0÷0

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sportsracer48
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0÷0

a few hundred years ago, the square root of -1 was undefined. If you asked somewhat what the square root of -9 was, they would say undefined. If you asked them if the two were different, they would say no. Then, someone decided to do something weird: he just said "let i be the square root of -1". This proved to be internal consistent, so people accepted it/ My question is, why hasn't anyone done this with the other big undefined number, n÷0. And yes, I know, it's the complex infinity, but really, it's undefined. Most calculators call it ERR, and everyone I know tries to steer clear of it whenever possible. Why not do a similar thing as people did with i, with ÷0? I don't know. perhaps it's stupid, but it might not be.

let Я=1/0 (pronounced yah [I like Cyrillic letters])
therefore 0Я=1, and Я is not a real number
Я^2= 1/0*1/0=1/0=Я, and Я is not an imaginary number in the normal sense.
the first thing I thought about is what Я^0 would be. It should be one, but if it isn't, then Я probably isn't a real thing.
Я/Я=1/0*0/1=0/0, so what we need is what is 0/0?

Multiplication by Я:
2Я=2*1/0=2/0
in fact:
nЯ=n*1/0=n/0

Therefore 0Я could be called 0/0, and we have defined 0Я as 1, so now I have proven two things:

Я^0=1 (I think that this is the first step towards proving that Я-complex numbers are internally consistent)
and
0/0=1 (wolfram alpha says that 0/0 is undefined, but I think I may have just defined it)

I have not had a very formal math education, so please forgive me if I'm obviously wrong, but if not, I think that this might just work.

phlip
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Re: 0-:0

There are two problems with defining 0/0:

Firstly, if you did so, it still wouldn't be "division" in the mathematical sense - it wouldn't be the inverse of multiplication. That is, we want "(ab)/b = a" to hold. But this is clearly impossible to do for b=0.
Division answers questions of the form "I picked a number and doubled it, and ended up with 10, what was my original number?"... the answer to which is easily found as 10/2 = 5. Division by 0 is trying to answer "I picked a number and multiplied it by 0 and ended up with 0. What number did I start with?" which is clearly unsolvable. Maybe it was 5 again, maybe not. Maybe it was your Ya, but probably not.

Secondly, if you want 0/0 = 1, or any other value, then you need to lose a lot of normal mathematical axioms that we like using. For instance, you'd need to lose at least one of substitution, additive identity or distribution, or you'll get:
1 = 0/0 = (0+0)/0 = 0/0 + 0/0 = 1 + 1 = 2
And from there you can prove that every number equals every other number. So you'd need to lose enough structure to lose one of those 3 steps, but all of those are quite important axioms.

You're not the first one to think of trying to give a definition to division by zero... and if you search the fora you'll see it's come up quite a few times. For example, this thread.

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sportsracer48
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Re: 0÷0

I disagree with your first idea, that what I did wasn't really division. The thing is that it's really similar to calling i the square root of -1, in that it seems impossible, but actually works, if you redefine what a number can be. I'm saying that 1/0 is something, and then seeing if that works out.

Your second idea, on the other hand, is a very good, point, and I shall have to think about how I could make it work with those axioms.

Jyrki
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Re: 0÷0

sportsracer48 wrote:I disagree with your first idea, that what I did wasn't really division. The thing is that it's really similar to calling i the square root of -1,

No. It is not similar at all. The existence of a square root of -1 does not violate any of the laws of algebra. The existence of 1/0 would do so.

sportsracer48 wrote:in that it seems impossible, but actually works, if you redefine what a number can be. I'm saying that 1/0 is something, and then seeing if that works out.

Your second idea, on the other hand, is a very good, point, and I shall have to think about how I could make it work with those axioms.

I hate to dampen your enthusiasm, but ... "Don't hold your breath." Here's another problem with the idea of defining "0/0=1": 1=0/0 = (0*0)/0 = 0*(0/0)=0*1=0. Or, if x is any number, then similarly: 1=0/0=(x*0)/0=x*(0/0)=x*1=x. So as a consequence of defining 0/0=1 all the numbers lose their identity and become equal to one another. While that kind of utopia allows easy solving of equations, such a concept of `a number' is hardly worth studying.

Any attempt to define division by zero is just a BAD IDEA. Back in the days, when an attempt to divide by zero caused a program (or the entire computer) to crash, some (bad) programmers were want of having division by zero defined any way whatsoever to avoid the crash. What they failed to understand that if you ever want to divide something by zero, then something has already gone horribly wrong.

