I would like to argue about .999

[uart]

decimals of fractions never accurately equal those fractions.

So are you implying that 0.5 doesn't equal $\frac{1}{2}$ then? Ok I think that you meant to say recurring decimals, but anyway you're still wrong.

Curd can you please tell us what you think $(1 - 0.999...)$ is equal to? If 0.999... and 1 are not equal then their difference (subtraction) must be non-zero right. So please try to write down that difference. You can use the "..." notation to indicate "repeating forever", we will understand what you mean.

When you actually try to do this you will find that you fall into a "trap" and hopefully that will help you understand why they are equal. Please give it a try.

 

[Stephen Tashi]

I like Fredrik's treatment. It is a question of defining .999... before starting to argue about what it's equal to.

On the ".9999... does not equal to 1" side, arguments usually hinge on the statement that any finite number of 9's gives a number less than 1. This is true, but irrelevant. If you are going to talk about an infinite number of 9's then talk about an infinite number of 9's, not a finite number of them.

 

[statdad]

doesn't matter if it's beautiful or not. if it is not reflective of reality then it's not accurate. what do physicists say? that's what will really count.

Okay, now we see you're not really interested in learning anything, you're simply trolling. The massive movement of the goalposts here seals that.

 

[Curd]

Here's an interesting proof, based on knowledge from 4th grade:

I'll pose it as questions, so as to allow the reader to reach their own conclusion.

Are there real numbers between any two unequal real numbers?
What is a number between .999... and 1?

You learn in 4th grade, in simpler terms, that the reals are dense. In any dense set, there are infinitely many members between any two members. All real numbers have a decimal expansion. There is no decimal expansion between .999... and 1, therefore there is no number between them. There is no number between them, therefore they are equal.

actually, there is a decimal expansion between .999... and 1. it's .9999... (the extra 9 and . signifying that it is still expanding and that there will always be a 9 between the two)

how can the two meet if .999... is expanding onward forever? it would have to stop expanding to reach 1.

also, for those of you whose ego's are attached to this issue (like the above poster and the one above him), please but out. I didn't come here for an attitude.

 

[Curd]

So are you implying that 0.5 doesn't equal $\frac{1}{2}$ then? Ok I think that you meant to say recurring decimals, but anyway you're still wrong.

Curd can you please tell us what you think $(1 - 0.999...)$ is equal to? If 0.999... and 1 are not equal then their difference (subtraction) must be non-zero right. So please try to write down that difference. You can use the "..." notation to indicate "repeating forever", we will understand what you mean.

When you actually try to do this you will find that you fall into a "trap" and hopefully that will help you understand why they are equal. Please give it a try.

actually no. please do not assert that I've said things that i have not. you could tell clearly from the pattern that i had set up with my earlier argument about .111... and it's relation to .999... that my argument only dealt with fractions who are represented by infinitely repeating digits after a decimal.


in such cases those decimals are only very precise approximations of the actual fraction. what I'm saying is that decimals can not always perfectly represent a fraction which is why ... has been added to some of them.


and as for your argument that there is a trap. 1-.999... = 1-.999... since there is no other way to write it that i know of. in words it equals an infinitely and increasingly small space between 1 and .999... although that space never does reach 0 because it is forever in the process of trying to reach zero without actually achieving it.


if this were not true then .999... could not go on for infinity as you (seem to) suggest it does. if it goes on for infinity then there must be the potential for an infinite space between .999... and 1 and ergo .999... can go on for infinity without reaching one. otherwise you have contradicted the concept of infinity.

 

[statdad]

"actually, there is a decimal expansion between .999... and 1. it's .9999..."
No, $$0.999 \dots$$ and $$0.9999 \dots$$ are precisely the same thing.
You could not state more clearly that you understand neither the notation nor the issue at hand.

"also, for those of you whose ego's are attached to this issue (like the above poster and the one above him), please but out. I didn't come here for an attitude."

And you could not state more clearly that you have no intention of trying to understand any of the explanations you've been presented.

 

[micromass]

Wow, Frederik, I really like your explanation! I'm going to save it and quote it every time the 0.999... issue comes up again.

As for the OP, it seems that we are misunderstanding each other. Can you please clarify the following:

1) What exactly do you mean with the notation 0.999...? How is this defined for you?
2) How does 0.999... occur in nature and physics which contradicts it being equal to 1?

