Comments on: .9999999… Is Equal to 1.000000

By: Grey matter

Grey matter — Thu, 10 Feb 2011 07:28:57 +0000

I think that our whole math system is screwed up.Maybe,somewhere in the universe,a race of aliens have devised a perfect math system which does not have any problems like what is 1/0?And how do deal with repeating decimals.

By: zkeletenz

zkeletenz — Wed, 27 Oct 2010 04:27:25 +0000

Thanks to you Travis, and to all others echoing the proof.

Mr. Lombard: I actually thought of the concept when I was a fetus. So, as they say in Intellectual Property parlance, I “swear behind your conception date.”

By: zkeletenz

zkeletenz — Wed, 27 Oct 2010 04:18:13 +0000

Pax: Yes, I am confident about my logic, but I am not a mathematician.

Thanks for the support, Hoax.

By: zkeletenz

zkeletenz — Wed, 27 Oct 2010 03:57:51 +0000

To respond more specifically to your reasoning, the example you provide is unfair, because it happens to coincide with the limits of the decimal system. 8/2 works with the decimal system perfectly, because it divides evenly by integers. The same cannot be said for 1/3.

To explain it on its own terms, with no need for analogy. Let’s say that you have a rifle with 10 horizontal aiming positions. Now, you have a target that lands at position 4 (8/2nds from your example) position. Can you hit it? Yes.

But can you hit a target at the 1/3rd horizontal position? Unfortunately, no. You need a better scope. You can change to a 100-point scope, or 1000 or 1 google plex. But you still have the same problem.

This is not a paradox – it is perfectly consistent. It is just another way of saying that you have a digital scale (10-point, 100, 1000, or whatever digital limit you are using). An infinitely-repeating digital expression will never perfectly express a fraction that does not divide evenly. That’s what infinite repetition means – it is NOT FINITE.

By: zkeletenz

zkeletenz — Wed, 27 Oct 2010 03:43:28 +0000

Thankyou, professor, this “namecalling” you have exhibited is yet another example of approximated terms to express more complex concepts. In the same way that your general, unreasoned term “Poppycock” gives insufficiently particular information concerning the concept you seek to advnace, the decimal expression .333 (repeated infinitely) gives insufficient expression to the concept of dividing into 1/3rd.

/takes his bow
//knows he is a wise-a&&

By: Professor Lessor

Professor Lessor — Wed, 11 Aug 2010 04:14:27 +0000

@post #16 to use a non-math term…poppycock. 0.33… is not an approximation for the fraction 1/3 any more than 8/2 is an approximation for 4.

By: Professor Lessor

Professor Lessor — Wed, 11 Aug 2010 04:09:39 +0000

And +1 for post #41. Complex analysis was one of my favorite math classes.

By: Professor Lessor

Professor Lessor — Wed, 11 Aug 2010 04:05:57 +0000

Additionally, the statement that you “must” start multiplication, addition, or subtraction from the right is simply false, (in fact many of the worlds fastest “human calculators” add and subtract from left to right). The only reason to go from right to left in our standard computation method is to “carry” to the next largest place value. In the case of something like .123 by 3. Try it from right to left and left to right. Because you don’t have to “carry” it’s just as easy to compute it in either order. For .33333… times 3 for example, since you have “3s forever” in the repeating decimal, you are guaranteed to multiply 3 in some decimal place value times the whole number 3, thus no carrying. This makes it perfectly okay to multiply from left to right, even in our standard “vertical” method of computation.

By: Professor Lessor

Professor Lessor — Wed, 11 Aug 2010 03:56:36 +0000

The assumption that there is a unique way to represent a number in decimal form is the main flaw in the argument for .9 repeating to be worth a different value that 1. This is simply not true. The decimal system is based on fraction fractions (that’s why the place values are named as tenths, hundredths, and so on). There are equivalent fractions that use different numbers to represent the same quantity, and, with repeating decimals, equivalent decimals using different numbers to represent the exact same values. On the topic of fractions .9 repeating can be expressed as an infinite sum of 9*(sigma) (1/10)^n where n ranges from 1 to infinity. In calculus this sum is easily proven to equal exactly 1. Q.E.D. as it were.

