.9999999… Is Equal to 1.000000

2003

When you add decimals together, you must start with the right most place value. In repeating decimals, there is no right most value, so the best that you can do is truncate and estimate. So .999… is NOT an exact value. You must convert to fractions to find the exact value.

.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.

Anyone who knows math is rolling their eyes right now.

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.”

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.

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.

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

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

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.

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.

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.

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.

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.

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

.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.

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.

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.

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

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.

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.

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.

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

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&&

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

Thanks for the support, Hoax.

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.