As for the issue you brought to our attentian: 0.999... and 0.9999... are EXACTLY the same thing. How many nines come after the 0 in the first case? infinitely many. And in the second case? again, infinitely many.

 

[Curd]

I like Fredrik's treatment. It is a question of defining .999... before starting to argue about what it's equal to.

On the ".9999... does not equal to 1" side, arguments usually hinge on the statement that any finite number of 9's gives a number less than 1. This is true, but irrelevant. If you are going to talk about an infinite number of 9's then talk about an infinite number of 9's, not a finite number of them.

i was talking about and infinite number of 9's (although i may have typed it improperly).

this could be an issue of semantics, but how could .999... go on for infinity if it could be stopped from doing so by 1? if 1 were a goal that an infinite number of .9's extending onward forever could reach then how would the possibility of reaching that goal also mean that there's no possibility of .999... going past 1?

 

[Curd]

As for the issue you brought to our attentian: 0.999... and 0.9999... are EXACTLY the same thing. How many nines come after the 0 in the first case? infinitely many. And in the second case? again, infinitely many.

:)

and ergo the will always be another 9 past the decimal to prevent .999... from achieving 1. that was my point.

 

[micromass]

:)

and ergo the will always be another 9 past the decimal to prevent .999... from achieving 1. that was my point.

No, I don't see how that could possibly follow from what I've said...

And do you care to answer my two questions? It would enhance the discussion a great deal...

 

[Char. Limit]

But Char Limit states his identity is proven below where he multiplies an infinite series by r!

Perfectly valid because it gives valid answers in reality as does the fact that 1/[infinity] = 0
and 1/[infinity] is the difference between .9... and 1.

What's wrong with multiplying an infinite series by a number? I don't see anything wrong with that.

Not to mention, I multiplied a finite series by a number. I didn't start working with infinite series until near the end, when I took the limit.

 

[Mark44]

this could be an issue of semantics, but how could .999... go on for infinity if it could be stopped from doing so by 1?

This is not about semantics. It seems to be about your not understanding what the notation .999... means. The phrase "go on for infinity" in your question is very imprecise. The notation ".999..." means that the 9's extend infinitely far. No matter how far out you go in the decimal places, there is a 9 digit there.

if 1 were a goal that an infinite number of .9's extending onward forever could reach then how would the possibility of reaching that goal also mean that there's no possibility of .999... going past 1?

You are apparently unfamiliar with the concept of infinite series, usually taught as a part of calculus. In an infinite series, infinitely many terms are added together. Some infinite series add up to a finite number (the series converges) but in other series, the more terms you add, the larger the sum grows, without bound. A series like .9 + .09 + .009 + ... grows larger as you add more terms, but the sum is bounded (by 1), and this can be proven.

An example of a series that diverges is 1 + 1/2 + 1/3 + ... + 1/n + ... can be proven to increase without bound. IOW, no matter how big a number M you specify, it's possible to add together a finite number of terms whose sum is larger than M.

 

[Mark44]

But Char Limit states his identity is proven below where he multiplies an infinite series by r!

Perfectly valid because it gives valid answers in reality as does the fact that 1/[infinity] = 0
and 1/[infinity] is the difference between .9... and 1.

Infinity is not a number in the reals, so it's not a number you can do arithmetic on. (Note that I am not talking about the extended reals.)

 

[Char. Limit]

I do want to say something here. Curd, from what I can tell, and I might be wrong, you seem to want mathematics to be grounded in reality. Do I have that right?

But the thing is, mathematicians don't really care too much about reality. They want a system that works, and works beautifully. Whether it makes intuitive sense is not a condition for them.

 

[HallsofIvy]

Curd, I am puzzled as to why you woulkd say things like

could you explain the above in layman's terms (i haven't seen anything like that in several years)?

layman as in a man who hasn't done calculus in over 5 years, took college algebra in a 20 day course 7 years ago (and therefore never had a solid foundation in it), and is now about a third of the way through going through his college algebra book again.

The only one that i remember vaguely is limits.

And yet keep asserting things like

actually, there is a decimal expansion between .999... and 1. it's .9999... (the extra 9 and . signifying that it is still expanding and that there will always be a 9 between the two)

and refusing to accept what mathematicians- who have studied and worked with these thing for years - tell you.

