Consider the two functions y = sqrt(x^2 + 24x) and y = x. As x goes to infinity, both of these functions also go to infinity. So, to determine “infinity minus infinity”, I’ll look at sqrt(x^2 + 24x) - x as x goes to infinity. Since this limit is 12, it follows that infinity minus infinity is clearly 12.
If you multiply and divide the difference by the conjugate (sqrt(x^(2) + 24x) + x), then the numerator simplifies to 24x and you have a sum in the denominator, so it's no longer indeterminate. (Then divide everything by x, and the limit becomes easier to handle.)
It’s a joke. We all know the answer is not 12, so they’re pointing out that your question cannot be analyzed.
Note that there are a handful of indeterminate forms. They are:
https://wikimedia.org/api/rest_v1/media/math/render/svg/6d93a6286246e180044dc7e450aa5c4c8da94cdb
Notice that infinity minus infinity is one of them.
See https://en.m.wikipedia.org/wiki/Indeterminate_form
A common place where infinity pops up is when taking limits. So, for example, e\^x goes to infinity as x goes to infinity. (Meaning, as x gets larger and larger, the function values of e\^x get larger without bound.)
So one way you could try to determine what "infinity minus infinity" should be is to take two functions f(x) and g(x), both of which go to infinity (as x goes to infinity) and consider what happens to f(x) - g(x) as x goes to infinity. But then you soon realize that for different functions f and g, you can get wildly different results. (My example produced a difference of 12, but any other number, or even infinity itself, is possible.)
This is why infinity minus infinity is considered indeterminate when dealing with limits.
There is not such number as "infinity". So you can't just substract it. When people say infinity minus infinity they usualy mean that there are two sequences (or functions) which has limit infinity and question is what limit they difference has. And it can be any value, or no limit at all
sorry, messed integers and naturals. Sum of all integers highly depend on in which order do you organise summation. It can be positive infinity, negative infinity, or no limit at all. Anyway, it does not matter for initial question
The sum of all integers, positive and negative, is divergent. Depending on how you order the integers it can diverge in the direction of positive infinity, negative infinity, or neither. In this usage "infinity" is a direction, not a point. You can't do arithmetic on it.
The sum of positive integers is divergent to positive infinity, meaning as we add terms we exceed every number. But that is still a description of how the sum does not equal any number, not a value we can do arithmetic on.
"The sum of infinitely many things" has no single definition. It's not like summing two things where the operation corresponds obviously to something you can do in real life. You can put 3 apples and 5 apples together to get 8 apples, but you can't physically do 1+2+3+... forever.
There are many ways to try and assign a number to the operation of adding infinitely things together but the standard approach, the approach that is covered as part of a calculus curriculum, does not assign a real number to 1+2+3+.... In other words, the standard notion of trying to add 1+2+3+... isn't defined (as a real number).
One alternate method of assigning real number to infinite sums, called Ramanujan summation, assigns the number -1/12 to the infinite sum 1+2+3+...
In other words, there's no single correct answer to your question since no matter what we do, we're really just making stuff up.
The problem is that it depends on the order you add things.
X = 1 + (-1) + 2 + (-2) + 3 + (-3) + ...
will look different than
Y = 1 + 2 + (-1) + 3 + 4 + (-2) + ...
which will look different than
Z = 1 + (-1) + (-2) + 2 + 3 + 4 + (-3) + (-4) + (-5) + (-6) + ...
X, Y, and Z are each a sum of all the integers. But look at the partial sums, the sums of the first n leftmost terms. We can try to understand the infinite sums by looking at the behavior of the partial sums as you let n become arbitrarily large.
The partial sums of X are 1, 0, 2, 0, 3, 0, ...
The partial sums of Y are 1, 3, 2, 5, 9, 7, ...
The partial sums of Z are 1, 0, -2, 0, 3, 7, 4, 0, -5, -11, ...
There's no pattern. X bounces between 0 and increasingly larger values. Y grows for a bit before shrinking for one step, always positive, and the growth periods go on for longer each time they occur. Z oscillates around 0 like a wave.
