# Are decimals flawed



## cheesy999 (May 7, 2011)

First: a maths problem i'm sure most of us know - 0.99 recurring x 10 - 0.99 recurring =9

this states that 0.99 recurring is = 1 and so does the equation

1/3=0.3'

however when you multiply out 0.3' you get 0.9' which means that 0.9' and 1 are the same, however 0.9' is not 1, but slightly less

my understanding of maths however, is that one numerical value cannot be the same as another

does this mean maths is flawed?

if so, is there a number system under which these problems do not occur?


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## 2DividedbyZero (May 7, 2011)

my sig has all the answers you need


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## cheesy999 (May 7, 2011)

2DividedbyZero said:


> my sig has all the answers you need



too small


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## lucas4 (May 7, 2011)

no, because recurring means there is no end to the number (IIRC)
u can't do it on a calculator or anything, because when ur typing in the number, it isnt actually recurring unless u type it as a fraction. 
i see what you are saying, but the margin of error is so minute, its practically insignificant.

at least thats the way i see it !

interesting point none the less


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## lucas4 (May 7, 2011)

cheesy999 said:


> First: a maths problem i'm sure most of us know - *9.99 recurring x 10 - 9.99 recurring =9*
> 
> this states that 0.99 recurring is = 1 and so does the equation



it = 90 btw


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## cheesy999 (May 7, 2011)

lucas4 said:


> it = 90 btw



sorry wasn't concentrating i'll fix that


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## lucas4 (May 7, 2011)

its an interesting point, and quite awkward to explain why 0.99 doesnt = 1.

i think the easiest way to explain it may be rounding error in the apparatus we use ??


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## Fourstaff (May 7, 2011)

Studying for a maths major here, so I think I have some credibility.

We know that 0.9 cannot be bigger than 1. Fix 0.9 recurring as a sequence, calling it a(n), where a(1) = 0.9, a(2) = 0.99 etc. Now we make this statement: For any positive number x, there exist an N such that N is a natural number (positive integer) and when n>N |1-a(N)|< x . From that statement, its obvious that a(n) tends to 1, therefore 0.99 recurring is 1. 

Analysis is a bitchy subject, and unless your job/life/education depends on it, its best avoided. We do silly things like finding the limit of 0/0 (0/0 tends to some random things depending on how you define the top and bottom zero).


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## cheesy999 (May 7, 2011)

lucas4 said:


> its an interesting point, and quite awkward to explain why 0.99 doesnt = 1.
> 
> i think the easiest way to explain it may be rounding error in the apparatus we use ??



not a rounding error

if x = 0.9'
therefore 10x = 9.9'
10x-x=9x = 9.9' - x

therefore 9x = 9


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## mlee49 (May 7, 2011)

It's all about being accurate as you need to be.

Decimals approximations are great for whole number divisions but when you try to approximate pi with decimals you'll fall short every time. However if you'd like to be accurate simply choose your degree of accuracy and there's a fraction to go with it!  22/7 or 355/113 ... it goes on!


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## Fourstaff (May 7, 2011)

cheesy999 said:


> not a rounding error
> 
> if x = 0.9'
> therefore 10x = 9.9'
> ...



10x - x = 9x =/= 9.9' - x 
9.9' - x = 9.9' - 0.9' = 9


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## cheesy999 (May 7, 2011)

mlee49 said:


> It's all about being accurate as you need to be.
> 
> Decimals approximations are great for whole number divisions but when you try to approximate pi with decimals you'll fall short every time. However if you'd like to be accurate simply choose your degree of accuracy and there's a fraction to go with it!  22/7 or 355/113 ... it goes on!



my point is

why use decimals if their not good?

i use fractions myself but because there a lot easier


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## Fourstaff (May 7, 2011)

cheesy999 said:


> my point is
> 
> why use decimals if their not good?
> 
> i use fractions myself but because there a lot easier



Decimal is good, but it can only do so much. Fractions are neat and easy to work with, but they are much less accurate than decimals.