 

[micromass]

Exactly, Halls. I admit that I don't know much about physics, but you don't see me saying things like "Einstein was wrong" or "There is no gravity"
I'm always getting very annoyed by people who don't understand mathematics and then say that all mathematicians don't know what they're doing, and exclaim that they can do it much better.

I don't mind people asking questions if they are willing to learn. But the OP doesn't seem to be like this. It appears that he already made his mind up. This makes me quite sad...

 

[Fredrik]

Curd, did you even read my post? It doesn't look like you did. You still seem to think that the statement "0.999...=1" means something like "if you add 9/100 to 9/10, and then add 9/1000 to the result of that, add 9/10000 to the result of that, and so on forever, the result will be 1". It doesn't! It means something very different and until you understand what that is, there's no way you will be able to understand this.

 

[Curd]

No, I don't see how that could possibly follow from what I've said...

And do you care to answer my two questions? It would enhance the discussion a great deal...

first off, i haven't read all your posts as i have other things to do. second off i will reply to them when i get time. third off i will probably wait to continue this argument once I've gotten back into my calculus book.

but the main point I'm making is that if 9's can go on for infinity after a decimal, then obviously there is an infinite space after the decimal and obviously if they can continue on forever they would necessarily have to be incapable of achieving 1 otherwise the infinite space that allows that expansion would have to be finite. Of course, this gets us into an argument about the meaning of infinity. if a space is infinitely large can it ever be filled. you may be right, perhaps it could be filled with another infinity.


I'll get back to this later when i have more time to read the calculus stuff.

 

[Fredrik]

but the main point I'm making is that if 9's can go on for infinity after a decimal, then obviously there is an infinite space after the decimal

Are you talking about the space occupied by the numbers if we try to write them all out? What makes you think this has anything to do with mathematics?

The meaning of "0.999..." is very different from anything that involves reserving "space" for each of the decimals.

Of course, this gets us into an argument about the meaning of infinity. if a space is infinitely large can it ever be filled. you may be right, perhaps it could be filled with another infinity.

This problem isn't about "filling" anything.

I'll get back to this later when i have more time to read the calculus stuff.

Make sure you understand the definition of the limit of a sequence. (The one I included at the end of post #35). It's the key to this whole thing. You may need to consult a calculus book to see more examples that can help you understand this definition, but everything you need to know about this particular problem is included in post #35.

Edit: After reading your comments again, I think it's clear that even though you've been saying repeatedly that you're talking about infinitely many decimals, you're actually not. You're making observations about 0.999...9 (with a finite number of nines), and then jumping to incorrect conclusions about 0.999... (with infinitely many nines).

 

[Hurkyl]

A decimal numeral has places. There's the one's place. Next to that on the left, there's the ten's place, then the hundred's place, and so on. On the other side (crossing the decimal point), there's the tenth's place, the hundredth's place, and so forth.

The decimal places are indexed by integers:

...
2 = hundreds
1 = tens
0 = ones
-1 = tenths
-2 = hundredths
...​

To state a decimal numeral, you have to say what digit is in every place. A common shorthand is to just write out a sequence of digits with a decimal point, such as

47.16​

which conventionally means there's a 4 in the tens' place, a 7 in the one's place, a 1 in the tenth's place, a 6 in the hundredth's place, and a 0 in every other place.


When a mathematician (or scientist or engineer or ...) writes 0.999..., they mean the numeral that has a 9 in every negatively indexed place (i.e. every place to the right of the decimal point), and a 0 in every other place.



Aside: in the use of decimals to name real numbers, the numerals allowed are precisely those that have repeating zeroes on the left. While ...999.000... is a decimal numeral, specifically the one with a 0 in every negatively indexed place and a 9 in the other places, it does not name a real number. (the term "decimal numeral" is often used so as to exclude numerals like this that do not have repeating zeroes on the left)

 

[Curd]

Are you talking about the space occupied by the numbers if we try to write them all out? What makes you think this has anything to do with mathematics?

The meaning of "0.999..." is very different from anything that involves reserving "space" for each of the decimals.


This problem isn't about "filling" anything.


Make sure you understand the definition of the limit of a sequence. (The one I included at the end of post #35). It's the key to this whole thing. You may need to consult a calculus book to see more examples that can help you understand this definition, but everything you need to know about this particular problem is included in post #35.