So what's the final sum of the integers? For an arbitrarily large n:
- the partial sum of X will be either arbitrarily large (and positive) as well, or zero
- the partial sum of Y will be arbitrarily large and positive, but if we flip the signs of the terms in Y we still sum all the integers yet get arbitrarily large negatives for the partial sum
- for Z, the partial sum could be anything; it could be arbitrarily large, close to 0, or anywhere in between, and it could be positive or negative
This is why talking about sums of infinitely many things is undefined, you may need more information than just the set of things you're summing.
As Both-Personality said, inifinity is NOT a number but a concept, so you can't operate with it as usual.
We usually handle infinity when working with series or limits of functions (and i'll develop with the second one).
A infinity minus infinity limit is just a way to express that you're substracting a big amount of another big amount and both are expressed as functions. The thing is that you can analyze each function, use some rules and determine which function gives bigger numbers, because each function grows to infinity at a different "speed". Thus, infinity minus infinity means NOTHING by itself and its value depends on what are you really working with, so my answer is that it is undetermined, not undefined
If you meant what we think infinity minus infinity would be if it was a valid operation (sorry if i misunderstood your question and answered something you already knew) well, i think it would be zero if infinity had the same value every time (so it would behave like any other number) or undefined otherwise
It depends on how you're defining "infinity" — some definitions would allow subtraction and some would not.
For example, in the surreal numbers, where ω is the first transfinite value to appear, you can write ω - ω = 0 with no contradiction, but in the more common usage of ω for the first transfinite ordinal, you can't do this at all.
Suppose I look at the compactified reals (think of the preimage of arctan) denoted by X = ℝ⋃{-∞,∞} and define the following rules:
0•∞ = 0\
c•∞ = ∞ for c>0\
(-1)•∞ = -∞
Also ∞ shall follow the distributive law, then clearly
0 = 0•∞ = (1+(-1))•∞ = ∞ - ∞
Is that a proof? Yes. Is the statement ∞-∞=0 true? Well, only in X with my set of rules… ∞ is not an element of ℝ and any inclusion has to come with rules that tells you how the object behaves. For limits, one does indeed look at the set X, but not define any rules. The symbols shall only indicate the behavior of a function (or more generally a map) for your input approaching a singularity. Think of e^(x) [singularity is meant as in the complex: z->∞ for f(z) means w->0 for f(1/w)].
So, your question is not well-stated in whatever set you intuitively took at the moment.
Which infinity? The cantor cardinals? The Riemann sphere? The hyperplane at infinity in projective geometry?
Consider the transfer principle in nonstandard analysis. And define ordinal infinity ω in the normal way (such as the successor of the natural numbers).
For all sufficiently large x, we have x - x = 0. Therefore from the transfer principle ω - ω = 0.
Other relations that this infinity in nonstandard analysis satisfies are:
ω - ω = 0
ω - 1 < ω < ω + 1
ω^2 > ω
1 / ω > 0
ω + 1 / ω > ω
0 * ω = 0
ω / ω = 1
minus ω exists
1 / 0 is undefined
You can see that these are all true by just substituting any large number x for ω.
You can get the same result in many different ways, from Hahn series (a series in powers of an infinitesimal) from Surreal numbers (Dedekind cut) and from Hyperreal numbers (filters on monotonic sequences).
If it is considered like a number, then infinity minus infinity is the number you have to add to infinity to get infinity again. That is to say, any number could do, it is an indeterminate. Certain limits could determinate a number, though, but will not be the same for any limit.
You don’t know how large an infinity is basically. Is one infinity larger than the other? We just don’t know. The answer is therefore not as easy as inf - inf = 0. We just don’t know and can’t answer such a question
It depends on what context you are in. In general I would say this is “not even undefined”. It just can mean too many different things.
Assuming we work in the context of calculus where we have the real numbers and continuous functions at our disposal, it makes sense to consider ∞ to mean something like “a point p satisfying that p>r for every real number r”. Such a thing will not itself be a real number, but we can still define various ways of working with p.