By: Minnesotastan

Minnesotastan — Fri, 30 Jul 2010 12:07:03 +0000

Vapur, you are quite correct that .333 does not equal 1/3. But part of the nucleus of this puzzle lies in the question of whether .3333…(infinite) is equal to 1/3. If not, how could one represent 1/3 in decimal form?

By: Vapur

Vapur — Fri, 30 Jul 2010 09:18:20 +0000

Logic comparison to an earlier post:

x = 0.999…
10x = 9.999…
10x – x = 9.999… – 0.999…
9x = 9
x = 1

vs.

x = 0.8
10x = 8.0
10x – x = 8 – 0.8
9x = 7.2
x = 0.8

Therefore, infinite numbers are introducing calculation problems. Infinity minus infinity does not equal zero.

By: Vapur

Vapur — Fri, 30 Jul 2010 08:51:36 +0000

After reading all these comments, I am still confused about the logic of your hypothesis. You start your argument by stating the equality of two different numbers (0.999… and 1); then, you make a comparison by converting the units to a fraction (1/3, 2/3, and 3/3). Considering that you are using a base 10 number system initially, the representation of the actual number for 1/3 is imperfect and results in an infinite calculation. Changing to the ternary number system can alleviate this issues, but that can introduce other problems when calculating with numbers that aren’t easily represented in that system. Depending on the calculation (with fractions versus decimals) it can appear that an infinite number is actually equal to some constant number. The truth of the matter is that 3/3 is equal to 1, while 0.333… multiplied by 3 is equal to 0.999… and not 1. The context is highly important since it changes the precision of your answer.

By: bpurcell

bpurcell — Tue, 27 Apr 2010 02:28:59 +0000

I was just explaining this to my 6th grade son, and he was extremely confused. I actually think it’s cute in a scary “American Education System” way that there are people here arguing that 0.333… = 1/3 is just an approximation for convenience. Just do the opposite; use long division to divide 1 by 3. Just because we can’t finish the problem in a finite amount of time doesn’t make it an approximation. It is what it is.

There are several proofs for 0.999… = 1, and the some are already here. Math deals with infinity and repeating decimals ALL THE TIME. If 0.333… was just an approximation, all of our upper mathematical theories would just collapse in failure.

And don’t even get started with e^(pi*i) + 1 = 0. That might make people’s heads explode.

By: Jromo

Jromo — Thu, 18 Mar 2010 15:39:09 +0000

wouldn’t .9… subtracted from 1 equal .0…1 ? In other words, point zero repeating with 1 infinitely to the right of the decimal point?

By: Brian-

Brian- — Tue, 16 Mar 2010 00:52:02 +0000

I can’t believe how many ignorant people opposing the fact that 0.9… = 1 have tried to justify their position.

By: Travis

Travis — Tue, 09 Mar 2010 16:23:41 +0000

So… Last time I checked, 3/3=1, not .999…, thus illustrating zkeletenz’ point nicely from the get-go. One need not confuse the issue further by trying to cram a square “mathematical work-around” peg into a round “mathematical proof” hole.

By: Video Game Dork

Video Game Dork — Tue, 09 Mar 2010 14:35:33 +0000

it is true that ‘asymptotic to 1 and equal to 1 are different’. However, .9 repeating is not ‘a’ symptotic to 1′. It is equal to one.

What people above said about ‘.3 repeating’ being the closest we can come to digitally displaying the value of 1/3, is true.

The concept of infinity is hard for the human mind to grasp; we can really only understand it abstractly. Thats why .3 repeating (and .9 repeating) cause such mental dissonance. Its kinda the same reason some people are so obsessed with finding digits of pi.