Edit: After reading your comments again, I think it's clear that even though you've been saying repeatedly that you're talking about infinitely many decimals, you're actually not. You're making observations about 0.999...9 (with a finite number of nines), and then jumping to incorrect conclusions about 0.999... (with infinitely many nines).

Alright, here's my view on the subject.

1) You don't know how to communicate

2) the easiest way to say that .999... equals one is to say that the space after the decimal is infinite in size but is being filled with and infinite amount of 9's therefore the requirement needed for it to be "bumped" up to 1 is met. Also, if 1/9 equals .111... then 1/1 should equal .999... it's the same concept but with a different appearance.


3) if the reaction to my original assertion had been strictly productive instead of egotistical then this thread may well have never occurred which links us back to the 1st item on this list. you need to learn to communicate more concisely and without an attitude (you being certain members that have responded).



and does not the ... mean the 9's continue on into infinity?

also, are there any cases where time can somehow be applied to this idea of ... so that at a certain point you have not actually reached 1?

 

[micromass]

Alright, here's my view on the subject.

1) You don't know how to communicate

2) the easiest way to say that .999... equals one is to say that the space after the decimal is infinite in size but is being filled with and infinite amount of 9's therefore the requirement needed for it to be "bumped" up to 1 is met. Also, if 1/9 equals .111... then 1/1 should equal .999... it's the same concept but with a different appearance.


3) if the reaction to my original assertion had been strictly productive instead of egotistical then this thread may well have never occurred which links us back to the 1st item on this list. you need to learn to communicate more concisely and without an attitude (you being certain members that have responded).



and does not the ... mean the 9's continue on into infinity?

also, are there any cases where time can somehow be applied to this idea of ... so that at a certain point you have not actually reached 1?

We are unable to communicate?? Really?? Of everybody who answered of this thread, I've understood them all very well! Especially Frederik, who made a very nice post about in what framework we should see 1=0.999...
I'm sorry to say, but the only one I've never understood was your posts. I've asked you several questions to clarify, but I've never got an answer. I think many people are confused about your posts...

All we were trying to do is to show you why 1=0.999... is true. In my opinion, it's not us who is unable to communicate. And calling us egoistical is way over the line.

 

[ramsey2879]

in such cases those decimals are only very precise approximations of the actual fraction. what I'm saying is that decimals can not always perfectly represent a fraction which is why ... has been added to some of them.

But the decimal .333... is also a decimal number, the ellipses "..." is just shorthand notation for the infinite string that constitutes this decimal number. Once you are talking of an infinite string of 3's there is no sense in asking of when does the sum reach 1/3 since it has by then reach 1/3. Any additional 3 in the string would be only adding a zero to the sum since n/[infinity] where n is a finite number = 0. That is a property of infinity. I see what you are saying but as long as you are talking in terms of decimal places of an infinite string any "additional decimal places" would effectively merely be adding zero to the sum. Also in reality you can not put a number on the number of decimal places to infinity or ever reach it to add to it. If you could you wouldn't be at infinity.

and as for your argument that there is a trap. 1-.999... = 1-.999... since there is no other way to write it that i know of. in words it equals an infinitely and increasingly small space between 1 and .999... although that space never does reach 0 because it is forever in the process of trying to reach zero without actually achieving it.


if this were not true then .999... could not go on for infinity as you (seem to) suggest it does. if it goes on for infinity then there must be the potential for an infinite space between .999... and 1 and ergo .999... can go on for infinity without reaching one. otherwise you have contradicted the concept of infinity.

I think I now see where you are coming from. In effect you may be right to say that the limit cannot be reached in reality because concept of ever reaching infinity is not reality. But when we put ellipses at the end of a decimal string we are invoking the concept of a decimal string that is indeed infinite contary to reality. This decimal string is most often precisely defined and in the case of .999... is precisely equal to 1.

 

[Robert1986]

Alright, here's my view on the subject.

1) You don't know how to communicate

Wow, as a (mostly) spectator in this thread, I find this claim quite funny. The guys on the forum have given several intuitive reasons to understand the claim and one solid proof. I don't see any bad communication on their part.