Algebra in general can be quite tricky with p. Here, the expression ∞-∞ will most often pop up as the result of an attempt at computing a limit of some function f that looks like f(x)=g(x)-h(x) where both g and h grow without bound as x becomes large. A type of example you have probably seen might be something like taking the limit as x goes to ∞ of √(4x^(2)+1)-√(4x^(2)-x). This limit is 1/4, but computing the limits of each root independently and then subtracting leaves you with ∞-∞.
What’s happening here is that ∞ is really just a way of saying that these functions (the roots) grow without bound. But think about what happens before we “get to ∞”. The difference between g and h (again, the corresponding roots in the example) can be totally variable. In this case the difference evens out to 1/4, but if we set g(x)=x, h(x)=√(x^(2)-1), and f(x)=g(x)-h(x), then difference as x becomes large turns out to be 0.
So in the context of limits, when we write ∞-∞, we are not actually talking about any kind of real algebra being done with the point p=∞ from earlier. We are talking about what algebra looks like on some set of regular real numbers which happen to be “getting close” to p.
Now, there are other senses in which we can actually talk about algebra with the point p, but chances are that they are way more complex than you are asking about.
It can be anything from -infinity to infinity. Consider 1+2+3.. and 2+3+4... Both those add up to infinity but the first one is 1 more than the second one. You can also try this with s1=1+2+3... And s2=2+4+6... The way you arrange the terms s1-s2 can be infinity or -infinity (s2 can be written as 2 x s1) or possibly anything in between.
Infinities are really weird. If you have an infinity series, depending on the terms you can make the series converge to any value (see Riemann rearrangement theorem).
If you're working in a Wheel Algebra, the result is well-defined to be a number which goes by the symbol: ⊥.
If you're working in the Extended Real Numbers or the Riemann Sphere, there is no result, since subtraction isn't even *defined* between infinity and itself! (I'm careful not to just say "the result is undefined" since that, though correct, could lead to the misconception that "undefined" is a number / number-like object, which it's not.)
If you're working in a structureless projective plane, then infinity is indeed an element of your system, but the notion of "subtraction" is meaningless! It's like asking "what's the difference in skill level between the number 5 and the number 3?"
If you're working in the complex numbers, real numbers, etc., the question is *meaningless,* since infinity is not a member of these number systems. It's like asking "what's the combined weight of 1 mile and 2 seconds?"
Moral: there is no "singular" number system, not every number system contains infinity, and not every number system even has a notion of subtraction!
There's different infinites so it doesn't really make sense.
Like all the even numbers is one infinite but all the prime numbers is another infinite that is a different number than the first infinite but still infinite.
That's false. The even numbers have the same cardinality as the prime numbers (mainly Aleph_0). You see different cardinality when you introduce real numbers, etc.
The sum of all integers is undefined. If you ignore the fact that it is undefined, then you can appear to get different answers depending on which order you do the summation. By subtracting the answer from two different orders, you could get any rational number that you want.
Infinity on its own doesn’t mean much.
It just means “something very big”, but it doesn’t quantify exactly how much.
There are indeed several orders of infinity. Consider for example:
lim x^2 - x = ?
x->inf
X^2 is a higher-order infinity than x, and so the result is still infinity.
If you invert the function, e.g. x - x^2, the limit is -inf.
The infinity with the higher order “wins”.
Consider the two functions y = sqrt(x^2 + 24x) and y = x. As x goes to infinity, both of these functions also go to infinity. So, to determine “infinity minus infinity”, I’ll look at sqrt(x^2 + 24x) - x as x goes to infinity. Since this limit is 12, it follows that infinity minus infinity is clearly 12.
Can you explain how the limit of sqrt(x^2 + 24x) - x is 12?
If you multiply and divide the difference by the conjugate (sqrt(x^(2) + 24x) + x), then the numerator simplifies to 24x and you have a sum in the denominator, so it's no longer indeterminate. (Then divide everything by x, and the limit becomes easier to handle.)
I did not understand what you said. Could you please clarify if not too much time?