2DividedbyZero said:


> my sig has all the answers you need



Your sig proves the => but not the <= which is what we want. You might as well say W1zzard is either a guy or a girl or both.


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## cheesy999 (May 7, 2011)

Fourstaff said:


> Decimal is good, but it can only do so much. Fractions are neat and easy to work with, but they are much less accurate than decimals.



how are they accurate if the system has errors?

i mean - 1/3 is precisely a 1/3 of the object


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## Fourstaff (May 7, 2011)

cheesy999 said:


> how are they accurate if the system has errors?
> 
> i mean - 1/3 is precisely a 1/3 of the object



The system has no errors


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## Frick (May 7, 2011)

cheesy999 said:


> how are they accurate if the system has errors?
> 
> i mean - 1/3 is precisely a 1/3 of the object



That is not an error. It's an infinite number so you can't express it in decimals.

I don't really get your point btw. Since when is 0.99 recuring = 1? And 1/3 has never been 0.3. At least not in my schools.


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## cheesy999 (May 7, 2011)

Frick said:


> That is not an error. It's an infinite number so you can't express it in decimals.



sounds like a problem with the system if you can't use it for something

i mean you couldn't do one calculation on an pentium processor and that was counted as an error - http://en.wikipedia.org/wiki/Pentium_FDIV_bug

there's one problem with a calculation in decimals - doesn't that mean their broken?


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## Fourstaff (May 7, 2011)

cheesy999 said:


> sounds like a problem with the system if you can't use it for something
> 
> i mean you couldn't do one calculation on an pentium processor and that was counted as an error - http://en.wikipedia.org/wiki/Pentium_FDIV_bug
> 
> there's one problem with a calculation in decimals - doesn't that mean their broken?



You will need to work with symbolic to ensure no rounding error problems, and x86 does not do its math symbolically. Well, in that sense fractions is better than decimals, contradicting what I said earlier :/


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## Kreij (May 7, 2011)

1/3 is not a number, it is a divisional proportion.
All fractions are estimates based on what they are derived from, and when converted to their decimal equivelents are rounded as to be useful.

When you do something 1/2 assed can it be equated to 0.5 assed? Not necessailry.


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## lucas4 (May 7, 2011)

cheesy999 said:


> sounds like a problem with the system if you can't use it for something
> 
> i mean you couldn't do one calculation on an pentium processor and that was counted as an error - http://en.wikipedia.org/wiki/Pentium_FDIV_bug
> 
> there's one problem with a calculation in decimals - doesn't that mean their broken?



i dont think it means they broken.

they are fit for their purpose, and we all know that they cannot be 100% accurate when we have a recurring answer or pi.
this is why we round our answers to a certain number of decimal places which is considered to be an accurate enough margin.

eg, sometimes rounding earlier answers to 4dp if u need a 2dp final answer is perfectly acceptable level of accuracy, whereas rounding to 4dp if we need a 10dp final answer isnt.
we know when to use the appropriate degree of accuracy which doesnt make them broken IMO


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## Fourstaff (May 7, 2011)

Kreij said:


> 1/3 is not a number, it is a divisional proportion.
> All fractions are estimates based on what they are derived from, and when converted to their decimal equivelents are rounded as to be useful.



An engineer's point of view.  We can sit here all day arguing who is right, but I have weapons in my arsenal to make me right.


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## lucas4 (May 7, 2011)

Fourstaff said:


> An engineer's point of view.  We can sit here all day arguing who is right, but I have weapons in my arsenal to make me right.



no need for a gun fight


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## cheesy999 (May 7, 2011)

Kreij said:


> 1/3 is not a number, it is a divisional proportion.
> All fractions are estimates based on what they are derived from, and when converted to their decimal equivelents are rounded as to be useful.
> 
> When you do something 1/2 assed can it be equated to 0.5 assed? Not necessailry.



so is there such thing as a number system where things like this are not a problem (say Roman numerals/attic numerals for example)


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## lucas4 (May 7, 2011)

cheesy999 said:


> so is there such thing as a number system where things like this are not a problem (say Roman numerals/attic numerals for example)



*swoosh* now the talk goes above my head


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## Frick (May 7, 2011)

IMO numbers and math is above such worldly things as usefullness. Just because we can't pin them down doesn't mean they're not useful. Think about pi.