Slime
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Re: 0÷0

Then, someone decided to do something weird: he just said "let i be the square root of -1".

Well, no, he said "define i such that i*i = -1," thus avoiding the whole issue of the square root, and simply proposing an interesting, isolated scenario. I believe everything interesting you know about imaginary numbers comes only from this fact and makes no use of the concept of the square root. (For instance, to prove that i^4 = 1, you substitute i^4 with i^2 * i^2 and plug in -1. You do not replace it with sqrt(-1)^4, because that doesn't make use of the definition in the original statement.)

So what I'm saying is, this is potentially interesting:

Let R be such that R * 0 = 1

But this is not:

Let R = 1 / 0

And if you want to draw any interesting conclusions from the former, you'd better not resort to the latter.

Torn Apart By Dingos
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Re: 0÷0

Jyrki wrote:
sportsracer48 wrote:I disagree with your first idea, that what I did wasn't really division. The thing is that it's really similar to calling i the square root of -1,

No. It is not similar at all. The existence of a square root of -1 does not violate any of the laws of algebra. The existence of 1/0 would do so.
Yes it does. sqrt(a*b)=sqrt(a)*sqrt(b) which holds wherever it is defined for real numbers (that is, for a,b>=0), doesn't hold for complex numbers (sqrt(-1*-1)=sqrt(1)=1 is not equal to sqrt(-1)*sqrt(-1)=i*i=-1). Depending on your use of language, you might not call that a "law of algebra", for which you might refer only to the axioms of arithmetic, but it surely is an algebraic identity which fails when you extend it to the complex numbers.

To the OP: The basic reason we choose not define 0/0 is because
1. It hasn't turned out to be very interesting, unlike what happened when we introduced i.
2. It introduces too many inconvenient exceptions to our algebraic rules. If we introduce a number infinity=1/0 to our number system, then whenever we have an identity (a/b)*b, we need to check that b is infinity before we simplify to a. And if we have an expression a-a, we need to check that a is not infinity before we simplify to 0. Et cetera. This is just annoying. Compare it to real numbers, when all we really need to check is that we never take a square root of a negative number, or that we don't divide by 0.

Sometimes it is convenient to add an extra number infinity to our number system, but then we leave a lot of new expressions undefined, such as infinity-infinity. An example where this is convenient is in integration theory of nonnegative functions. There we can introduce a number infinity, and define 0*infinity=0. This makes sense because we may wish to write <imath>\int_0^0\infty dx=0*infinity=0</imath> or <imath>\int_0^\infty 0 dx=infinity*0=0</imath>.

jestingrabbit
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Re: 0÷0

Something that I only learnt quite recently (damn you missed Galois theory lectures) was that i was introduced so that the real solutions of some cubics could be written down exactly. It was later that the theory of complex numbers was developed. If this new Я (which I am pronouncing like a pirate because I am mature like that) solves a similar problem in a useful way, it will be adopted.

As it is, what you have most closely resembles the real projective line

http://en.wikipedia.org/wiki/Real_projective_line

where Я=∞. You can see that even there, they don't define ∞*0, and even there, you get things like a*(b+c) = a*b +a*c not always being true (because one side can be defined whilst the other isn't).
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nykevin
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Re: 0÷0

Torn Apart By Dingos wrote:Yes it does. sqrt(a*b)=sqrt(a)*sqrt(b) which holds wherever it is defined for real numbers (that is, for a,b>=0), doesn't hold for complex numbers (sqrt(-1*-1)=sqrt(1)=1 is not equal to sqrt(-1)*sqrt(-1)=i*i=-1). Depending on your use of language, you might not call that a "law of algebra", for which you might refer only to the axioms of arithmetic, but it surely is an algebraic identity which fails when you extend it to the complex numbers.

The problems demonstrated above use very few axioms, mostly the distributive law and some identity laws. Introducing this "yah" would require us to drop one of those axioms.

That's a problem. Those axioms are very basic and very important.

The square root thing you demonstrate doesn't require us to modify any fundamental axioms, nor does it break most of higher mathematics.

Antimony-120
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Re: 0÷0

nykevin wrote:The square root thing you demonstrate doesn't require us to modify any fundamental axioms, nor does it break most of higher mathematics.

It most certainly doesn't break anything. On the contrary, it makes things more interesting! And it clarifies quite a lot. Like why square roots on the real number line are double valued, but cubic roots single valued, quadratic roots double valued, quintic roots single value etc. etc. All that falls out of a single beautiful relation in the complex plane.