By: pax

pax — Mon, 08 Mar 2010 13:50:37 +0000

Now I’m just confused, so I’d rather not say anything except, Mike T., that is a very clever joke.

By: badwolf42

badwolf42 — Sun, 07 Mar 2010 19:27:15 +0000

To my engineering mind, I see this as an asymptote. (Which is typically accepted to never actually reach the asymptotic value, but gets ‘close enough’ for physicists.)
In this case it’s a function of dividing by 10 and adding to the summation. The function is asymptotic to 1. To me, asymptotic to 1 and equal to 1 are different, but not for any practical purpose.

By: Mike T.

Mike T. — Sun, 07 Mar 2010 07:16:28 +0000

An infinite number of mathematicians walk into a bar. The first orders one beer, the next one orders half a beer, the next one orders a quarter of a beer, the next an eighth, the next a sixteenth — then the bartender interrupts them by saying “You’re all idiots” and pouring them two beers.

By: Video Game Dork

Video Game Dork — Sun, 07 Mar 2010 06:55:39 +0000

.9 repeating is equal to one. Thats really its definition. Since .3 repeaing is ‘one third’, three thirds is a whole.

I think I encountered this idea in 1st grade or something when I told the teacher I’d found the smallest number EVER! and said it was 1 – .9 repeating. She thought it was cute and told me whats up.

By: Another Jake

Another Jake — Sun, 07 Mar 2010 00:42:58 +0000

I’m going to amend that statement by saying that since 3/3 can be represented as “1″, 0.9-bar is an unnecessary and meaningless numerical representation.

By: Another Jake

Another Jake — Sun, 07 Mar 2010 00:30:02 +0000

It’s a simple error if you ask me. 0.3-bar + 0.6-bar = 1, not 0.9-bar.

By: rideicon

rideicon — Sun, 07 Mar 2010 00:03:34 +0000

to state that .9999 = 1 because 1/3 = .33333 is just ignorance.
.3333 is as close as we can write 1/3 in numbers without having a fraction, but .999 repeating is not 1,

By: Minnesotastan

Minnesotastan — Sat, 06 Mar 2010 18:20:12 +0000

I probably should let this thread just die. A not-surprisingly-similar discussion went on for ages at the original math blog; I counted 18 PAGES of comments – and that was just for the first DAY – until the comment section was closed. Let me offer an attempt at closure.

It’s true that .33 is not the same as 1/3, but .333… (with the ellipsis) IS the same. [[It's awkward to use the ellipsis because the proper conventional mathematical symbol is a vinculum above the repeating decimals, as is shown in the figure, but that isn't available in the character set of the font I'm using here, so I'll continue to use the ellipsis to designating a repeating string of digits.]]

But back to the argument. .333… IS the same as 1/3 because that’s the way 1/3 is expressed in decimals. One-third could be expressed in other ways in other base systems, but in a base10 system the only choice is .333… (or with the vinculum).

Understanding that, just say to yourself .333… is a way to REPRESENT 1/3. And .666… is a way to REPRESENT 2/3. Then .999… is equally a way in the decimal system to REPRESENT the number 1. [You can also represent 1 as 1.0 or 1.00. Other ways to represent it are 7/7 or 10 to the power 0.]

.999… IS the number one.

And it’s not correct to say that “an infinitely repeating number can not be applied to the finite things we deal with on a regular basis.” To prove it, tonight I will eat .333… of a pizza.

By: Grant Yoder

Grant Yoder — Sat, 06 Mar 2010 17:36:50 +0000

No offense, but that’s like saying “y=x/2 WILL reach 0″…. 1 is the limit of .99999999, not the value.

By: dorkafork

dorkafork — Sat, 06 Mar 2010 17:17:05 +0000

No.

0.3333… is not 1/3rd and 2/3rd is not 0.66666 – those are just an approximates we use for convenience.

