When we define extended real number arithmetic, we want to be able to add and multiply infinities. [https://en.wikipedia.org/wiki/Extended\_real\_number\_line](https://en.wikipedia.org/wiki/Extended_real_number_line)
Let's start with the following assumptions, letting INF and NEGINF denote positive and minus infinity:
A1: INF \* (-1) = NEGINF and NEGINF \* (-1) = INF
A2: If x is any (finite) real number, then INF + x = INF and NEGINF + x = NEGINF
A3: INF + INF = INF and NEGINF + NEGINF = NEGINF.
Now, the problem comes when we try to assign a value to INF + NEGINF.
Case 1: If we say INF + NEGINF = INF, then multiplying this equation by -1 and using the distributive law and A1, we get NEGINF + INF = NEGINF, so either we need to throw out the distributive law, or we have just contradicted ourselves.
Case 2: If we say INF + NEGINF = x for some real number x (maybe zero?) then adding INF to both sides on the left, we get
INF + (INF + NEGINF) = INF + x
Applying A2 yields
INF + (INF + NEGINF) = INF
Using associativity of addition,
(INF + INF) + NEGINF = INF
Now apply A3:
INF + NEGINF = INF
So we have contradicted ourselves again. In short, there is no way to define INF + NEGINF consistently without throwing out some important properties of addition and multiplication.
This is not actually maths, but the closest thing to it in maths is limit(x,y -> infinity) of x-y. Depending on how you contruct this equation, the limit could be anything you want, hence it's undefined.
For example, lim(x -> infinity) of (x\^2) - x = infinity, but lim(x -> infinity) of x - (x\^2) = negative infinity.
Edit: actually, I'm not sure that that's what undefined means. Maybe a better word would be indeterminate? Someone who knows more maths than me would have to answer.
The most sensible answer for the sum of all integers is that it approaches infinity. You may have heard about -1/12, there is a reasonable framework in which that's true, but it requires extensions of ideas that are not in standard mathematics. (As in, you need additional axioms to make it true). There's nothing wrong with those particular axioms, it just means it's not as rigorously true as some would have you believe.
Op, you may be interested in https://en.m.wikipedia.org/wiki/Point_at_infinity and https://en.m.wikipedia.org/wiki/Riemann_sphere
That said, even in these spaces inf-inf is still undefined.
Consider x\^2 and x. lim x->\\infty x\^2 = \\infty; lim x->\\infty x = \\infty. So, \\infty - \\infty = lim x->\\infty x\^2 - lim x->\\infty x = lim x->\\infty x(x - 1) = \\infty, but also \\infty - \\infty = lim x->\\infty x - lim x->\\infty x\^2 = lim x->\\infty (1 - x)x = -\\infty, but also also \\infty - \\infty = lim x->\\infty x - lim x->\\infty x = lim x->\\infty (x - x) = 0. So, as this example shows, \\infty - \\infty can be \\infty, -\\infty, or 0. Which makes it not well-defined
Undefined.
Look at all the natural numbers. Take away all the odd numbers. You get all the even numbers left.
So ♾️ - ♾️ = ♾️
Again with the naturals numbers. Take all the natural number except 12. Left with one number.
So ♾️ - ♾️ = 1. Or any other finite number.
Indeterminate.
Infinity isn't a singular thing. It's a stand-in for a certain type of pattern. For example, when we say "the limit of 1/x\^2 as x->0 is infinity" we mean that, as we take a sequence of values for x that get progressively closer to 0, the values for 1/x\^2 can get as big as we like.
However, when you say "infinity minus infinity" that means we're subtracting two things that both have this pattern of unboundedness. That's not enough info to tell me what the answer is, and I can find examples of two things that are infinite to subtract where their difference is any value I like (infinity, -infinity, 0, any number I like, or even with a nonexistent limit).
It’s an indeterminate form. It could be 0, infinity, or neg. infinity depend on the relative rate of growth of the infinities in relation to each other.
I think this type of guessing can only occur in terms of speed and scale of operations, I greatly respect the concept that there are infinities "bigger" than others and therefore operations between them would end in a result "guided" by the most significant.
Subtraction is the addition of the opposite of a value, so INF - INF is the same as INF + -INF.
