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Rock_InhabitantWhen you divide by zero, the conventional answer is undefined. But I think this is lazy and lacking closure so I need some opinions. On one hand, the smaller the fraction you divide by, the bigger the quotient is. So as the divisor approaches zero, the quotient approaches infinity. So by this definition, 0/0=infinity. But on the other hand, if someone asks you to divide a bunch of apples by zero, you could use logic and just say the quotient is 0, because the number of apples you get after dividing the bunch by zero is zero. Visualize dividing an apple into zero pieces. There are no pieces, so it must be zero. Out of these two theories I can’t decide which one makes the most sense. I can’t attach a poll cuz mobile but I want to break away from the social propaganda of “undefined”
Rock_InhabitantWhen you divide by zero, the conventional answer is undefined. But I think this is lazy and lacking closure so I need some opinions. On one hand, the smaller the fraction you divide by, the bigger the quotient is. So as the divisor approaches zero, the quotient approaches infinity. So by this definition, 0/0=infinity. But on the other hand, if someone asks you to divide a bunch of apples by zero, you could use logic and just say the quotient is 0, because the number of apples you get after dividing the bunch by zero is zero. Visualize dividing an apple into zero pieces. There are no pieces, so it must be zero. Out of these two theories I can’t decide which one makes the most sense. I can’t attach a poll cuz mobile but I want to break away from the social propaganda of “undefined”
SendyMcSendyfaceyou need to eat some mushrooms big dawg
DirtYStylEI thought you said this question is important
since you substituted 0 with an apple. An apple is 1 apple so it’s not 0. It’s just a locgial flaw in your argument
Rock_InhabitantI lowk want to but not before I’m 25ish so I don’t mess up my brain
SteezyYeeterWhen you ask if my momma is fat, the conventional answer is yes. But I think this is understated and lacking sophistication so I need some opinions. On one hand, the smaller the portion left on the plate, the bigger she is. So as the amount of food on the plate approaches zero, her size approaches infinity. So by this definition, she is fat. But on the other hand, if someone gave her a million apples, you could use logic and say she is fat, because the number of apples you get after giving them to her is going to be zero. Visualize her eating a million apples. There are no apples left, so she must be fat. Out of these two theories I can’t decide which one makes the most sense. I can’t attach a poll cuz mobile but I want to break away from the social propaganda of “yo momma is fat”
SlushSeasonDude there's like a million proofs of this from actual mathematicians
But I think this is lazy and lacking closure so I need some opinions.
So as the divisor approaches zero, the quotient approaches infinity.
Out of these two theories I can’t decide which one makes the most sense.
ArabianYou're right, this is lazy, and people are not good at explaining it because people are, for lack of a better word, uninformed.
This is what we call an "analytical" answer. In fact, one needs to define several things before we reach for this particular hammer:
1. Division: well-defined division is usually defined to be an operation definitive of a https://en.wikipedia.org/wiki/Division_ring or https://en.wikipedia.org/wiki/Field_(mathematics) for which the term `n/0` is actually undefined because division is only defined for non-zero elements of the field or division ring. It can be constructed syntactically, sure, but the term is ill-typed if there's a zero in the denominator, so it should be rejected outright. Let's suppose it's defined because the mathematician in particular here doesn't give a shit about any type theory or the laws they're building upon.
2. Approaching: by "approaches", surely you mean, "tends toward", which refers to a sequence approaching a bound, which are constructed using an Axiom of Dependent (Countable) Choice (DCC), which tells you that every countable, bounded set has a unique supremum (a leaset upper bound) which either does or does not belong to the set. In this setting, one constructs a sequence of numbers (say, s_n = {1/n}_(n in N)), which enumerates a set {1, 1/2, 1/3,...} that has unique greatest lower bound (a consequence of the DCC) that is contained in the sequence r_n = {1/n}_(n in R, n>=1). Now, usually, at this point, any Real Analysis text book will also point out that we need something called the Archimedian Property (https://en.wikipedia.org/wiki/Archimedean_property) to give you the necessary fact that in order to have a unique greatest lower bound, the term s_m for some m must always exists as a lower bound for each decreasing subsequence of r_n, and then you know that r_n tends to 0 because s_n tends to 0. Likewise, on the interval (0,1), one needs to make an Archimedean argument to show r_n does the opposite cuz it's bounded by every natural.
3. Absolute Convergence: In the case bove, you can say the unique upper bound is "the" limit for the sequence because the sequence because the sequence converges absolutely. That is, for every r_n, and arbitrary distance from 0, there is some point in the sequence that puts in within the interval (0, s_N], and since this works for at every point in the sequence of r_n's, you know that 0 is a unique limit point. This is called "absolute convergence". (in contrast to just "convergence. Take the sequence {1, -1, 1, -1...} alternating infinitely, and the subsequence of odd indices converges to 1 and the subsequence of evens to -1, but together they do *not* converge. Multiple limit points exist!).
