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The required number represented by ??? is [color="blue"]818[/color].

[u]Explanation[/u]:
The equation is:

∞ - ∞ = ???

Rotating the lhs by 90 degrees and reading up to down, we get the number 818.

Consequently, the required number represented by ??? is [color="blue"]818[/color].

Edited by K Sengupta
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The required number represented by ??? is [color="blue"]818[/color].

[u]Explanation[/u]:
The equation is:

∞ - ∞ = ???

Rotating the lhs by 90 degrees and reading up to down, we get the number 818.

Consequently, the required number represented by ??? is [color="blue"]818[/color].

Clever. I like that answer!

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are you saying infinity - infinity = a variable that isnt defined... or is mu defined in zen

anyway if i remember right you can't really simplify, it could be something like

summation of e^x (x is positive integers)- summation X (x is positive integers) which would equal infinity

or the other way around which would equal -infinity or it could = 0.

it really depends mostly on comparing derivatives

it could even equal a number such as 1 such as in

summation X(all positive integers)- summation X(x>2 and integer)=1

tho im probably missing something with the puzzle or its going to turn out to be some pun or something

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If there's anything I ever learned in math it's that if you have some thing, and you take away all of that something, you have no more of it. If I have a :D and I subtract :D, I don't get :). I get zero. SOOOO, unless you want to argue that Infinity can never equal itself, Infinity minus infinity equals zero.

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any number minus itself equals 0. Rule number 1 of basic subtraction. Good riddle!!

Ok, so, some people have already hinted at this in their answers...but I thought I would write my thoughts on this riddle down:

First of all, "Infinity" is NOT a number...as a result, subtraction is not clearly defined (as one poster said earlier). While it is true that any number minus itself equals 0, infinity is not a number. Hence, the basic rules of subtraction do not apply.

Infinity is an IDEA, or a concept. You cannot count to infinity...there is no "infinity plus 1"...It's funny that infinity is so hard for many people to wrap their mind around, since it truly is simpler as a concept than anything finite (since you don't have to define any boundaries...it's just infinite).

Let's say we wanted to try to treat this infinity as a number for this riddle...we can then prove that there are actually infinite answers to this equation:

Infinity - Infinity = INFINITY

1. There are infinite positive even numbers...so the sum of all positive even numbers is "infinity"

2. There are infinite positive odd numbers...so the sum of all positive odd numbers is "infinity"

3. There are infinite positive whole numbers...so the sum of all positive whole numbers is "infinity"

4. So, the riddle states infinity - infinity = ????. If we substitute we get "sum of all positive whole numbers minus sum of all positive odd numbers equals what??" And the answer to that would be "the sum of all positive even numbers"...which is INFINITY

OR Infinity - Infinity = 4

1. There are infinite positive whole numbers...so the sum of all positive whole numbers is "infinity"

2. There are infinite positive whole numbers that are NOT the number 4...so the sum of all these numbers is "infinity"

3. So, again, we subtract the sum of all the positive whole numbers that are not the number 4 from the sum of all the positive whole numbers, and we get 4 as our answer

OR Infinity - Infinity = 74629274739

1. There are infinite positive whole numbers...so the sum of all positive whole numbers is "infinity"

2. There are infinite positive whole numbers that are NOT the number 74629274739...so the sum of all these numbers is "infinity"

3. So, again, we subtract the sum of all the positive whole numbers that are not the number 74629274739 from the sum of all of the positive whole numbers, and we get 74629274739 as our answer

So, you can see that trying to apply "basic" mathematical operations to INFINITY is pointless...

Since infinity is an idea, or concept, the equation basically becomes "an idea - an idea = ???" which would be "no ideas" or "no thought"...

Therefore, my answer is either a completely mindless person or Nirvana (or some other Zen-like state in which there are no thoughts)

Even the title of this riddle "Number infinity" is essentially an oxymoron. I'm actually really hoping there is some twist or cool trick that the OP was intending (like the 818 answer that was given)...

