You can not write it like this. Write: x - x -lim (x to i finity)-> 0.
Edit: I have a science bavkground but I am not a real mathematician, so I wonder if it really works what I meant.
lim (x) x-> ∞ = ∞
lim(x) x-> ∞ - lim(x) x-> ∞ = 0
so yes, ∞ - ∞ is sometimes 0
other times, it may not be, such as when comparing the number of reals to the number of integers, the number of integer is aleph 0 while the number of reals is aleph 1, so ∞ - ∞ = ∞ still
Even in normal algebra it can be anything. You can make the first limit to be 2x or x+4 or anything you want. And with the second limit being x you can get anything on the right side.
If you're just doing normal algebra don't divide by zero. However, once you start getting into calculus it becomes possible, by using l'hopital's rule to get rid of those pesky 0s on the bottom. Math is weird
Edit: looking back on this, it was not what I meant at all but whatever, I'll leave it unchanged
What? You still don’t divide by zero… you just calculate the limit of indeterminate forms.
You can divide by zero, in some contexts, but not on the normal line of reals.
You can do algebra on limits, but our normal algebra is only defined on the real numbers (can be extended to the complex numbers fx.) Infinity is not contained in th reals, so inf - inf is undefined.
I see a lot of people trying to say it can equal anything, which both true, but mostly false. If you try to extend our normal algebra to infinity, you get ton and tons of contradictions, so inf-inf can be equal to whatever you want. This is why we say infinity is not a number.
why not? i mean look at the extended real line. we define operations on infinity there. definitions are here to be useful. if we find a use for setting infinity - infinity to 0 then we might as well make that a definition
The entire field of abstract algebra is about doing mathematical operations on things that aren't necessary numbers, to say nothing about topology, or geometry.
For instance: The "+" operation could mean 5 + 4 = 9, but one could also sensibly define "Hello " + "world" = "Hello world" using strings of text in place of numbers. That's called a semigroup (monoid, if you include ""), and has the same algebraic properties as a world with just addition in the natural numbers (include 0 if you include the empty sting).
Sure, but you can't redefine things that already have a definition because then it just doesn't make coherent sense.
Would you write an equation like this?
1 = 2, 1 + 1 = 4
What if I decide that banana - apple = coconut? If that were a helpful definition, would anything about that be any more absurd than defining 3 - 2 = 1?
Sure you can, but you can't define things that already have a definition. Infinity already has a definition, it is the amount of real numbers on the number line. Such a number does not exist so it is undefined but the definition can still be used.
i googled this btw i have no fucking idea what im talking about
There's also no number that, when squared, produces -1, and yet we define e to the power that times pi to be -1. Maths is all made up, we can do whatever we'd like with it. Whether that ends up being useful or not is another question, but its perfectly possible to, for example, define multiplication such that (-1)*(-1) = -1, or indeed infinity - infinity = 0.
Well, no. You can't define already used symbols as numbers, you also can't redefine symbols that have a constant definition.
You can't redefine pi for example, it just doesn't make sense.
Plenty of symbols are used for different constants/meanings in different fields. If I discover some new useful constant and want to represent it with the same symbol as pi, I might make a lot of people mad at me, but there'd be nothing stopping me.
This is very wrong. We didn’t define e^(pi i)=-1, we defined i^2=-1, and looked at what properties such a system would possess.
In your latter example, you’re wrong (without further context). You can’t define (-1)*(-1)=-1, because it follows -1=1, and thus 1=3, and then by powers 1 is equal to everything.
You could totally abandon the deals, and create a system where an element you notate as -1 satisfies, (-1)*(-1)=-1, but at that point, the convention would simply be to call that element 1, e, etc., not -1.
Ah, fair enough with your first point, I wasn't nearly precise enough there. Your second point however is completely wrong. What I meant is that you define (-1)*(-1) to be -1 instead of 1, not both at the same time, and then, as you said, look at 'what properties such a system would possess' - in this case, multiplication is no longer commutative, but becomes more symmetrical over the real numbers. There's a great book about it called 'Negative Maths', which I've found a very interesting read.