In the example given above, the existance for positive real numbers of the identity sqrt(a*b) = sqrt(a)*sqrt(b) is merely a special case of the general laws of roots. That is to say, we may have "broken" that one identity, but in doing so we introduced a new one that simplifies to that one in the special case of two positive reals. It's a little like complaining about the cosine law "breaking" the identity "a^2 + b^2 = c^2". Yes, it does in fact "break" the identity, but it gives us a new identity that simplifies to the old one AND is applicable to more cases.

In the case of Я it doesn't do so. The identity (ab)/b = a is not replaced by a more general identity that reduces to (ab)/b = a in the case of real numbers. You could, at best, posit it as an exception, but to do that we need to define ANOTHER new quatity that has such a magical property, and even then said quantity is dubious, since it can be shown to equal every number on the real line (actually every number on the extended complex plane if we choose to look at it).

To understand the difference, mathmaticians knew that i was begging to be there. They saw that it looked like a perfectly reasonable number, acted like one, and in short was well defined. The reason it took so long to come into vogue was because it FEELS wrong to have a root that doesn't exist as a real number(hence "imaginary"). i was a number that was logically called for but rejected because it didn't jibe with intuition. By contrast you're Я is a number that feels like it should exist, but logically spews out inconsitencies.

It can be hard to let go of a pet hypothosis, but take comfort in the fact that many very intelligent people have tried very hard to make it work. You're not stupid for trying, but it is not to be.
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Kayangelus
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Re: 0÷0

Torn Apart By Dingos wrote:
Jyrki wrote:
sportsracer48 wrote:I disagree with your first idea, that what I did wasn't really division. The thing is that it's really similar to calling i the square root of -1,

No. It is not similar at all. The existence of a square root of -1 does not violate any of the laws of algebra. The existence of 1/0 would do so.
Yes it does. sqrt(a*b)=sqrt(a)*sqrt(b) which holds wherever it is defined for real numbers (that is, for a,b>=0), doesn't hold for complex numbers (sqrt(-1*-1)=sqrt(1)=1 is not equal to sqrt(-1)*sqrt(-1)=i*i=-1). Depending on your use of language, you might not call that a "law of algebra", for which you might refer only to the axioms of arithmetic, but it surely is an algebraic identity which fails when you extend it to the complex numbers.

To the OP: The basic reason we choose not define 0/0 is because
1. It hasn't turned out to be very interesting, unlike what happened when we introduced i.
2. It introduces too many inconvenient exceptions to our algebraic rules. If we introduce a number infinity=1/0 to our number system, then whenever we have an identity (a/b)*b, we need to check that b is infinity before we simplify to a. And if we have an expression a-a, we need to check that a is not infinity before we simplify to 0. Et cetera. This is just annoying. Compare it to real numbers, when all we really need to check is that we never take a square root of a negative number, or that we don't divide by 0.

Sometimes it is convenient to add an extra number infinity to our number system, but then we leave a lot of new expressions undefined, such as infinity-infinity. An example where this is convenient is in integration theory of nonnegative functions. There we can introduce a number infinity, and define 0*infinity=0. This makes sense because we may wish to write <imath>\int_0^0\infty dx=0*infinity=0</imath> or <imath>\int_0^\infty 0 dx=infinity*0=0</imath>.

The algebraic law would actually be (a*b)^n=(a^n)*(b^n)
By which: (-1*-1)^(1/2)=1^(1/2)=1 OR -1
(-1)^(1/2)*(-1)^(1/2)=(P i)(P i)=P(-1)=1 OR -1, where P=plus or minus

The square root function is not the same as taking something to the 1/2 power. And hence there is no such algebraic rule for it. What you wrote is just a trick that is useful when we are only interested in positive numbers

Torn Apart By Dingos
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Re: 0÷0

nykevin wrote:The square root thing you demonstrate doesn't require us to modify any fundamental axioms, nor does it break most of higher mathematics.

It doesn't require us to drop any of the field axioms, but a priori, there's no reason they should be our only axioms. One could imagine a different set of axioms for the real numbers, one of which is the silly identity sqrt(a*b)=sqrt(a)*sqrt(b). I was just saying that there are true identities of the real numbers which we need to drop if we introduce the complex numbers.

