn't the number itself (which is only a concept, an idea representing a way of understanding reality) that is or isn't equal to another number; it is that our ways of expressing the number (either .99999...... or 1.0) represent the same value.

The sum of the vanishingly small 9s approaches 1 as they go toward infinity. That apparent paradox is built into the system that we use to express the numbers.

Quote: JyBrd0403

1 - 2/2 = 0

this is great :-)))))))))))))

As a postscript, I'm reading "The Information", the latest book by James Gleick. In it he discusses the cuneiform writing of the ancient Babylonians. Their tablets were full of accounting records, things like receipts for barley and livestock, and they were recorded in sexagesimal: base 60.

Quote: Doc

YDR11: Even though we disagree, I have found no reason not to maintain a polite dialog.

In contrast, JyBrd0403 has made numerous posts in this thread that make no sense at all. On several occasions yesterday, he noted that he was drunk.....

LOL, ok fair enough mate.