"Two things are infinite: the universe and human stupidity; and I'm not sure about the universe." -Albert Einstein

Edited by Pickett
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Ok, so, some people have already hinted at this in their answers...but I thought I would write my thoughts on this riddle down:

First of all, "Infinity" is NOT a number...as a result, subtraction is not clearly defined (as one poster said earlier). While it is true that any number minus itself equals 0, infinity is not a number. Hence, the basic rules of subtraction do not apply.

Infinity is an IDEA, or a concept. You cannot count to infinity...there is no "infinity plus 1"...It's funny that infinity is so hard for many people to wrap their mind around, since it truly is simpler as a concept than anything finite (since you don't have to define any boundaries...it's just infinite).

Let's say we wanted to try to treat this infinity as a number for this riddle...we can then prove that there are actually infinite answers to this equation:

Infinity - Infinity = INFINITY

1. There are infinite positive even numbers...so the sum of all positive even numbers is "infinity"

2. There are infinite positive odd numbers...so the sum of all positive odd numbers is "infinity"

3. There are infinite positive whole numbers...so the sum of all positive whole numbers is "infinity"

4. So, the riddle states infinity - infinity = ????. If we substitute we get "sum of all positive whole numbers minus sum of all positive odd numbers equals what??" And the answer to that would be "the sum of all positive even numbers"...which is INFINITY

OR Infinity - Infinity = 4

1. There are infinite positive whole numbers...so the sum of all positive whole numbers is "infinity"

2. There are infinite positive whole numbers that are NOT the number 4...so the sum of all these numbers is "infinity"

3. So, again, we subtract the sum of all the positive whole numbers that are not the number 4 from the sum of all the positive whole numbers, and we get 4 as our answer

OR Infinity - Infinity = 74629274739

1. There are infinite positive whole numbers...so the sum of all positive whole numbers is "infinity"

2. There are infinite positive whole numbers that are NOT the number 74629274739...so the sum of all these numbers is "infinity"

3. So, again, we subtract the sum of all the positive whole numbers that are not the number 74629274739 from the sum of all of the positive whole numbers, and we get 74629274739 as our answer

So, you can see that trying to apply "basic" mathematical operations to INFINITY is pointless...

Since infinity is an idea, or concept, the equation basically becomes "an idea - an idea = ???" which would be "no ideas" or "no thought"...

Therefore, my answer is either a completely mindless person or Nirvana (or some other Zen-like state in which there are no thoughts)

Even the title of this riddle "Number infinity" is essentially an oxymoron. I'm actually really hoping there is some twist or cool trick that the OP was intending (like the 818 answer that was given)...

"Two things are infinite: the universe and human stupidity; and I'm not sure about the universe." -Albert Einstein

nicely put!! so since

infinity-infinity can be any no from -infinity to 0 to +infinity as u illustrated (sum of all nos - sum of all nos except 'x'), or depending on the context of the use of infinity can even be any irrational or imaginary number, infinity - infinity=[set of all possible numbers] the answer can only be expressed in a set..

Edited by Sapientia
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You cannot count to infinity...
1, 2, 3, 4, ... ?

They are called the "counting" numbers, and their cardinality is Aleph0.

Other infinities, like the reals, have a higher cardinality and are uncountable.

So you can count to some infinities, but not all. ;)

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1, 2, 3, 4, ... ?

They are called the "counting" numbers, and their cardinality is Aleph0.

Other infinities, like the reals, have a higher cardinality and are uncountable.

So you can count to some infinities, but not all. ;)

I'm not so sure about that. You can count to some really high numbers, but there is theoretically no limit. While counting up to 900 septillion may seem like infinity, it isn't really. There may be a finite number of "counting" numbers used in daily situations, but if you are talking about about whole numbers, there is definitely an infinite number of them. I would very much like to have you explain to me what you are getting at because I am not convinced. I don't think you can count to any infinity. You would just have to make up new names for the really high numbers.

I do agree, though, that infinity - infinity = infinity. It's just like infinity/infinity does not = 1.