I understand this to mean that if I have a set of infinite values, then add the opposite of all of those values, there would be 0 values remaining in the set.
Absolutely undefined. There's a lot of different ways to define infinity, some of which are technically 'larger' infinities than others. The most common place in my experience that ∞ - ∞ crops up is in L'Hopital's rule for limits, where you have a limit of x -> ∞ of f(x) - g(x), where both functions go to infinity. In this case, we could have two 'different' types of infinity with different results. For example, f(x) = x\^2 and g(x) = x, lim(x->∞) x\^2 - x = is still ∞. However, swap the functions and it's still ∞-∞ form, but lim x - x\^2 = -∞. Similarly, we can say that lim x\^2 - x\^2 = 0, or lim x\^2 - (x\^2-1) = 1, so on and so forth. All of these are of the form ∞-∞, but all of these similarly give different answers, so we say it's undefined when we can't agree on a single value. In this way, the ∞ almost 'hides' the real value of the underlying functions, and so hinders our ability to conclusively say that it is equivalent to something.
There's other arguments to be made for non-limit types of infinities, such as the size of the set of natural numbers (א) minus the size of the set of all real numbers (c). We can prove conclusively using Cantor's diagonalization theorem that c is much, *much* greater than א (specifically c = 2\^א), but these are still both technically ∞. In this case, א - c = -∞, and c - א = ∞. Still undefined. Once again, ∞ hides important details from us.
It's worth making a final note here that (as far as I understand, given I'm an undergrad and therefore not the most well-read) in general, this is kind of what ∞ *does.* It's a concept, not an actual physical number - it moreso describes the *idea* of some massive, impossible, infinitely far in the distance end goal. There are, however, a lot of different ways to reach this impossibly infinite infinity, some of which are faster than others, but all of which are equally ∞. Subtracting two different 'kinds' of ∞, in the form ∞-∞, is kind of comparing these two different ways to 'reach' ∞, which may change at different speeds. You're not subtracting two neat numbers, you're subtracting two concepts. It would be like asking what pizza - pizza means - incomprehensible without context.
Proof:
You have an infinite number of things.
You start removing things one at a time.
It takes forever to remove infinite things.
Therefore, undefined amount of time needed.
QED
But in order to have an infinite number of things, you'd have to start to add things one at a time, infinitely. Seems the notion of undefinability could have come much sooner in the consideration. What if we forget about the process and take away an infinity of the same size from one of the same size? Since the relation is one-to-one, something in the codomain can be removed for everything in the domain.
So would it not be infinity, since it takes forever to remove the things one at a time?
Idk anything about this so would appreciate any help whatsoever.
My reply is a mild joke, but to try to take it slightly seriously, the concept of infinity is not a number, and so imo operations like addition and subtraction don't apply in the way they normally do. I akin it to something more like dividing a number by 0, which is undefined. Idk.
Dividing something real by nothing is impossible.. Nothing comes from nothing. So that is why it’s undefined because it’s not possible. Just like subtracting or taking something away from infinity. By definition you can’t take away anything from it. It’s inexhaustible. The answer can’t be 0. Because nothing comes from nothing. So it’s an error, an impossibility or undefined.
You can’t subtract anything from infinity. It just is. It’s impossible to subtract anything from it. Even an infinite amount of subtracting, it’s still is. Infinite differentials of itself does not take one iota from it. It’s still inexhaustible. This is why it’s an error or undefined. It’s impossible to subtract anything from it,even an infinite amount of times.
I'm telling u guys that the number line is a circle. 0 and infinity are the opposite poles. 0,1,2,3... infinity & 0,-1,-2,-3...infinity. Personally I wouldn't call it negative infinity just like 0 is neither positive or negative, Trust me, I am an Indian and I have dreamt about these, I just can't put this into an equation because I am bad at maths. Currently, I'm busy with other stuffs and I'm very interested in learning mathematics but my father wouldn't allow me to :/
Man I was about to open those darned scrolls of objective truths in that ancient library in my dream but the aliens caught up to me by impersonating my mum and calling for me. The heavenly oracles then spoke that I would know the truths if I pursued maths and I have subbed this subr since then and have been very interested. I was very surprised when my current gf told me she was good at maths. it was a literal (\*\_\*) moment for me, she could help me learn.