Kayangelus wrote:The algebraic law would actually be (a*b)^n=(a^n)*(b^n)
By which: (-1*-1)^(1/2)=1^(1/2)=1 OR -1
(-1)^(1/2)*(-1)^(1/2)=(P i)(P i)=P(-1)=1 OR -1, where P=plus or minus

The square root function is not the same as taking something to the 1/2 power. And hence there is no such algebraic rule for it. What you wrote is just a trick that is useful when we are only interested in positive numbers

I don't know what the hell you're talking about. Fix a principal branch of the square root function, and they are exactly the same. No one uses the notation a^b for a multi-valued function. What does "algebraic rule" mean to you, if not an algebraic identity?

Kayangelus
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Re: 0÷0

Torn Apart By Dingos wrote:
nykevin wrote:The square root thing you demonstrate doesn't require us to modify any fundamental axioms, nor does it break most of higher mathematics.

It doesn't require us to drop any of the field axioms, but a priori, there's no reason they should be our only axioms. One could imagine a different set of axioms for the real numbers, one of which is the silly identity sqrt(a*b)=sqrt(a)*sqrt(b). I was just saying that there are true identities of the real numbers which we need to drop if we introduce the complex numbers.

And we are saying no, we disagree with you, and your example is wrong.

Torn Apart By Dingos wrote:
Kayangelus wrote:The algebraic law would actually be (a*b)^n=(a^n)*(b^n)
By which: (-1*-1)^(1/2)=1^(1/2)=1 OR -1
(-1)^(1/2)*(-1)^(1/2)=(P i)(P i)=P(-1)=1 OR -1, where P=plus or minus

The square root function is not the same as taking something to the 1/2 power. And hence there is no such algebraic rule for it. What you wrote is just a trick that is useful when we are only interested in positive numbers

I don't know what the hell you're talking about. Fix a principal branch of the square root function, and they are exactly the same. No one uses the notation a^b for a multi-valued function. What does "algebraic rule" mean to you, if not an algebraic identity?
[/quote]

Well I'm glad to know you believe that most of my physics class, and some of my math/physics professors don't exist... as well as some of my highschool math teachers and the people who's work I have seen in math classes there...

And no they are not the same. The square root sign is defined to only take the positive solution (at least that is how I learned it, and how people here use it). If as you say sqrt(a) is the same as a^(1/2), then in trying to show that imaginary numbers break the "rule" you gave, you failed to understand how the square root function works.

To me, an algebraic rule is an identity that is always true. Kinda like (a*b)^(1/2)=a^(1/2)*b^(1/2)

Antimony-120
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Re: 0÷0

Torn Apart By Dingos wrote:I don't know what the hell you're talking about. Fix a principal branch of the square root function, and they are exactly the same. No one uses the notation a^b for a multi-valued function. What does "algebraic rule" mean to you, if not an algebraic identity?

I think his point was that sqrt(1) = -+1, so the result sqrt(1) = sqrt(-1)*sqrt(-1) = -1 does in fact hold.

Edit: Oh, ninja'd. Yeah, I was always taught that the square root sign means both values, so it is equivalent to the exponent 1/2. But your point about the negative result is fine.

And, fine, don't use a^b for a multivalued function (I, and everyone I've ever seen, do it nigh constantly, but okay). Then use the right bloody value of your multivalued function. It's a little like taking only the priciple branch of arcsin and claiming that clearly there is no answer to arcsin(pi*5/4). If you use the wrong branch on a multivalued function it spits out nonsense, but that's just you being wrong, not the function,
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Torn Apart By Dingos
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Re: 0÷0

Arguing on the internet, how did I fall into this trap again?

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Re: 0÷0

Torn Apart By Dingos wrote:Arguing on the internet, how did I fall into this trap again?
By being condescendingly wrong in a thread where most of the other posters know way more than you, evidently.
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sportsracer48
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Re: 0÷0

I (unlike some who have this idea) do concede that there are some very large flaws with this idea. However, someone much cleverer than me could very well make it work.

Antimony-120
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Re: 0÷0

sportsracer48 wrote:I (unlike some who have this idea) do concede that there are some very large flaws with this idea. However, someone much cleverer than me could very well make it work.

They've tried, I meant it when I said you're not stupid for thinking of it. It's an alluring idea, and some very intelligent and well regarded mathematicians have investigated it. It doesn't work.

For a more visual idea, consider, instead of +infinity and -infinity, the complex plane. infinity is now a circle girding the plane, rather than two seperate points, which changes the topoology quite a bit. Importantly it suggests that Я = -Я, which is something of an issue, considering usually that's given as proof that something is equal to zero (for good reason).