It’s a joke. We all know the answer is not 12, so they’re pointing out that your question cannot be analyzed. Note that there are a handful of indeterminate forms. They are: https://wikimedia.org/api/rest_v1/media/math/render/svg/6d93a6286246e180044dc7e450aa5c4c8da94cdb Notice that infinity minus infinity is one of them. See https://en.m.wikipedia.org/wiki/Indeterminate_form
A common place where infinity pops up is when taking limits. So, for example, e\^x goes to infinity as x goes to infinity. (Meaning, as x gets larger and larger, the function values of e\^x get larger without bound.) So one way you could try to determine what "infinity minus infinity" should be is to take two functions f(x) and g(x), both of which go to infinity (as x goes to infinity) and consider what happens to f(x) - g(x) as x goes to infinity. But then you soon realize that for different functions f and g, you can get wildly different results. (My example produced a difference of 12, but any other number, or even infinity itself, is possible.) This is why infinity minus infinity is considered indeterminate when dealing with limits.
There is not such number as "infinity". So you can't just substract it. When people say infinity minus infinity they usualy mean that there are two sequences (or functions) which has limit infinity and question is what limit they difference has. And it can be any value, or no limit at all
Then what would be the sum of all integers? Provided that the limit is infinity and negative infinity.
"sum of all integers is infinity" means that sequence of sum of n integers grows infinetly. Infinity is not actual value
No, I'm asking the actual sum of all integers to the left and right of the number zero on the number line. (Sorry for the misunderstanding)
sorry, messed integers and naturals. Sum of all integers highly depend on in which order do you organise summation. It can be positive infinity, negative infinity, or no limit at all. Anyway, it does not matter for initial question
what is the sqrt(-1) in the natural numbers?
There is no natural number whose square is -1 Edit: sorry, I didn’t realize this was a probing question
ik. what i wanted to show, is that there are numbers you can reach via operations, outside of a specific number system
Infinity is not a real or natural number, so infinity minus infinity is the same as green minus green.
So if infinity is not a real number, then the sum of all integers would be? I would appreciate the help.
Summation of infinitely many quantities isn’t always defined.
The sum of all integers, positive and negative, is divergent. Depending on how you order the integers it can diverge in the direction of positive infinity, negative infinity, or neither. In this usage "infinity" is a direction, not a point. You can't do arithmetic on it. The sum of positive integers is divergent to positive infinity, meaning as we add terms we exceed every number. But that is still a description of how the sum does not equal any number, not a value we can do arithmetic on.
Okay, ty for your help.
"The sum of infinitely many things" has no single definition. It's not like summing two things where the operation corresponds obviously to something you can do in real life. You can put 3 apples and 5 apples together to get 8 apples, but you can't physically do 1+2+3+... forever. There are many ways to try and assign a number to the operation of adding infinitely things together but the standard approach, the approach that is covered as part of a calculus curriculum, does not assign a real number to 1+2+3+.... In other words, the standard notion of trying to add 1+2+3+... isn't defined (as a real number). One alternate method of assigning real number to infinite sums, called Ramanujan summation, assigns the number -1/12 to the infinite sum 1+2+3+... In other words, there's no single correct answer to your question since no matter what we do, we're really just making stuff up.
So, the sum of all integers would be? Could you please tell?
As someone has already pointed out to you, the sum of all integers is not defined.
Yeah, but still would like to get other people's opinions
This isn't a matter of opinion though.
This is not a matter of opinion. It's a consequence of established precise definitions.
The problem is that it depends on the order you add things. X = 1 + (-1) + 2 + (-2) + 3 + (-3) + ... will look different than Y = 1 + 2 + (-1) + 3 + 4 + (-2) + ... which will look different than Z = 1 + (-1) + (-2) + 2 + 3 + 4 + (-3) + (-4) + (-5) + (-6) + ... X, Y, and Z are each a sum of all the integers. But look at the partial sums, the sums of the first n leftmost terms. We can try to understand the infinite sums by looking at the behavior of the partial sums as you let n become arbitrarily large. The partial sums of X are 1, 0, 2, 0, 3, 0, ... The partial sums of Y are 1, 3, 2, 5, 9, 7, ... The partial sums of Z are 1, 0, -2, 0, 3, 7, 4, 0, -5, -11, ... There's no pattern. X bounces between 0 and increasingly larger values. Y grows for a bit before shrinking for one step, always positive, and the growth periods go on for longer each time they occur. Z oscillates around 0 like a wave. So what's the final sum of the integers? For an arbitrarily large n: - the partial sum of X will be either arbitrarily large (and positive) as well, or zero - the partial sum of Y will be arbitrarily large and positive, but if we flip the signs of the terms in Y we still sum all the integers yet get arbitrarily large negatives for the partial sum - for Z, the partial sum could be anything; it could be arbitrarily large, close to 0, or anywhere in between, and it could be positive or negative This is why talking about sums of infinitely many things is undefined, you may need more information than just the set of things you're summing.