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## cheesy999 (May 7, 2011)

lucas4 said:


> *swoosh* now the talk goes above my head



other numbers used by other people

Roman Numerals - you may recognise as I II III IV V VI VII VIII IX X XI XII from a clock

Attic numerals were the number system used by the ancient greeks

Under ours

1+1=2       10+5 = 15

under Roman

I+I=II       X+V=XV


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## Bo$$ (May 7, 2011)

But this all depends on what aspect you are working problems out, engineers a different methodology for compensating for decimals than, pure mathematicians or physicists require as their work and whole thought process is different.


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## Fourstaff (May 7, 2011)

Bo$$ said:


> But this all depends on what aspect you are working problems out, engineers a different methodology for compensating for decimals than, pure mathematicians or physicists require as their work and whole thought process is different.









This sums up nicely  Notice how mathematics is further away from sociology than physics is.


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## Bo$$ (May 7, 2011)

Fourstaff said:


> http://imgs.xkcd.com/comics/purity.png
> 
> This sums up nicely  Notice how mathematics is further away from sociology than physics is.



i was looking for that one 

where do the engineers come, in between the meths and physics
edit: maths*


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## erocker (May 7, 2011)

Numbers along with decimals are not flawed. As a matter of fact they are perfect and can go in infinite directions.


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## newtekie1 (May 7, 2011)

Decimals aren't flawed, rounding is...


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## HTC (May 7, 2011)

newtekie1 said:


> Decimals aren't flawed, rounding is...



Yup!

Reminds me of that classic story about the turtle and the rabbit: turtle was 100m in front of the rabbit and the rabbit was chasing the turtle. For ever 10m the rabbit ran, the turtle "ran" 1m.

Did the rabbit ever caught the turtle or not?


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## Kreij (May 8, 2011)

rounding isn't flawed, ask any good carpenter. If it fits without a gap, it's right.


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## FordGT90Concept (May 8, 2011)

cheesy999 said:


> if so, is there a number system under which these problems do not occur?


Fractional notation.  Instead of turning 1/3 into 0.3333333, leave it 1/3.  I even use fractional in some of my programs when 1/3+2/3 can't equal 0.99999999.  This is also how the best graphing calculators work (e.g. TI-89 and TI-92).




cheesy999 said:


> why use decimals if their not good?


In the context of computers, because they are designed to handle decimals and not fractions.  Fractions take up more memory and require more steps to perform mathematical operations on.  With decimals, they can fly through the 128-bit FPU in the CPU in just a few CPU cycles.  Moreover, often the precision of fractions isn't required in a lot of computing tasks (like graphics).


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## theJesus (May 8, 2011)

cheesy999 said:


> why use decimals if their not good?


Because 4/3 of the population has trouble with fractions.


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## mlee49 (May 8, 2011)

newtekie1 said:


> Decimals aren't flawed, rounding is...



You'd be surprised how much is basically rounded. Some digital signal processing is not truly 1 or exactly 0 but because it's 0.99999 it's practically 1.


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## twilyth (May 8, 2011)

It's too bad no one here realizes that this is an attempt at illustrating Godel's incompleteness theorems

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems


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## Meaker (May 8, 2011)

2DividedbyZero said:


> my sig has all the answers you need



You do like dividing by zero don't you.


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## 1freedude (May 8, 2011)

*number systems*

Binary and hex...no decimals for obvious reasons.  Clock systems usually have finite divisions, without a . and numbers behind it.

I deal with addition of decimals to the third place at work.  We are adding areas in square inches.  Very simple stuff, until it hits a maximum...local or absolute.  To put it bluntly, the people that add these numbers don't give a shit about fractional repetition or rounding.  If either of these maximums are exceeded, we (the company) have to spent a ridiculous amount of money to correct it.