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I'm not so sure about that.

You can count to some really high numbers, but there is theoretically no limit.

While counting up to 900 septillion may seem like infinity, it isn't really.

I would very much like to have you explain to me what you are getting at because I am not convinced.

I don't think you can count to any infinity.

You would just have to make up new names for the really high numbers.

Well I think you have stated the idea already, but let me put it into these words.

There are just two cases.

  1. 1, 2, 3, 4, ... ends at some number N, beyond which one cannot count.
  2. There is no such number N.
To show that 1, 2, 3, 4, ... does not reach to infinity, one must show that N exists.
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ive never thought of it like this but if it ends at N it isnt infinity so y would it even be considered in this question or are you guys discussing whether if it ends in N in can be infinity?

as for the counting numbers or Natural they arent countable they just have a lower cardinality then most other

infinite sets. so for most it would be smaller, however it is possible to have a set of all numbers between the numbers 0 and 1 and this is the same cardinality as counting numbers (just put the counting numbers as decimals and voila) yet would be smaller. I think higher cardinality means bigger (whether bigger negative or positive), but often thoughts so simple in math are wrong. Anyway the point is

im pretty sure infinity - infinity can equal whatever it doesn't always equal infinity as even though infinity is not a number it does only represent a set idea. So i guess you could really only simplify this with a pair of bounds if you don't know which infinity is bigger (that concept is weird isn't it).

the number null works in computers as infinity causes a brain fart

0 is a very limited possibility

a constant is a limited possiblity

negative infinity and positive infinity are likely

I think this agrees with some of what each of you are saying and disagrees with some, but ill be interested to see how this turns out.

Edited by final
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Well I think you have stated the idea already, but let me put it into these words.

There are just two cases.

  1. 1, 2, 3, 4, ... ends at some number N, beyond which one cannot count.
  2. There is no such number N.
To show that 1, 2, 3, 4, ... does not reach to infinity, one must show that N exists.

Yes, the set of numbers in the series 1, 2, 3, 4, ... does have infinite cardinality, but the set of numbers counted will always be finite and therefore have a cardinality less than Aliph0. So you can count towards infinity, but you cannot count to infinity. "Towards" implies only the direction, but "to" implies reaching the destination.

Notice that Aliph0 / 2 = Aliph0. If you happen to ever reach infinity in a finite amount of time, you should have reached it in half the time. Repeating this logic, you would need to count an infinite amount of numbers in an infinitesimal amount of time, or, effectively, less than the amount of time required to count to 1.

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as for the counting numbers or Natural they arent countable they just have a lower cardinality then most other

infinite sets.

The counting numbers / natural numbers are trivially countable. The definition of countable is that there exists a 1-to-1 mapping between the set and a subset (including the possibility of the whole set) of the natural numbers. Just some quick trivia, but the rational numbers (all numbers that can be expressed as the division of one whole number by another) is also countable.

so for most it would be smaller, however it is possible to have a set of all numbers between the numbers 0 and 1 and this is the same cardinality as counting numbers (just put the counting numbers as decimals and voila) yet would be smaller. I think higher cardinality means bigger (whether bigger negative or positive), but often thoughts so simple in math are wrong.

The set of all numbers between 0 and 1 is not countable. No matter how you try and create a 1-to-1 mapping between them and the natural numbers, you'll miss some (a lot... but who's counting? pun not initially intended).

I think this agrees with some of what each of you are saying and disagrees with some, but ill be interested to see how this turns out.

I'd say that the answer is undefined as given, but that the range of possibilities are as you say (infinity, negative infinity, or any real number). If given two formulae, sets, limits, etc that spawned the two infinities, I'd be inclined to say that you could calculate the difference and get the (possibly finite) result. I guess it's because I disagreed with a calculus teacher back in the day that integrating 1/x between -1 and 1 was undefined... it should be 0, right? It's rotationally symmetric! grr! ;)

(hmm....now that I think about it....could it be proven with Lesbesgue integration? I know very little about that, but do know it integrates by cutting up the space differently (horizontal cuts vs. vertical cuts))

This is sort of the same problem as shown in the following example...