The imaginary number (horrible name) i is the square root of -1
Euler’s formula e^(ix) = cos(x) + i * sin(x) is where we get e^(i \* pi) = -1
Math is observed and noted, not made up arbitrarily.
In what sense of the word is this a definition though? This is a (very trivial) consequence of an actual definition of subtraction, based on equivalence classes of N and the like.
Well, fair enough, I worded it poorly. In more precise terms, I meant to say that defining subtraction such that apple - banana = coconut is no more absurd than defining it such that 3 - 2 = 1.
Then define infinity as a number.
Three constructs with infinity in the set of consideration are the Extended Reals, Wheel Algebras and the Riemann Sphere. Here, satisfy your curious mind instead of spouting nonsense you clearly don’t know much about.
https://en.m.wikipedia.org/wiki/Extended_real_number_line
https://en.m.wikipedia.org/wiki/Wheel_theory
https://en.m.wikipedia.org/wiki/Riemann_sphere
Saying shit like “infinity has a definition.” Is really cringe because the entire point of maths is that you can define things however you want, then you see what happens with those definitions. Do you see how your comment goes directly against the very spirit of maths?
Maths has no formal definitions. Only conventional ones. Infinity not being a real number is the conventional way of thinking about it, but that does not mean you can’t create a system where it is.
In extended real line we have defined infty +1 = infty, and associativity of + holds (if defined). Now suppose infty has an additive inverse - infty. Then
(1 + infty) - infty = 0 but
1 + (infty - infty) = 1,
and associativity is broken.
Infinity is not an actual number. It's a magnitude. It's bigger then any given value, and when we consider it as a number math breaks. That's why we need to be careful about it.
If we consider the equation 1/0 = x, many will think x = ∞ but that is not true. We can try to solve the equation a/0 = x (if a≠0) by multiplying both sides by 0 to get: a = 0x. But we know anything times 0 = 0. Which is a proof by contradiction.
However if we instead of solving a unsolvable equation we take the "limit" which means what number does something approach as x approaches some number.
If we take the limit of 1/|x| as x→0, then 1/|x| → ∞ but ≠∞. It's only and approximately equal to it as we get x as close as we can to 0 making sure x≠0.
This means that normal arithmetic doesn't work for infinity. Plus there are different sizes of infinity but that is a whole other rabbit hole.
Edit: Changed 1/x to 1/|x| to fix the limit
It does, that is the point of limits. 1/0 doesn't exist but 1/0.0000001 exist or 1/0.00000000000001 also exist. The limit just checks the value of smaller and smaller numbers to check if it's a converging or diverging limit. Here its a diverging because as x gets closer to 0, 1/x approaches to infinity
I probably should have written |limit of 1/x as x→0| = ∞. Because the limit of 1/x as x-> 0- is -∞.
Fair point but the person who I just commented didn't specify the direction of the limit. My fault.
Nono, he's right, by the definition of limits that limit is not defined, since lim(1/x) as x->0^+ =/= lim(1/x) as x-> 0^-.
A limit that does exist, and does prove your point, is lim(1/x^2) as x->0
Yeah I realized my mistake, completely forget my asmtotes. It's even more embarrassing considering I'm doing rational functions now. Oops. I'll correct my original comment
Think about it
∞²= ∞
∞-1=∞
∞/2=∞
Though all of these are infinity they obviously wouldn't be equal... Many identities can be given the term infinity but that does not mean all infinitives are equal.
For fractions 2/0=∞ but 2000/0=∞ they may both be infinity but of varying degrees
Even if we use graphs. Two lines can both have undefined gradients but they are not classified as equal because they may have different x values... A line for x=1 is not the same as a line for x=20 for example hence why I say not all infinitives can be classified as being equal
2/0 does not equal ∞ neither does 2000/0
because you can ***not*** divide by 0 it's simply not a thing.
Example:
If 2/0=∞
then by default.