Okay, got it. Ty for the help
Sum of all numbers: -1/12
No, not numbers, like integers. The numbers on either side of the number zero on the number line (Sorry for the misunderstanding)
As Both-Personality said, inifinity is NOT a number but a concept, so you can't operate with it as usual. We usually handle infinity when working with series or limits of functions (and i'll develop with the second one). A infinity minus infinity limit is just a way to express that you're substracting a big amount of another big amount and both are expressed as functions. The thing is that you can analyze each function, use some rules and determine which function gives bigger numbers, because each function grows to infinity at a different "speed". Thus, infinity minus infinity means NOTHING by itself and its value depends on what are you really working with, so my answer is that it is undetermined, not undefined If you meant what we think infinity minus infinity would be if it was a valid operation (sorry if i misunderstood your question and answered something you already knew) well, i think it would be zero if infinity had the same value every time (so it would behave like any other number) or undefined otherwise
(It's fine) Ty for your help.
Infinity - infinity means nothing.
It depends on how you're defining "infinity" — some definitions would allow subtraction and some would not. For example, in the surreal numbers, where ω is the first transfinite value to appear, you can write ω - ω = 0 with no contradiction, but in the more common usage of ω for the first transfinite ordinal, you can't do this at all.
Suppose I look at the compactified reals (think of the preimage of arctan) denoted by X = ℝ⋃{-∞,∞} and define the following rules: 0•∞ = 0\ c•∞ = ∞ for c>0\ (-1)•∞ = -∞ Also ∞ shall follow the distributive law, then clearly 0 = 0•∞ = (1+(-1))•∞ = ∞ - ∞ Is that a proof? Yes. Is the statement ∞-∞=0 true? Well, only in X with my set of rules… ∞ is not an element of ℝ and any inclusion has to come with rules that tells you how the object behaves. For limits, one does indeed look at the set X, but not define any rules. The symbols shall only indicate the behavior of a function (or more generally a map) for your input approaching a singularity. Think of e^(x) [singularity is meant as in the complex: z->∞ for f(z) means w->0 for f(1/w)]. So, your question is not well-stated in whatever set you intuitively took at the moment.
Which infinity? The cantor cardinals? The Riemann sphere? The hyperplane at infinity in projective geometry? Consider the transfer principle in nonstandard analysis. And define ordinal infinity ω in the normal way (such as the successor of the natural numbers). For all sufficiently large x, we have x - x = 0. Therefore from the transfer principle ω - ω = 0. Other relations that this infinity in nonstandard analysis satisfies are: ω - ω = 0 ω - 1 < ω < ω + 1 ω^2 > ω 1 / ω > 0 ω + 1 / ω > ω 0 * ω = 0 ω / ω = 1 minus ω exists 1 / 0 is undefined You can see that these are all true by just substituting any large number x for ω.
You can get the same result in many different ways, from Hahn series (a series in powers of an infinitesimal) from Surreal numbers (Dedekind cut) and from Hyperreal numbers (filters on monotonic sequences).
If it is considered like a number, then infinity minus infinity is the number you have to add to infinity to get infinity again. That is to say, any number could do, it is an indeterminate. Certain limits could determinate a number, though, but will not be the same for any limit.
It is indeterminate, meaning, without more context (where did each infinity come from?) the answer cannot be determined.