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## MilkyWay (May 8, 2011)

Our numerals are called Arabic numerals.
0.333^ Its basically infinite isn't it?

Apparently they try to explain this by saying its not 1 but 0.999^, its almost one! Think of it this way if you look at 0.999^=1 then your saying 1 doesn't exist nothing is quite complete and 1 is just a stand in for 0.999^ because we have to round up numbers because you cannot calculate an infinite number because it never ends.

(0.99^x10)-0.99^=9

1/3= .333...
2/3= .666...
3/3= .999...

but 3/3 =1?

HOLY FUCK! MY MINDS BLOWN!


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## FordGT90Concept (May 8, 2011)

1/3 is infinite .3333
2/3 is infinite .6666
3/3 is 1 (3 can go into 3 once).

The problem with decimial is preserving the infnite (fractional has no infinite to preserve).  The same issue arises in pi and imaginary.  If you somehow preserve the infinite, 1/3 + 2/3 is 1; however, if the infinite is lost and it isn't rounded (.67 instead of .66), you get an infinite .9999.  1/3 + 2/3 is always 1 unless you convert to decimial and lose the infinite.


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## twilyth (May 8, 2011)

I've seen this example before and whether it points out a flaw in our number system or not, I don't think it matters.  Just read the first paragraph of the the link I posted previously.  Godel's incompleteness theorems say that no purely axiomatic, deductive system can derive all true conclusions consistent with the axioms.

In the case of natural numbers, this means that no numeric system can deductively prove all true statements about natural numbers.  If you find a system that does, then virtually by definition, the system is flawed.

I don't know how much sense that explanation made, but once you get it, your mind will be blown.  Basically Godel proved that no purely deductive system based on a certain set of axioms can ever describe it's subject matter completely.  If it does, then it is self contradictory in some respect.  And remember, this isn't just another theory, he proved this to an absolute certainty.


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## Fourstaff (May 8, 2011)

twilyth said:


> I've seen this example before and whether it points out a flaw in our number system or not, I don't think it matters.  Just read the first paragraph of the the link I posted previously.  Godel's incompleteness theorems say that no purely axiomatic, deductive system can derive all true conclusions consistent with the axioms.
> 
> In the case of natural numbers, this means that no numeric system can deductively prove all true statements about natural numbers.  If you find a system that does, then virtually by definition, the system is flawed.
> 
> I don't know how much sense that explanation made, but once you get it, your mind will be blown.  Basically Godel proved that no purely deductive system based on a certain set of axioms can ever describe it's subject matter completely.  If it does, then it is self contradictory in some respect.  And remember, this isn't just another theory, he proved this to an absolute certainty.



There are indeed proofs that says our number theory will never be made complete. We can use fancy axioms involving infinity and stuff like that to have a complete number theory, but we know that the axioms are shaky at best, false at worst. Welcome to the harsh world, where our understanding of the universe is based on an incomplete system.


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## cheesy999 (May 8, 2011)

twilyth said:


> I've seen this example before and whether it points out a flaw in our number system or not, I don't think it matters.  Just read the first paragraph of the the link I posted previously.  Godel's incompleteness theorems say that no purely axiomatic, deductive system can derive all true conclusions consistent with the axioms.
> 
> In the case of natural numbers, this means that no numeric system can deductively prove all true statements about natural numbers.  If you find a system that does, then virtually by definition, the system is flawed.
> 
> I don't know how much sense that explanation made, but once you get it, your mind will be blown.  Basically Godel proved that no purely deductive system based on a certain set of axioms can ever describe it's subject matter completely.  If it does, then it is self contradictory in some respect.  And remember, this isn't just another theory, he proved this to an absolute certainty.





Fourstaff said:


> There are indeed proofs that says our number theory will never be made complete. We can use fancy axioms involving infinity and stuff like that to have a complete number theory, but we know that the axioms are shaky at best, false at worst. Welcome to the harsh world, where our understanding of the universe is based on an incomplete system.



so you guys are saying these contradictions are unavoidable and will always happen no matter how someone changes the number systems?