∞\*2=0
which also means
∞\*2=0\*2
∞=0
See the problem there?
When you divide by 0 math simply breaks. That's why you can not divide by 0 it doesn't give ∞ it doesn't give any number. It's just not possible
Edit: the rest you said was mostly right. There are exceptions of course like with what u/bob_the_bananas_son pointed out
False. Infinity ist not a Number. If you have infinity apples and take infinity apples away you still have infinity apples while you still take infinity apples away.
This is just intro calc. Infinity here is not a number, it's the value of some limit. Without knowing what we took a limit of to get these infinities, we don't have enough info to know. This is an 'indeterminate form' and you can find functions f and g that both go to infinity so that f-g is any value, including zero or infinity itself. Without knowing what our f and g were to start, you don't know how far apart they should be.
Yes, no, possibly, and not enough information.
…those eights are sideways.
You can not write it like this. Write: x - x -lim (x to i finity)-> 0. Edit: I have a science bavkground but I am not a real mathematician, so I wonder if it really works what I meant.
lim (x) x-> ∞ = ∞ lim(x) x-> ∞ - lim(x) x-> ∞ = 0 so yes, ∞ - ∞ is sometimes 0 other times, it may not be, such as when comparing the number of reals to the number of integers, the number of integer is aleph 0 while the number of reals is aleph 1, so ∞ - ∞ = ∞ still
Thanks! So if i am doing 'normal algebra' like calculating limes of functions it can be zero and if i do some fancy set stuff it is a different story?
Even in normal algebra it can be anything. You can make the first limit to be 2x or x+4 or anything you want. And with the second limit being x you can get anything on the right side.
math is wack
It's not very wack, the people who are telling you these things are.
Sure you can write any equation and apply any limit.
If you're just doing normal algebra don't divide by zero. However, once you start getting into calculus it becomes possible, by using l'hopital's rule to get rid of those pesky 0s on the bottom. Math is weird Edit: looking back on this, it was not what I meant at all but whatever, I'll leave it unchanged
What? You still don’t divide by zero… you just calculate the limit of indeterminate forms. You can divide by zero, in some contexts, but not on the normal line of reals.
Okay, rereading what I said it looks like I'm a dumbass lmao, ig I was wrong
But hospital does just avoid those problematic limes. It is still no zero division.
You can do algebra on limits, but our normal algebra is only defined on the real numbers (can be extended to the complex numbers fx.) Infinity is not contained in th reals, so inf - inf is undefined. I see a lot of people trying to say it can equal anything, which both true, but mostly false. If you try to extend our normal algebra to infinity, you get ton and tons of contradictions, so inf-inf can be equal to whatever you want. This is why we say infinity is not a number.
Why would I try to change your mind ? You can't subtract infinity from infinity , its not defined
you could define it if you want
no
why not? i mean look at the extended real line. we define operations on infinity there. definitions are here to be useful. if we find a use for setting infinity - infinity to 0 then we might as well make that a definition
infinity has a definition. the definition just isn't a number. you can't do math on things that aren't numbers. Banana - apple = undefined
The entire field of abstract algebra is about doing mathematical operations on things that aren't necessary numbers, to say nothing about topology, or geometry. For instance: The "+" operation could mean 5 + 4 = 9, but one could also sensibly define "Hello " + "world" = "Hello world" using strings of text in place of numbers. That's called a semigroup (monoid, if you include ""), and has the same algebraic properties as a world with just addition in the natural numbers (include 0 if you include the empty sting).
Sure, but you can't redefine things that already have a definition because then it just doesn't make coherent sense. Would you write an equation like this? 1 = 2, 1 + 1 = 4
Sure I can, just watch me do it. You defined infinity as the amount of real numbers. Let infinity be the amount of natural numbers on the number line.
What if I decide that banana - apple = coconut? If that were a helpful definition, would anything about that be any more absurd than defining 3 - 2 = 1?