No, they’re exactly equal to 1/3 & 2/3. That is the entire purpose the bar notation was created, to allow an exact notation for fractions like 1/3 in decimal notation. If bar notation was not used in that way, mathematicians would have had to invent some other way to represent 1/3 in decimal.

Furthermore they are not “approximates we use for convenience”. Approximates are decidedly inconvenient for countless engineering and scientific problems. Significant digits are important.

By: OddNumber

OddNumber — Sat, 06 Mar 2010 15:03:27 +0000

This is just sloppy use of significant digits…

By: duncan

duncan — Sat, 06 Mar 2010 11:35:45 +0000

No.

0.3333… is not 1/3rd and 2/3rd is not 0.66666 – those are just an approximates we use for convenience. There are many fractions that can’t be perfectly expressed by the decimal system. The fact that the decimals must go on forever is a symptom of that fact. Whoever originally thought this up is just using this gotcha either on purpose or unwittingly to make the 0.999 == 1.0 point.

There is no such point of contention.

By: Sig Orn

Sig Orn — Sat, 06 Mar 2010 10:45:41 +0000

and this is why economists can’t predict anything about the future, they deal with numbers that aren’t.

By: ND

ND — Sat, 06 Mar 2010 05:51:16 +0000

I am a mathematician, and this comment stream has been frustrating to read. Yes, 0.9(bar)=1. They are just two different ways of expressing the same number. It’s neither a failure of mathematics nor language, though it is an interesting thing to note. Glad to see neatorama take an interest in the subject nearest to my heart.

By: dennis miller

dennis miller — Sat, 06 Mar 2010 04:18:54 +0000

When reading this explaination I can’t help but think of the Abott & Costello routine where Abott explains why “7 x 13 equals 28″. Just because we lack the means to express an idea shouldn’t then mean our “closest possible answer” is then the absolute truth. I would point to the very Sciency, “horse shoes and hand grenades” theorem.

By: alquimista

alquimista — Sat, 06 Mar 2010 02:37:22 +0000

The expression 0.999… = 1 will always be a matter of contention for a few reasons.

a. The expression is a convention. Its generally agreed to be true. The convention is not without its uses, but as conventions go, you will always have nay sayers.

b. 0.000…1 cannot exist as a non-terminating decimal. Essentially, it terminates at the decimal place where the 1 occurs. This means that you cannot subtract 0.999… from 1. If you don’t believe me, try solving 0.999… + X = 1 within the constraints of Dedekind cuts.

c. Until someone within the mathematical community has the spine to stand up and say 1/0 = [infinity], one will always have to defend the idea that certain non-terminating decimals are rational while a non-terminating decimal like Pi is irrational.

Those who simply dismiss this age old discussion as a waste of time, fail to recognize the importance in accurately defining basic principals of math.

Keep the discussion going Netorama. I’m glad to see someone in the main stream bringing this discourse back to light.

By: Hoax

Hoax — Sat, 06 Mar 2010 01:43:00 +0000

…on another note, .999 repeating APPROACHES the value of 1 but it can NEVER BE 1. a little of the limit theorem from calculus would keep you puzzling over this for months.

By: Hoax

Hoax — Sat, 06 Mar 2010 01:38:31 +0000

zkeletenz is correct. The repeating decimal values are called approximations. Therefore it IS true that 1/3 is approximately .333…. and 2/3 is approximately .666…. and 1 is approximately .999…. the proper sign to use looks like two stacked tildes, or a wavy equal sign.

By: pax

pax — Sat, 06 Mar 2010 01:06:38 +0000

@zkeletenz, really? Nobody told me that so I just took it as 100% true.
I’ve asked three people about this (two are my math professors and one is my really smart friend) and they said it wasn’t exactly equal to it (my current professor told me she forgot why.).
I thought since repeating decimals can be turned into x/9 as fractions, .99999999999 became 9/9 thus equal to one, but zkeletenz’ argument nullifies this, I guess.