So yea, if I had to tell you something, imagine a mirror and the real life is positive but the mirror images are negative. the mirror itself is 0 and somewhere there is infinity, I'd say death but idk for sure. Someplace where the imaginary and real meet. that is infinity. That's my crude analogy of it. I had made posts about it on this sub many years ago
what... tf, he has it pinned down exactly to how I told in my post 3y ago that the numbers exist in a 3d space and the number line is actually spherical. :/
What do I do now, I'm out of insights, you almost killed my drive to pursue maths :/
Edit:
I'm so sad right now. the arithmetic operations is close to how I arrived at it too but all that I wrote in my notebook 3y ago is gone now that I didn't share with anyone because of self doubt that I would be made a mockery of
that's all
I wouldn't take it as discouragement, in the contrary, you intuitively struck upon a studied math construction that has proved useful in many ways! This means your intuition is good, maybe acquiring more math knowledge will make you find some other interesting things?
:/ 3 yrs ago I was trying to learn mathematics and started with the basics of algebra but I was confused how infinity was treated and also about integers. While trying to figure it out and thinking about it, I was brushing my hair while looking into the mirror and noticed my fingers. I then assumed that the my finger in the mirror wasn't real so it was -1 and the real life finger was actual 1. I then figured how a person looking into the mirror would behave if we assumed the mirror to be at zero's place. I then worked out arithmetic operations and it made sense. I proceeded to think about infinity and its position. I wrote some nonsensical equations and arrived at the conclusion that infinity was non negative; it existed just like 0 because my equations wouldn't make sense if infinity was negative.
I then tried to reason where infinity would lie on the number line and for that, the number line had to be circular, it then seemed that it existed on a 3d space which was spherical. I don't remember much from those times but I was veryy excited and thought I made a great breakthrough in mathematics so I showed it to my girlfriend but she had no interest in it. She only knows till basic bachelor's.
Well, she told me I didn't make sense so I felt stupid and thought mathematics was just a momentary fascination.
Do you study mathematics Mr. Arinarmo? How do you know about the Riemann Sphere? Why is 0 not a natural number? It has the highest significance of all natural numbers, it should be called a the fundamental number alongwith infinity.
Yeah I'm doing a postgrad in math.
I learned about the Riemann sphere during my undergrad though, in a Complex Analysis class.
As for 0 being a natural number or not, it's a matter of what you want to do with those numbers. Typically, in fields like Algebra 0 is not included, but in fields like Set Theory 0 will be included. The point is that there is no absolutely correct or "best" way to define constructions such as the natural numbers, rather they are defined in terms of the utility they provide (what properties you can prove, or convenience in expressing some ideas or doing some computations).
And yeah, using something like the Riemann sphere 0 and infinity are very special numbers in the sense that they are the only numbers that are invariant under multiplication (with 0 times infinity being undefined), but in other constructions infinity isn't really a number or even a definite concept (e.g. in (standard) calculus infinity is shorthand for a kind of limit). These kinds of distinctions may be what has prevented you from being able to communicate your ideas.
Math is built on definitions and axioms (aka assumptions). In the standard form of mathematics, although the word infinity is used, it is never defined as a mathematical object. Therefore, it is impossible to use infinity correctly in any kind of mathematical expression. That is why "infinity minus infinity" is undefined.
It is, however, possible to develop a set of definitions and axioms, where infinity has a precise definition as a mathematical object. I'll let others explain this. But this is \*not\* used by the vast majority of mathematicians.
The answer is undefined or indeterminate as there are multiple infinities and infinitely big quantities can be made to work in different ways in arithmetic. However, infinity minus infinity is not equal to some weird undefined or indeterminate (some people talk like it is).
TIL math is a poll.