0 = (0+0+0+...) = ((-1+1) + (-1+1) + (-1+1) + ... ) = (-1 + (1+-1) + (1+-1) + (1+-1) + ...) = (-1+0+0+0+...) = -1

But this also equals ((1+1+1+...) + (-1+-1+-1+...)) = infinity + -infinity = infinity-infinity = ....?

Arranging two infinite sums in different ways can give you drastically different answers. So I guess without knowing not only the degree but the actual "value" of infinity, a difference may be impossible to find.

You know.....Godel became mentally unstable after thinking about this kinda stuff too long... (I know, I know, correlation != causation, single data point, etc.). It is my intention to sit down and play video games for several hours (for my sanity's sake, of course).

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If there's anything I ever learned in math it's that if you have some thing, and you take away all of that something, you have no more of it. If I have a :D and I subtract :D, I don't get :). I get zero. SOOOO, unless you want to argue that Infinity can never equal itself, Infinity minus infinity equals zero.

whiles it's off topic seeing that ksengupta came up with a clever answer, I want to point out that your logic is off due to the fact that you are treating infinity, a concept as a number.

for example, what is 3- infinity? that of course equals -infinty and whats - infinity + infinity? rearanging that you get infinity - infinity which according to you equals 0. that poses an algebraic anomaly though because you are in effect saying that (3- infinity) + infinity then equals 0 and not 3.

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thankyou i had forgotten the countable meant a 1 to 1 mapping

I would argue if you exclude 1 then map every number to its decimal or rather the other way around (for example .123759869 would be mapped to 123759869 ) shows the numbers between 0 and 1 only have one more item.

any ideas?

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i remember now the problem is .0000... numbers they cannot be mapped so the the set is the size of a set of sets of countable numbers or rather size of countable numbers squared....

anyway im surprised such an easy topic started such talk

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nope back to my original thought because eventhough you can look at it as .123 and.0123 as the hundred twenty third number in the first and second set of countable numbers (the first being all numbers with no starting 0's the second having one 0) which makes it look like the squared size.

You can just as easily look at it as the counting numbers can only have trailing zeros (i.e. 1000, 230), these numbers ([0,1])can only have 0's in front of it (other then middle zeros, i.e. .0001, .032) therefore they should have a one to one ratio. You can look at this as just the number backwards (230 .032) or as just switching the 0's from the end to the beginning (23450, .02345). either way only one number will be mapped to one number therefore it is a 1 to 1 mapping. It is very simple to prove that it is a one to one mapping by contradiction.

so were you wrong or is there a flaw somewhere?

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nope back to my original thought because eventhough you can look at it as .123 and.0123 as the hundred twenty third number in the first and second set of countable numbers (the first being all numbers with no starting 0's the second having one 0) which makes it look like the squared size.

You can just as easily look at it as the counting numbers can only have trailing zeros (i.e. 1000, 230), these numbers ([0,1])can only have 0's in front of it (other then middle zeros, i.e. .0001, .032) therefore they should have a one to one ratio. You can look at this as just the number backwards (230 .032) or as just switching the 0's from the end to the beginning (23450, .02345). either way only one number will be mapped to one number therefore it is a 1 to 1 mapping. It is very simple to prove that it is a one to one mapping by contradiction.

so were you wrong or is there a flaw somewhere?

(Hope I'm not totally misunderstanding what you're trying to say here) what you're pointing out is that the number of finite decimal strings between 0 and 1 is countable. This agrees with what EventHorizon said about rational numbers being countable, since all finite decimals are rational numbers (they equate to fractions, like 0.345256346 = 345256346/1000000000)

What you left out were those numbers which cannot be expressed as finite decimal strings. Some of these are rational too (like 1/3), but those that aren't, the irrational ones like sqrt(0.5), are uncountable.

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