2) the easiest way to say that .999... equals one is to say that the space after the decimal is infinite in size but is being filled with and infinite amount of 9's therefore the requirement needed for it to be "bumped" up to 1 is met. Also, if 1/9 equals .111... then 1/1 should equal .999... it's the same concept but with a different appearance.

au contraire; this is probably one of the worst ways to say .999...=1.

3) if the reaction to my original assertion had been strictly productive instead of egotistical then this thread may well have never occurred which links us back to the 1st item on this list. you need to learn to communicate more concisely and without an attitude (you being certain members that have responded).

Oh dear; I cannot believe that I am reading this. Everyone in this thread has been concise. I see no failure to be concise on the part of any of the posters. They were patient with you; you just refused to learn.

I think it is funny that you essentially claim to not know what you are talking about, but then argue with everyone, and then have the nerve to call them egotistical.

 

[uart]

This thread illustrates exactly why $0.999... \neq 1$ threads are usually banned.

 

[uart]

actually no. please do not assert that I've said things that i have not.

Actually yes. I quoted you verbatim and you said "decimals of fractions never accurately equal those fractions". If you are referring to only recurring decimal then use the word "recurring". And you've got the nerve to say that others here are poor communicators.

you could tell clearly from the pattern that i had set up with my earlier argument about .111... and it's relation to .999... that my argument only dealt with fractions who are represented by infinitely repeating digits after a decimal.

Yes that's true. That's why I was trying to help you word your question more correctly ok.

and as for your argument that there is a trap. 1-.999... = 1-.999... since there is no other way to write it that i know of.

Ok you didn't get as far as I expected. Most "layman" when confronted with this usually go something like,

1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 = 0.001

therefore 1 - 0.999... = 0.00...1
where the "..." means repeats forever.

Usually at this point even the layperson can see the absurdity of what they've just written in 0.00...1. Essentially this is saying that there is a decimal point followed by a never ending number of zeros then at the end of this never ending string of zeros we put a one.

Clearly that one on the end is redundant so therefore 0.00...1 = 0.00... which is just an inefficient way of writing zero.

The difference between 1 and 0.999... is zero and therefore they are equal.

 

[Mark44]

But the decimal .333... is also a decimal number, the ellipses "..." is just shorthand notation for the infinite string that constitutes this decimal number. Once you are talking of an infinite string of 3's there is no sense in asking of when does the sum reach 1/3 since it has by then reach 1/3. Any additional 3 in the string would be only adding a zero to the sum since n/[infinity] where n is a finite number = 0. Where are these additional 3's that you're talking about that don't contribute to the sum?

Infinity is not a number in the real number system, so you can't divide by it or otherwise do arithmetic with it. Limits are what you need to be working with if a variable gets large without bound.

That is a property of infinity. I see what you are saying but as long as you are talking in terms of decimal places of an infinite string any "additional decimal places" would effectively merely be adding zero to the sum. Also in reality you can not put a number on the number of decimal places to infinity or ever reach it to add to it. If you could you wouldn't be at infinity.

 

[Fredrik]

1) You don't know how to communicate

Are you serious? What exactly makes you say that? I really want to know.

2) the easiest way to say that .999... equals one is to say that the space after the decimal is infinite in size but is being filled with and infinite amount of 9's therefore the requirement needed for it to be "bumped" up to 1 is met. Also, if 1/9 equals .111... then 1/1 should equal .999... it's the same concept but with a different appearance.

This doesn't make any sense. The actual reason why 0.999...=1 is explained in post #35.

For every r>0, all but a finite number of members of the sequence 0.9, 0.99, 0.999,... are between 1-r and 1+r.​

See #35 for a more thorough explanation.

3) if the reaction to my original assertion had been strictly productive instead of egotistical then this thread may well have never occurred which links us back to the 1st item on this list. you need to learn to communicate more concisely and without an attitude (you being certain members that have responded).

You have displayed an attitude problem in just about every one of your posts. The problem is with you, not with the people who have tried to help you. No one gave you "attitude" before you started insulting everyone.

and does not the ... mean the 9's continue on into infinity?

Yes, but what does that mean? You clearly don't know the answer to that, and as long as you don't, there's no way you can understand this problem. This is why you should read post #35, where it's explained.

also, are there any cases where time can somehow be applied to this idea of ... so that at a certain point you have not actually reached 1?