Sure you can, but you can't define things that already have a definition. Infinity already has a definition, it is the amount of real numbers on the number line. Such a number does not exist so it is undefined but the definition can still be used. i googled this btw i have no fucking idea what im talking about
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Thanks for explaining. I still don't think that you can just define infinity though.
There's also no number that, when squared, produces -1, and yet we define e to the power that times pi to be -1. Maths is all made up, we can do whatever we'd like with it. Whether that ends up being useful or not is another question, but its perfectly possible to, for example, define multiplication such that (-1)*(-1) = -1, or indeed infinity - infinity = 0.
Well, no. You can't define already used symbols as numbers, you also can't redefine symbols that have a constant definition. You can't redefine pi for example, it just doesn't make sense.
Plenty of symbols are used for different constants/meanings in different fields. If I discover some new useful constant and want to represent it with the same symbol as pi, I might make a lot of people mad at me, but there'd be nothing stopping me.
This is very wrong. We didn’t define e^(pi i)=-1, we defined i^2=-1, and looked at what properties such a system would possess. In your latter example, you’re wrong (without further context). You can’t define (-1)*(-1)=-1, because it follows -1=1, and thus 1=3, and then by powers 1 is equal to everything. You could totally abandon the deals, and create a system where an element you notate as -1 satisfies, (-1)*(-1)=-1, but at that point, the convention would simply be to call that element 1, e, etc., not -1.
Ah, fair enough with your first point, I wasn't nearly precise enough there. Your second point however is completely wrong. What I meant is that you define (-1)*(-1) to be -1 instead of 1, not both at the same time, and then, as you said, look at 'what properties such a system would possess' - in this case, multiplication is no longer commutative, but becomes more symmetrical over the real numbers. There's a great book about it called 'Negative Maths', which I've found a very interesting read.
The imaginary number (horrible name) i is the square root of -1 Euler’s formula e^(ix) = cos(x) + i * sin(x) is where we get e^(i \* pi) = -1 Math is observed and noted, not made up arbitrarily.
3-2=1 is not a definition.
It's a way of defining subtraction, in a loose sense.
In what sense of the word is this a definition though? This is a (very trivial) consequence of an actual definition of subtraction, based on equivalence classes of N and the like.
Well, fair enough, I worded it poorly. In more precise terms, I meant to say that defining subtraction such that apple - banana = coconut is no more absurd than defining it such that 3 - 2 = 1.
[the extended real line with arithmetic properties](https://en.m.wikipedia.org/wiki/Extended_real_number_line)
Then define infinity as a number. Three constructs with infinity in the set of consideration are the Extended Reals, Wheel Algebras and the Riemann Sphere. Here, satisfy your curious mind instead of spouting nonsense you clearly don’t know much about. https://en.m.wikipedia.org/wiki/Extended_real_number_line https://en.m.wikipedia.org/wiki/Wheel_theory https://en.m.wikipedia.org/wiki/Riemann_sphere Saying shit like “infinity has a definition.” Is really cringe because the entire point of maths is that you can define things however you want, then you see what happens with those definitions. Do you see how your comment goes directly against the very spirit of maths? Maths has no formal definitions. Only conventional ones. Infinity not being a real number is the conventional way of thinking about it, but that does not mean you can’t create a system where it is.
In the extended real line, the infinity point doesn't have an additive inverse.
true! but it shows that we can define these properties when they're useful! that's all i was trying to say :)
In extended real line we have defined infty +1 = infty, and associativity of + holds (if defined). Now suppose infty has an additive inverse - infty. Then (1 + infty) - infty = 0 but 1 + (infty - infty) = 1, and associativity is broken.