By: clinton robert labombard

clinton robert labombard — Fri, 05 Mar 2010 22:26:12 +0000

It’s nice to see something I figured out when I was a kid finally catching on.

By: zkeletenz

zkeletenz — Fri, 05 Mar 2010 21:14:14 +0000

The mistake here is assuming that the decimals .333… and .666… are accurate expressions of 1/3 and 2/3, respectively. They are not. They are simply the closest digital expressions can come to expressing analog. And that’s why your CD/MP3 will never sound as good as granpa’s phonograph. Sorry!

By: Anonymustard

Anonymustard — Fri, 05 Mar 2010 17:16:52 +0000

Is there something about this that isn’t entirely obvious?

By: hsdell

hsdell — Fri, 05 Mar 2010 17:11:15 +0000

Not a math geek by any stretch, but seems like it is associated with Xeno’s Paradox (for you Godel, Escher, Bach fans).

http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

By: KIK000

KIK000 — Fri, 05 Mar 2010 16:31:14 +0000

it’s not a failure of math and it’s not a failure of language either.
It is what it is, a theorem if you will; and a infinitely repeating number can not be applied to the finite things we deal with on a regular basis. If we wanted to talk about infinite things this would apply and perhaps be useful but how many times have you tried to measure something infinite in the past 6 months? and how many true statements can we, as the human race, make to describe things we think to be infinite?
Human beings tend to think what’s easy to understand is correct and what is hard to comprehend must be wrong. Just because you can’t find a use for this or because it seems like a difficult concept does not mean it is invalid or wrong or not worth talking about it. If it does not interest you.. walk away.

By: Boredas

Boredas — Fri, 05 Mar 2010 15:34:05 +0000

Stuart McCracken – good copy paste from many many websites,

This is old, pointless and not funny geeky maths to even the maths geeks.

roll on by, old news is old new – blizzard even did it as a joke one time its that popular – so come on….

By: Edward

Edward — Fri, 05 Mar 2010 15:18:54 +0000

Isn’t the space between theory and practice a wonderful playground? Everyone can be right. My brother and I have had variations on this discussion for 40 years.

By: AA

AA — Fri, 05 Mar 2010 15:15:57 +0000

selective ignorance is not funny

http://xkcd.com/169/

By: kingthorin

kingthorin — Fri, 05 Mar 2010 14:55:06 +0000

If this statement is true “But, you say, there’s a one at the end that string of zeroes. No, there isn’t, because the string of 9s doesn’t end.”

Then the same can be said for 0.000000…1 you never reach the 1 at the end therefore you haven’t really done the subtraction properly. To put it your way “the string of 0s [before the one] doesn’t end.”

By: Stuart McCracken

Stuart McCracken — Fri, 05 Mar 2010 14:47:39 +0000

The way that I understand it best is:

x = 0.999..
10x = 9.999..
10x – x = 9.999.. – 0.999..
9x = 9
x = 1

Since we assumed that x = 0.999.., and have shown that x = 1, it must be the case that 0.999.. = 1

By: Craig

Craig — Fri, 05 Mar 2010 14:46:52 +0000

Anyone who knows math is rolling their eyes right now.

By: Merrick

Merrick — Fri, 05 Mar 2010 14:36:38 +0000

It doesn’t seem to make sense because point nine repeating is just plain not a number. You can’t have point nine repeating of anything. You can have .9 of something, or .99, or .9999, but what would .9 repeating, as a number, even mean? That’s why math doesn’t seem to work on it right.

By: roycommi

roycommi — Fri, 05 Mar 2010 14:36:03 +0000

.9999.. is not equal to one.

The fact that you can rationalize that it is thru mathematics does not in fact prove the author’s point.

I believe the fault here is assuming that mathematics simply reflects some pure objective fact, when it doesn’t, or something.

By: Gauldar

Gauldar — Fri, 05 Mar 2010 14:34:29 +0000

Is has there ever been a practical application of this? I’ve heard this plenty of times, but I have yet to see it used in any beneficial manor.