Not in mathematics. If you consider the problem of physically writing the decimal expansion on paper, then time is obviously an issue, but then we are no longer talking about mathematics. Time has absolutely nothing to do with the issue of why 0.999...=1.

 

[Calrid]

Infinity is hard to comprehend, so some people are tragically incapable of accepting that at infinity anything equals anything. No one can really visualise an infinity but what they can do is accept that as a limit it makes sense that certain values converge to an exact form.

If you can't accept that the infinite is basically an indefinite value that is without bound, then you are looking at it finitely and hence you are not going to grasp the implication of what infinity actually means. If you were to count to infinity it would take you forever, literally and if I stopped you in ten years you still would be infinitely far away.

Perhaps if you could put 1 billion 9s after the decimal and tried to do an equation you'd see an incredibly small amount of deviation from the exact value, now imagine that there are an infinite amount of these .9s how small is it now?

 

[MathematicalPhysicist]

As homage to this thread or any such thread I believe we should post a thread along the lines:"I would like to kill (you) about .999..."

Any in favour?

 

[ramsey2879]

Infinity is not a number in the real number system, so you can't divide by it or otherwise do arithmetic with it. Limits are what you need to be working with if a variable gets large without bound.

Where in my post did I say to multiply or do any arithmetic with infinity? In fact I actually said the contrary, to wit infinity was boundless. I was responding to the Op's argument that fractions such as .333... or .999... never add up to 1/3 or 1 becauase if they did the concept of infinity would be contradicted. I also said
" I think I now see where you are coming from. In effect you may be right to say that the limit cannot be reached in reality because concept of ever reaching infinity is not reality. But when we put ellipses at the end of a decimal string we are invoking the concept of a decimal string that is indeed infinite contary to reality. This decimal string is most often precisely defined and in the case of .999... is precisely equal to 1."
The definition of course comes from the theorey of limits.
I think the major thought I contributed in my post was the idea that infinity is not a concept well structured by the "reality" of being reachable as the Op seems to be insisting on.

 

[Mark44]

Right here...

But the decimal .333... is also a decimal number, the ellipses "..." is just shorthand notation for the infinite string that constitutes this decimal number. Once you are talking of an infinite string of 3's there is no sense in asking of when does the sum reach 1/3 since it has by then reach 1/3. Any additional 3 in the string would be only adding a zero to the sum since n/[infinity] where n is a finite number = 0. That is a property of infinity.

You are dividing some finite number n by infinity.

 

[ramsey2879]

Right here...

You are dividing some finite number n by infinity.

To get the limit of A/n as n goes to infinity you have to know what A/[infinity] is. P.S. My high school teacher said A/[infinity] was 0.

 

[micromass]

To get the limit of A/n as n goes to infinity you have to know what A/[infinity] is. P.S. My high school teacher said A/[infinity] was 0.

I think your high school teacher was wrong then... (unless he was talking about extended reals, but I doubt it).
And to calculate limits of A/n, you don't need to know what A/[infinity] is. You can simply get it from the definition:

$$\forall \epsilon>0:\exists n_0\forall n\geq n_0:~|A/n|<\epsilon$$

The existence of the n₀ in question follows from the axiom of Archimedes. So there's no need for calculations with infinity. Indeed, the whole purpose of limits is to avoid calculations with infinites, as these are ill-defined. Of course, you can still do it with infinites in your intuition (that's how I do it), but it's not rigourous mathematics.

 

[Hurkyl]

To get the limit of A/n as n goes to infinity you have to know what A/[infinity] is. P.S. My high school teacher said A/[infinity] was 0.

No, to get the limit of A/n as n goes to infinity, you need to know the Archimedean principle.

To compute the limit of A/n as n goes to infinity instead by comparing to A/[infinity], you need a lot more information. One set of information would be

• A number system containing an element called [infinity] along with all real numbers
• Knowledge that A/[infinity] = 0 if A is finite
• Knowledge that division in this new number system gives the same results as division in the real numbers, when both numbers are real
• Knowledge that division is continuous in this new number system (at least, at (A, [infinity]))
• Knowledge that the limit of n as n goes to infinity converges to [infinity]
• Knowledge that limits computed in this new number system agree with limits computed in the real numbers when it would make sense.

(For the record, my thought processes probably would compute the limit by invoking continuity of division in the projective real numbers before any other approach)