4. Limits: "tends toward" is a sly way of saying "absolutely converges to a limit", which is what we're doing here.
This kind of intuitively makes sense by the 1st point above, because it's just... never possible to divide by zero. You fucked up, it doesn't work. You can only talk about fractions that are epsilon-close to a division by zero, and talk about a limit *at* zero (which doesn't exist in the set of solutions - limits need only adhere closely to the working sequence), but you can't actually talk about the limit.
People who fuck this up are fucking it up because it's complicated and we elide all sorts of stuff like that in our education. Like, why does 0! = 1? Because ! is shorthand for "permutations of", which is combinatorially the number of non-repeating arrangements of a collection of stuff in a particular finite indexing scheme. Of which there is only one way to permute nothing: just give me back nothing (a vacuously true thing to do). Or, those stupid meme math "questions" like "what's 6 / 5(1 + 2)", which are not actual math terms because they're syntactically invalid. The 5(1 + 2) is invalid syntax and it needs to be parenthesized to be either 6 / (5 * (1 + 2)) or (6/5)*(1 + 2), and the question should be rejected outright until it's made unambiguous! It's just stupid. There's all sorts of type-theoretic, syntactic, and semantic bullshit in the curriculum.
**This post was edited on Oct 31st 2024 at 1:52:56am
ArabianYou're right, this is lazy, and people are not good at explaining it because people are, for lack of a better word, uninformed.
This is what we call an "analytical" answer. In fact, one needs to define several things before we reach for this particular hammer:
1. Division: well-defined division is usually defined to be an operation definitive of a https://en.wikipedia.org/wiki/Division_ring or https://en.wikipedia.org/wiki/Field_(mathematics) for which the term `n/0` is actually undefined because division is only defined for non-zero elements of the field or division ring. It can be constructed syntactically, sure, but the term is ill-typed if there's a zero in the denominator, so it should be rejected outright. Let's suppose it's defined because the mathematician in particular here doesn't give a shit about any type theory or the laws they're building upon.
2. Approaching: by "approaches", surely you mean, "tends toward", which refers to a sequence approaching a bound, which are constructed using an Axiom of Dependent (Countable) Choice (DCC), which tells you that every countable, bounded set has a unique supremum (a leaset upper bound) which either does or does not belong to the set. In this setting, one constructs a sequence of numbers (say, s_n = {1/n}_(n in N)), which enumerates a set {1, 1/2, 1/3,...} that has unique greatest lower bound (a consequence of the DCC) that is contained in the sequence r_n = {1/n}_(n in R, n>=1). Now, usually, at this point, any Real Analysis text book will also point out that we need something called the Archimedian Property (https://en.wikipedia.org/wiki/Archimedean_property) to give you the necessary fact that in order to have a unique greatest lower bound, the term s_m for some m must always exists as a lower bound for each decreasing subsequence of r_n, and then you know that r_n tends to 0 because s_n tends to 0. Likewise, on the interval (0,1), one needs to make an Archimedean argument to show r_n does the opposite cuz it's bounded by every natural.
3. Absolute Convergence: In the case bove, you can say the unique upper bound is "the" limit for the sequence because the sequence because the sequence converges absolutely. That is, for every r_n, and arbitrary distance from 0, there is some point in the sequence that puts in within the interval (0, s_N], and since this works for at every point in the sequence of r_n's, you know that 0 is a unique limit point. This is called "absolute convergence". (in contrast to just "convergence. Take the sequence {1, -1, 1, -1...} alternating infinitely, and the subsequence of odd indices converges to 1 and the subsequence of evens to -1, but together they do *not* converge. Multiple limit points exist!).
4. Limits: "tends toward" is a sly way of saying "absolutely converges to a limit", which is what we're doing here.
This kind of intuitively makes sense by the 1st point above, because it's just... never possible to divide by zero. You fucked up, it doesn't work. You can only talk about fractions that are epsilon-close to a division by zero, and talk about a limit *at* zero (which doesn't exist in the set of solutions - limits need only adhere closely to the working sequence), but you can't actually talk about the limit.
People who fuck this up are fucking it up because it's complicated and we elide all sorts of stuff like that in our education. Like, why does 0! = 1? Because ! is shorthand for "permutations of", which is combinatorially the number of non-repeating arrangements of a collection of stuff in a particular finite indexing scheme. Of which there is only one way to permute nothing: just give me back nothing (a vacuously true thing to do). Or, those stupid meme math "questions" like "what's 6 / 5(1 + 2)", which are not actual math terms because they're syntactically invalid. The 5(1 + 2) is invalid syntax and it needs to be parenthesized to be either 6 / (5 * (1 + 2)) or (6/5)*(1 + 2), and the question should be rejected outright until it's made unambiguous! It's just stupid. There's all sorts of type-theoretic, syntactic, and semantic bullshit in the curriculum.
**This post was edited on Oct 31st 2024 at 1:52:56am
KungKalmarThis is some weird crackhead mathematics that no one will understand. You can't just come like that and destroy his bright ideas. This is how the world punishes new independent thinkers and intellectuals.