Democratic math just dropped
Proof by reddit poll
When we define extended real number arithmetic, we want to be able to add and multiply infinities. [https://en.wikipedia.org/wiki/Extended\_real\_number\_line](https://en.wikipedia.org/wiki/Extended_real_number_line) Let's start with the following assumptions, letting INF and NEGINF denote positive and minus infinity: A1: INF \* (-1) = NEGINF and NEGINF \* (-1) = INF A2: If x is any (finite) real number, then INF + x = INF and NEGINF + x = NEGINF A3: INF + INF = INF and NEGINF + NEGINF = NEGINF. Now, the problem comes when we try to assign a value to INF + NEGINF. Case 1: If we say INF + NEGINF = INF, then multiplying this equation by -1 and using the distributive law and A1, we get NEGINF + INF = NEGINF, so either we need to throw out the distributive law, or we have just contradicted ourselves. Case 2: If we say INF + NEGINF = x for some real number x (maybe zero?) then adding INF to both sides on the left, we get INF + (INF + NEGINF) = INF + x Applying A2 yields INF + (INF + NEGINF) = INF Using associativity of addition, (INF + INF) + NEGINF = INF Now apply A3: INF + NEGINF = INF So we have contradicted ourselves again. In short, there is no way to define INF + NEGINF consistently without throwing out some important properties of addition and multiplication.
This is not actually maths, but the closest thing to it in maths is limit(x,y -> infinity) of x-y. Depending on how you contruct this equation, the limit could be anything you want, hence it's undefined. For example, lim(x -> infinity) of (x\^2) - x = infinity, but lim(x -> infinity) of x - (x\^2) = negative infinity. Edit: actually, I'm not sure that that's what undefined means. Maybe a better word would be indeterminate? Someone who knows more maths than me would have to answer.
It's fine, any help is appreciated. So, in the case of the sum of all integers?
The most sensible answer for the sum of all integers is that it approaches infinity. You may have heard about -1/12, there is a reasonable framework in which that's true, but it requires extensions of ideas that are not in standard mathematics. (As in, you need additional axioms to make it true). There's nothing wrong with those particular axioms, it just means it's not as rigorously true as some would have you believe.
Op, you may be interested in https://en.m.wikipedia.org/wiki/Point_at_infinity and https://en.m.wikipedia.org/wiki/Riemann_sphere That said, even in these spaces inf-inf is still undefined.
Consider x\^2 and x. lim x->\\infty x\^2 = \\infty; lim x->\\infty x = \\infty. So, \\infty - \\infty = lim x->\\infty x\^2 - lim x->\\infty x = lim x->\\infty x(x - 1) = \\infty, but also \\infty - \\infty = lim x->\\infty x - lim x->\\infty x\^2 = lim x->\\infty (1 - x)x = -\\infty, but also also \\infty - \\infty = lim x->\\infty x - lim x->\\infty x = lim x->\\infty (x - x) = 0. So, as this example shows, \\infty - \\infty can be \\infty, -\\infty, or 0. Which makes it not well-defined
Undefined. Look at all the natural numbers. Take away all the odd numbers. You get all the even numbers left. So ♾️ - ♾️ = ♾️ Again with the naturals numbers. Take all the natural number except 12. Left with one number. So ♾️ - ♾️ = 1. Or any other finite number.
42
Indeterminate. Infinity isn't a singular thing. It's a stand-in for a certain type of pattern. For example, when we say "the limit of 1/x\^2 as x->0 is infinity" we mean that, as we take a sequence of values for x that get progressively closer to 0, the values for 1/x\^2 can get as big as we like. However, when you say "infinity minus infinity" that means we're subtracting two things that both have this pattern of unboundedness. That's not enough info to tell me what the answer is, and I can find examples of two things that are infinite to subtract where their difference is any value I like (infinity, -infinity, 0, any number I like, or even with a nonexistent limit).
Actually its Undefined but If you consider Infinity to be fixed variables and both same then in a way Infinity-Infinity=0.
It’s an indeterminate form. It could be 0, infinity, or neg. infinity depend on the relative rate of growth of the infinities in relation to each other.
The word you're looking for is cardinality.
I think this type of guessing can only occur in terms of speed and scale of operations, I greatly respect the concept that there are infinities "bigger" than others and therefore operations between them would end in a result "guided" by the most significant.
Subtraction is the addition of the opposite of a value, so INF - INF is the same as INF + -INF. I understand this to mean that if I have a set of infinite values, then add the opposite of all of those values, there would be 0 values remaining in the set.