Really, it's been investigated, and disproven for several reasons. Like many wrong hypotheticals in the sciences, having the wrong hypothetical shows the intelligence to see something that looks interesting, and the creativity to play with it, both of which are promising attributes. It's just that in this case the hypothetical is, in fact, wrong.

But keep trying! A good scientist has a lot of these ideas that don't pan out.
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Qaanol
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Re: 0÷0

Jyrki wrote:
sportsracer48 wrote:I disagree with your first idea, that what I did wasn't really division. The thing is that it's really similar to calling i the square root of -1,

No. It is not similar at all. The existence of a square root of -1 does not violate any of the laws of algebra. The existence of 1/0 would do so.

Just to be clear, even with something as “mundane” as extending to ℂ, you still lose one property of ℝ. Namely, you lose ordering, and with it the trichotomy property.

Similarly, when you extend to the quaternions you lose commutativity, and to the octonions you lose associativity. After that things get real dicey when you lose more properties. The sedenions and beyond all have zero divisors.
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Jyrki
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Re: 0÷0

@Torn Apart By Dingos: Yeah, I should have specified that with "laws of algebra" I meant the axioms of a field. Didn't make it specific, because it wasn't clear to me whether that would mean anything to the OP. My bad.

Qaanol wrote:
Jyrki wrote:
sportsracer48 wrote:I disagree with your first idea, that what I did wasn't really division. The thing is that it's really similar to calling i the square root of -1,

No. It is not similar at all. The existence of a square root of -1 does not violate any of the laws of algebra. The existence of 1/0 would do so.

Just to be clear, even with something as “mundane” as extending to ℂ, you still lose one property of ℝ. Namely, you lose ordering, and with it the trichotomy property.

Similarly, when you extend to the quaternions you lose commutativity, and to the octonions you lose associativity. After that things get real dicey when you lose more properties. The sedenions and beyond all have zero divisors.

Yup. Losing the ordering makes some things tricky, but that is more of a topological property as opposed to algebraic.

But why extend the reals only? If you drop completeness, you have a very rich playground. Fields of any dimension over Q or skewfields of any square dimension over their center Q are out there!

andrewxc
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Re: 0÷0

Antimony-120 wrote:
sportsracer48 wrote:I (unlike some who have this idea) do concede that there are some very large flaws with this idea. However, someone much cleverer than me could very well make it work.

For a more visual idea, consider, instead of +infinity and -infinity, the complex plane. infinity is now a circle girding the plane, rather than two seperate points, which changes the topoology quite a bit. Importantly it suggests that Я = -Я, which is something of an issue, considering usually that's given as proof that something is equal to zero (for good reason).

I had the idea of seeing the real plane as a sphere, instead, where the origin is at one pole and infinity is at the other. Essentially, when you "zoom out", you are pulling at the sphere to see more of the real plane, and the material will keep coming forever, no matter how much you zoom out or pull at it.
This is a useful definition when you think about functions like [imath]f(x) = {1 \over x}[/imath], which "go to infinity" under the Cartesian system, but would be continuous (and would overlap, like ribbon around a present at infinity) under the infinite sphere system.
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mike-l
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Re: 0÷0

andrewxc wrote:I had the idea of seeing the real plane as a sphere, instead, where the origin is at one pole and infinity is at the other. Essentially, when you "zoom out", you are pulling at the sphere to see more of the real plane, and the material will keep coming forever, no matter how much you zoom out or pull at it.
This is a useful definition when you think about functions like [imath]f(x) = {1 \over x}[/imath], which "go to infinity" under the Cartesian system, but would be continuous (and would overlap, like ribbon around a present at infinity) under the infinite sphere system.
Not particularly useful for mathematics, but fun to draw on ping-pong balls with dry-erase markers. Lots of math can (an is) done in such ways. Check out the projective plane and manifolds. For example, 1/x is a perfectly fine function to talk on the manifold that is the circle.
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Cleverbeans
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Re: 0÷0

andrewxc wrote:I had the idea of seeing the real plane as a sphere, instead, where the origin is at one pole and infinity is at the other. Essentially, when you "zoom out", you are pulling at the sphere to see more of the real plane, and the material will keep coming forever, no matter how much you zoom out or pull at it.

Sounds a lot like the Riemann Sphere, which is definitely one of my favorite mathematical objects for the reasons you mentioned among others.
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andrewxc
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Re: 0÷0

Blargh. I came up with the basic idea for the Riemann Sphere without realizing it. Mine was [imath]\Re^2[/imath], rather than [imath]\Re x \Im[/imath], but still the math works out right.
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