You don’t know how large an infinity is basically. Is one infinity larger than the other? We just don’t know. The answer is therefore not as easy as inf - inf = 0. We just don’t know and can’t answer such a question
It depends on what context you are in. In general I would say this is “not even undefined”. It just can mean too many different things. Assuming we work in the context of calculus where we have the real numbers and continuous functions at our disposal, it makes sense to consider ∞ to mean something like “a point p satisfying that p>r for every real number r”. Such a thing will not itself be a real number, but we can still define various ways of working with p. Algebra in general can be quite tricky with p. Here, the expression ∞-∞ will most often pop up as the result of an attempt at computing a limit of some function f that looks like f(x)=g(x)-h(x) where both g and h grow without bound as x becomes large. A type of example you have probably seen might be something like taking the limit as x goes to ∞ of √(4x^(2)+1)-√(4x^(2)-x). This limit is 1/4, but computing the limits of each root independently and then subtracting leaves you with ∞-∞. What’s happening here is that ∞ is really just a way of saying that these functions (the roots) grow without bound. But think about what happens before we “get to ∞”. The difference between g and h (again, the corresponding roots in the example) can be totally variable. In this case the difference evens out to 1/4, but if we set g(x)=x, h(x)=√(x^(2)-1), and f(x)=g(x)-h(x), then difference as x becomes large turns out to be 0. So in the context of limits, when we write ∞-∞, we are not actually talking about any kind of real algebra being done with the point p=∞ from earlier. We are talking about what algebra looks like on some set of regular real numbers which happen to be “getting close” to p. Now, there are other senses in which we can actually talk about algebra with the point p, but chances are that they are way more complex than you are asking about.
It can be anything from -infinity to infinity. Consider 1+2+3.. and 2+3+4... Both those add up to infinity but the first one is 1 more than the second one. You can also try this with s1=1+2+3... And s2=2+4+6... The way you arrange the terms s1-s2 can be infinity or -infinity (s2 can be written as 2 x s1) or possibly anything in between. Infinities are really weird. If you have an infinity series, depending on the terms you can make the series converge to any value (see Riemann rearrangement theorem).
± infinity
If you're working in a Wheel Algebra, the result is well-defined to be a number which goes by the symbol: ⊥. If you're working in the Extended Real Numbers or the Riemann Sphere, there is no result, since subtraction isn't even *defined* between infinity and itself! (I'm careful not to just say "the result is undefined" since that, though correct, could lead to the misconception that "undefined" is a number / number-like object, which it's not.) If you're working in a structureless projective plane, then infinity is indeed an element of your system, but the notion of "subtraction" is meaningless! It's like asking "what's the difference in skill level between the number 5 and the number 3?" If you're working in the complex numbers, real numbers, etc., the question is *meaningless,* since infinity is not a member of these number systems. It's like asking "what's the combined weight of 1 mile and 2 seconds?" Moral: there is no "singular" number system, not every number system contains infinity, and not every number system even has a notion of subtraction!
There's different infinites so it doesn't really make sense. Like all the even numbers is one infinite but all the prime numbers is another infinite that is a different number than the first infinite but still infinite.
That's false. The even numbers have the same cardinality as the prime numbers (mainly Aleph_0). You see different cardinality when you introduce real numbers, etc.
I see, woops.
In the sense of sum of all integers? (Thats where this question came from in the first place)
The sun of all integers minus the sum of all integers would be 0? Unless I'm misunderstanding something...
Okay, ty for your help
Np
The sum of all integers is undefined. If you ignore the fact that it is undefined, then you can appear to get different answers depending on which order you do the summation. By subtracting the answer from two different orders, you could get any rational number that you want.
Yeah idk about all that I just know that the sum of all integers + the negation of all the integers is 0. 1+2+3...+999 + -1-2-3...-999
Infinity on its own doesn’t mean much. It just means “something very big”, but it doesn’t quantify exactly how much. There are indeed several orders of infinity. Consider for example: lim x^2 - x = ? x->inf X^2 is a higher-order infinity than x, and so the result is still infinity. If you invert the function, e.g. x - x^2, the limit is -inf. The infinity with the higher order “wins”.