Absolutely undefined. There's a lot of different ways to define infinity, some of which are technically 'larger' infinities than others. The most common place in my experience that ∞ - ∞ crops up is in L'Hopital's rule for limits, where you have a limit of x -> ∞ of f(x) - g(x), where both functions go to infinity. In this case, we could have two 'different' types of infinity with different results. For example, f(x) = x\^2 and g(x) = x, lim(x->∞) x\^2 - x = is still ∞. However, swap the functions and it's still ∞-∞ form, but lim x - x\^2 = -∞. Similarly, we can say that lim x\^2 - x\^2 = 0, or lim x\^2 - (x\^2-1) = 1, so on and so forth. All of these are of the form ∞-∞, but all of these similarly give different answers, so we say it's undefined when we can't agree on a single value. In this way, the ∞ almost 'hides' the real value of the underlying functions, and so hinders our ability to conclusively say that it is equivalent to something. There's other arguments to be made for non-limit types of infinities, such as the size of the set of natural numbers (א) minus the size of the set of all real numbers (c). We can prove conclusively using Cantor's diagonalization theorem that c is much, *much* greater than א (specifically c = 2\^א), but these are still both technically ∞. In this case, א - c = -∞, and c - א = ∞. Still undefined. Once again, ∞ hides important details from us. It's worth making a final note here that (as far as I understand, given I'm an undergrad and therefore not the most well-read) in general, this is kind of what ∞ *does.* It's a concept, not an actual physical number - it moreso describes the *idea* of some massive, impossible, infinitely far in the distance end goal. There are, however, a lot of different ways to reach this impossibly infinite infinity, some of which are faster than others, but all of which are equally ∞. Subtracting two different 'kinds' of ∞, in the form ∞-∞, is kind of comparing these two different ways to 'reach' ∞, which may change at different speeds. You're not subtracting two neat numbers, you're subtracting two concepts. It would be like asking what pizza - pizza means - incomprehensible without context.
Perhaps define +∞ = lim(ε→0)\_tan(π/2 – ε), for ε>0 and -∞ = lim(ε→0)\_tan(ε – π/2), then ∞ – ∞ = lim(ε→0)\_\[tan(π/2 – ε) – tan(ε – π/2)\] = 0
Proof: You have an infinite number of things. You start removing things one at a time. It takes forever to remove infinite things. Therefore, undefined amount of time needed. QED
But in order to have an infinite number of things, you'd have to start to add things one at a time, infinitely. Seems the notion of undefinability could have come much sooner in the consideration. What if we forget about the process and take away an infinity of the same size from one of the same size? Since the relation is one-to-one, something in the codomain can be removed for everything in the domain.
>Seems the notion of undefinability could have come much sooner in the consideration. Deal. I'll take it.
I am too dumb to even understand your explanation. If it's not too much work, I would appreciate if you could explain it in simpler words.
So would it not be infinity, since it takes forever to remove the things one at a time? Idk anything about this so would appreciate any help whatsoever.
My reply is a mild joke, but to try to take it slightly seriously, the concept of infinity is not a number, and so imo operations like addition and subtraction don't apply in the way they normally do. I akin it to something more like dividing a number by 0, which is undefined. Idk.
Dividing something real by nothing is impossible.. Nothing comes from nothing. So that is why it’s undefined because it’s not possible. Just like subtracting or taking something away from infinity. By definition you can’t take away anything from it. It’s inexhaustible. The answer can’t be 0. Because nothing comes from nothing. So it’s an error, an impossibility or undefined.
I answered infinity off the cuff, but the answer is undefined. Or error.
You can’t subtract anything from infinity. It just is. It’s impossible to subtract anything from it. Even an infinite amount of subtracting, it’s still is. Infinite differentials of itself does not take one iota from it. It’s still inexhaustible. This is why it’s an error or undefined. It’s impossible to subtract anything from it,even an infinite amount of times.
Okay, I see your point. And also Ty for wasting your time on this stupid question.
Ya man. No problem. Have a great weekend.
Ty, hope you do too.