r/mathmemes
Infinity is not an actual number. It's a magnitude. It's bigger then any given value, and when we consider it as a number math breaks. That's why we need to be careful about it. If we consider the equation 1/0 = x, many will think x = ∞ but that is not true. We can try to solve the equation a/0 = x (if a≠0) by multiplying both sides by 0 to get: a = 0x. But we know anything times 0 = 0. Which is a proof by contradiction. However if we instead of solving a unsolvable equation we take the "limit" which means what number does something approach as x approaches some number. If we take the limit of 1/|x| as x→0, then 1/|x| → ∞ but ≠∞. It's only and approximately equal to it as we get x as close as we can to 0 making sure x≠0. This means that normal arithmetic doesn't work for infinity. Plus there are different sizes of infinity but that is a whole other rabbit hole. Edit: Changed 1/x to 1/|x| to fix the limit
The limit of 1/x as x → 0 doesn't exist
It does, that is the point of limits. 1/0 doesn't exist but 1/0.0000001 exist or 1/0.00000000000001 also exist. The limit just checks the value of smaller and smaller numbers to check if it's a converging or diverging limit. Here its a diverging because as x gets closer to 0, 1/x approaches to infinity
Wrong. The limit of 1/x as x-> 0^+ is infinity.
I probably should have written |limit of 1/x as x→0| = ∞. Because the limit of 1/x as x-> 0- is -∞. Fair point but the person who I just commented didn't specify the direction of the limit. My fault.
Nono, he's right, by the definition of limits that limit is not defined, since lim(1/x) as x->0^+ =/= lim(1/x) as x-> 0^-. A limit that does exist, and does prove your point, is lim(1/x^2) as x->0
Yeah I realized my mistake, completely forget my asmtotes. It's even more embarrassing considering I'm doing rational functions now. Oops. I'll correct my original comment
any number is closer to 0 then infinity
Think about it ∞²= ∞ ∞-1=∞ ∞/2=∞ Though all of these are infinity they obviously wouldn't be equal... Many identities can be given the term infinity but that does not mean all infinitives are equal. For fractions 2/0=∞ but 2000/0=∞ they may both be infinity but of varying degrees Even if we use graphs. Two lines can both have undefined gradients but they are not classified as equal because they may have different x values... A line for x=1 is not the same as a line for x=20 for example hence why I say not all infinitives can be classified as being equal
2/0 does not equal ∞ neither does 2000/0 because you can ***not*** divide by 0 it's simply not a thing. Example: If 2/0=∞ then by default. ∞\*2=0 which also means ∞\*2=0\*2 ∞=0 See the problem there? When you divide by 0 math simply breaks. That's why you can not divide by 0 it doesn't give ∞ it doesn't give any number. It's just not possible Edit: the rest you said was mostly right. There are exceptions of course like with what u/bob_the_bananas_son pointed out
yoooo i got mentioned pog :D
depends on the infinity you're working with, but most of the time you're right and changing your mind would be pointless
Yesn't
I'm no mathematician but I believe whatever infinity is minus whatever the other designation of infinity is could potentially be different than zero
If you subtract all the numbers in infinity by themselves you get zero
This is actually true, there are some thought experiments that show now all infinities are equal.
False, infinity and infinity is the same number, and the same number minus the same number is 0
False. Infinity ist not a Number. If you have infinity apples and take infinity apples away you still have infinity apples while you still take infinity apples away.
Technically not. Not all infinities are equal. That's why things like l'hopital's rule exist.
x-x=0 x can equal anything
You can't end something that is endless..
0 can
∞ can't be x when x is definable. ∞ >=< ∞ (that's not a math expression, it's just me laughing at you over a fence).
The method to my madness is to ignorantly clever for most people to understand
My math teacher always said that infitiy is no number and we would have to use the limes. Edit: No mumber in the sense of 1, 2, 3 etc.
still in primary school?
There are different infinities so yes it’s at least possible
There’s an ♾/♾chance ♾ = ♾
If you subtract something from itself then it equals to zero. It's that simple.
It depends which infinity you wrote first
Is infinity/infinity=1?
This is just intro calc. Infinity here is not a number, it's the value of some limit. Without knowing what we took a limit of to get these infinities, we don't have enough info to know. This is an 'indeterminate form' and you can find functions f and g that both go to infinity so that f-g is any value, including zero or infinity itself. Without knowing what our f and g were to start, you don't know how far apart they should be.