We didn't invent 'i' to "solve sqrt(-1)". This is an extremely common misconception about maths and how it progressed that unfortunately people get led into believing by lazy teachers every day
So what did happen?
Why don't we define “imaginary” numbers for every “impossibility”? (2012)
21–30 of 98 posts
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#22Earlier quoted context omitted.
Those are all just normal imaginary numbers. The question is why, when we can't answer a question, we don't just invent a symbol, say it's the answer to the question, and call it a day. It's a stupid question, but it's not related to your response.
The question has 300+ upvotes. That’s a proxy for how “good” it is. A person is curious about an aspect of mathematics and posed a well stated question. It is not a stupid question. From their perspective mathematicians appear to do something and they wonder why it can’t be done in other situations. Such a question is the basis of understanding. It is by wondering such things that enables one to gain true understandi…
> Peoples’ curiosity ought not be labeled as stupid.
Correct. That is why I feel more comfortable asking "stupid" questions to chatGPT. I clarified a lot of concepts in economics through repeatedly asking questions about each concept that pop up in its answers and trying to push it to the limits of what can be defined, explained, etc. One cannot be sure of the truthfulness or soundness of the answers, but they may help.
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#23Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…
Riemann Sphere:
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#24Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…
The Isabelle/HOL theorem prover assigns 0 to x/0 for all x, without contradiction.
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#25Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…
The Isabelle/HOL theorem prover assigns 0 to x/0 for all x, without contradiction.
I believe you but that’s kind of mind blowing. How do they avoid the seemingly-obvious corollary that 0*0 = X, for all values of X? That is, just multiplying both sides of “x/0 = 0” by zero.
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#26Earlier quoted context omitted.
"it becomes possible to "prove" that any number is equal to any other number." There are multiple ways to define what division by zero means. Which definition leads to this outcome? How?
Division by zero is not defined anywhere on math. The closest thing you'd get to it is to 1. define a limit (lim x->a of ƒ(x) exists if and only if given any ε > 0 there exists a δ > 0 such that ...)[1]. 2. chose a function ƒ(x) such that on a given "a", ƒ(a) = ƒ(a)/0. 3. prove that the limit exists and is finite. Now if we defined division by zero it would look like this: Axiom: For every element x of the real numbe…
"_____ is not defined anywhere in math"
is a kind of sentence that is almost never true, and even if it were, it would be impossible to prove that someone hadn't jotted a valid definition on a napkin somewhere. In this case it is certainly not true (as others have mentioned: https://en.wikipedia.org/wiki/Riemann_sphere ). Now, specifying a definition for division by zero does require you to be careful about how the other operations extend to this new number, but there are perfectly consistent (and useful!) ways to do so.
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#27We didn't invent 'i' to "solve sqrt(-1)". This is an extremely common misconception about maths and how it progressed that unfortunately people get led into believing by lazy teachers every day
So what did happen?
Square roots of negative numbers came up when solving cubic equations, even if the final solutions were all real. This meant the square root of a negative number was not something nonsensical the way you might claim for x^2 = -1, but actually...real in some sense.
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#28For polynomial equations, the construction works in quite some generality, and is known as quotient ring: https://en.wikipedia.org/wiki/Quotient_ring Given any polynomial P (e.g. x^2 + 1) over a filed F (e.g. reals) we can form: `R = F[X]/P` This is an algebraic "set" that supports addition, substraction, multiplication and has 0,1 but not division in general. Elements are elements of F and a new symbol X that satisf…
Nit: I think everywhere you write x^2-1 you actually meant x^2+1.
yes!
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#29For polynomial equations, the construction works in quite some generality, and is known as quotient ring: https://en.wikipedia.org/wiki/Quotient_ring Given any polynomial P (e.g. x^2 + 1) over a filed F (e.g. reals) we can form: `R = F[X]/P` This is an algebraic "set" that supports addition, substraction, multiplication and has 0,1 but not division in general. Elements are elements of F and a new symbol X that satisf…
> If the polynomial P is invertible, i.e. has degree 1 Should be degree 0: only constant polynomials are invertible. E.g. x+1 is not invertible, and modding it out doesn't result in the zero ring. The example is a bit confusing, because $x=x+1$ is equivalent to $0=1$, which has degree 0.
Yes! Fixing
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#30Earlier quoted context omitted.
The Isabelle/HOL theorem prover assigns 0 to x/0 for all x, without contradiction.
I believe you but that’s kind of mind blowing. How do they avoid the seemingly-obvious corollary that 0*0 = X, for all values of X? That is, just multiplying both sides of “x/0 = 0” by zero.
By specifying that x/y*y is only equal to x if y≠0, I guess?