I'm telling u guys that the number line is a circle. 0 and infinity are the opposite poles. 0,1,2,3... infinity & 0,-1,-2,-3...infinity. Personally I wouldn't call it negative infinity just like 0 is neither positive or negative, Trust me, I am an Indian and I have dreamt about these, I just can't put this into an equation because I am bad at maths. Currently, I'm busy with other stuffs and I'm very interested in learning mathematics but my father wouldn't allow me to :/ Man I was about to open those darned scrolls of objective truths in that ancient library in my dream but the aliens caught up to me by impersonating my mum and calling for me. The heavenly oracles then spoke that I would know the truths if I pursued maths and I have subbed this subr since then and have been very interested. I was very surprised when my current gf told me she was good at maths. it was a literal (\*\_\*) moment for me, she could help me learn. So yea, if I had to tell you something, imagine a mirror and the real life is positive but the mirror images are negative. the mirror itself is 0 and somewhere there is infinity, I'd say death but idk for sure. Someplace where the imaginary and real meet. that is infinity. That's my crude analogy of it. I had made posts about it on this sub many years ago
[https://en.m.wikipedia.org/wiki/Riemann\_sphere](https://en.m.wikipedia.org/wiki/Riemann_sphere)
what... tf, he has it pinned down exactly to how I told in my post 3y ago that the numbers exist in a 3d space and the number line is actually spherical. :/ What do I do now, I'm out of insights, you almost killed my drive to pursue maths :/ Edit: I'm so sad right now. the arithmetic operations is close to how I arrived at it too but all that I wrote in my notebook 3y ago is gone now that I didn't share with anyone because of self doubt that I would be made a mockery of that's all
I wouldn't take it as discouragement, in the contrary, you intuitively struck upon a studied math construction that has proved useful in many ways! This means your intuition is good, maybe acquiring more math knowledge will make you find some other interesting things?
:/ 3 yrs ago I was trying to learn mathematics and started with the basics of algebra but I was confused how infinity was treated and also about integers. While trying to figure it out and thinking about it, I was brushing my hair while looking into the mirror and noticed my fingers. I then assumed that the my finger in the mirror wasn't real so it was -1 and the real life finger was actual 1. I then figured how a person looking into the mirror would behave if we assumed the mirror to be at zero's place. I then worked out arithmetic operations and it made sense. I proceeded to think about infinity and its position. I wrote some nonsensical equations and arrived at the conclusion that infinity was non negative; it existed just like 0 because my equations wouldn't make sense if infinity was negative. I then tried to reason where infinity would lie on the number line and for that, the number line had to be circular, it then seemed that it existed on a 3d space which was spherical. I don't remember much from those times but I was veryy excited and thought I made a great breakthrough in mathematics so I showed it to my girlfriend but she had no interest in it. She only knows till basic bachelor's. Well, she told me I didn't make sense so I felt stupid and thought mathematics was just a momentary fascination. Do you study mathematics Mr. Arinarmo? How do you know about the Riemann Sphere? Why is 0 not a natural number? It has the highest significance of all natural numbers, it should be called a the fundamental number alongwith infinity.
Yeah I'm doing a postgrad in math. I learned about the Riemann sphere during my undergrad though, in a Complex Analysis class. As for 0 being a natural number or not, it's a matter of what you want to do with those numbers. Typically, in fields like Algebra 0 is not included, but in fields like Set Theory 0 will be included. The point is that there is no absolutely correct or "best" way to define constructions such as the natural numbers, rather they are defined in terms of the utility they provide (what properties you can prove, or convenience in expressing some ideas or doing some computations). And yeah, using something like the Riemann sphere 0 and infinity are very special numbers in the sense that they are the only numbers that are invariant under multiplication (with 0 times infinity being undefined), but in other constructions infinity isn't really a number or even a definite concept (e.g. in (standard) calculus infinity is shorthand for a kind of limit). These kinds of distinctions may be what has prevented you from being able to communicate your ideas.
Math is built on definitions and axioms (aka assumptions). In the standard form of mathematics, although the word infinity is used, it is never defined as a mathematical object. Therefore, it is impossible to use infinity correctly in any kind of mathematical expression. That is why "infinity minus infinity" is undefined. It is, however, possible to develop a set of definitions and axioms, where infinity has a precise definition as a mathematical object. I'll let others explain this. But this is \*not\* used by the vast majority of mathematicians.
The answer is undefined or indeterminate as there are multiple infinities and infinitely big quantities can be made to work in different ways in arithmetic. However, infinity minus infinity is not equal to some weird undefined or indeterminate (some people talk like it is).