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## Fourstaff (May 8, 2011)

cheesy999 said:


> so you guys are saying these contradictions are unavoidable and will always happen no matter how someone changes the number systems?



In a nutshell, yes.


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## cheesy999 (May 8, 2011)

Fourstaff said:


> In a nutshell, yes.



i'd thank you but the button to do so isn't there any more


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## slyfox2151 (May 8, 2011)

cheesy999 said:


> i'd thank you but the button to do so isn't there any more




i thanked him for you


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## qubit (May 8, 2011)

Decimals are not flawed.

Decimals and vulgar fractions simply have different properties. For example, you can't express an irrational number such as PI or e using vulgar fractions, but you can with decimals.

Similarly, decimals that repeat endlessly often can be expressed exactly using vulgar fractions.

There's other significant differences in properties, but I don't know what they are off the top of my head.

Search decimals and vulgar fractions on Wikipedia for more details.


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## cheesy999 (May 8, 2011)

qubit said:


> you can't express an irrational number such as PI or e using vulgar fractions, but you can with decimals.



can't you just express them in terms of pie?


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## qubit (May 8, 2011)

cheesy999 said:


> can't you just express them in terms of pie?



eh?  That makes no sense.


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## cheesy999 (May 8, 2011)

qubit said:


> eh?  That makes no sense.



1.17pi etc (tpu won't let me write the symbol for some reason)


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## pvtbert (May 16, 2011)

0,99(infinite 9's) x 10 = 9,999(infinite 9's)*0* isn't it?


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## Bo$$ (May 16, 2011)

qubit said:


> eh?  That makes no sense.



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


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## Bo$$ (May 16, 2011)

cheesy, that would make them i little more complicated, in terms of pi


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## qubit (May 16, 2011)

Bo$$ said:


> http://en.wikipedia.org/wiki/Pi



I know what PI is. Cheesy's question made no sense and I was trying to get him to clarify it.


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## cheesy999 (May 16, 2011)

Bo$$ said:


> cheesy, that would make them i little more complicated, in terms of pi



yeah, very complicated 7/pi etc, means any addition or anything has to be done as an equation doesn't it?


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## Bo$$ (May 16, 2011)

i mean to manuplate them without a calculator, hence the use of rounding, try doing 34dp without a calculator, it will take ages

after all this talk of maths, i must return to my M1 and C1 as my exam is on wednesday


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## streetfighter 2 (May 16, 2011)

Interesting fact: The set of all integers is a smaller infinity than the set of all real numbers between any two integers.
(source)

EDIT:  
It suddenly occurs to me that 0.9' is a rational number because it is equal to (∞-1)/∞.  Furthermore (∞-1) = ∞, so 0.9' = 1. QED
It can be shown that --
1/9 = 0.1'
2/9 = 0.2'
3/9 = 0.3'
. . .
9/9 = 1
Another way it can be shown is with limits --
lim (1/9)^(1/n) = 1

In other words, 0.9' isn't a failure of decimal point numbers, it's a failure of us comprehending infinity.


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## Arctucas (May 16, 2011)

cheesy999 said:


> not a rounding error
> 
> if x = 0.9'
> therefore 10x = 9.9'
> ...



Umm... I know I am late to the party here, but that is incorrect.

10 · 0.9 = 9 ≠ 9.9. 


Therefore, 10 · 0.9 - 0.9 = 9 · 0.9 = 8.1 
                 (10x)   - (x)  =  (9x)  =  (9x)


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## Fourstaff (May 16, 2011)

streetfighter 2 said:


> Interesting fact: The set of all integers is a smaller infinity than the set of all real numbers between any two integers.
> (source)



Don't make me shoot you. Infinity is hard to work with as it is without touching cardinality and cardinal numbers. But you are right, there exists different sizes of infinity, if you will. Enjoy getting mad like Cantor if you want to play with infinity.


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