The answer to that question didn’t change, what changed is how you might interpret the question.
If I asked “what are the REAL roots of x² + 2x + 2” the answer is still “none”. And prior to imaginary numbers being widely used, that is how the question would have been understood.
Mathematics involves making choices about what set of rules we’re working with. If you don’t allow the concept of negative numbers, the equation “x+1=0” has no solution. If you give me an apple, then I have no apples, how many apples did I have before? The question describes an impossible situation, and that’s a perfectly valid way to view it.
Different sets of rules can change what’s possible but don’t invalidate conclusions based on other sets of rules. We just need to specify what set of rules we’re working with.
My entire point is that before they weren’t saying “real” versus “imaginary”. You’re proving my point. In the other fields mentioned you could make the same argument about the interpretation changing but the book still being useful.
The other fields are attempting to describe reality. While Newtonian physics is useful, as an approximation, it’s also quite clearly wrong. You can imagine a universe which follows those rules but it’s not this universe, and that’s why it’s wrong. Mathematics doesn’t care about this universe, so you can pick whatever rules you want. Imaginary numbers are not “more accurate”, they don’t invalidate any previous understanding. They are an imaginary concept with interesting properties. For mathematics, that’s enough.
Imaginary numbers are not “more accurate”, they don’t invalidate any previous understanding. They are an imaginary concept with interesting properties. For mathematics, that’s enough.
No. Imaginary numbers have the worst name. Like the Schrodinger’s Cat thought experiment it was something meant to mock the concept originally but stuck once real applications were found. Imaginary and complex numbers describe very real processes in nature and are not just some weird artifact of trying to get the square root of a negative number.
Here is an interesting video on the topic that also covers some of the applications used to describe things in nature. https://youtu.be/cUzklzVXJwo
I agree, because really all numbers are imaginary. Numbers are also wonderfully useful for describing nature, and it’s amazing how what might start as a quest for completeness and elegance ends up reflecting something about the real world. Each extension on our use of numbers is an augmentation, an extended toolkit to solve different problems, but doesn’t negate anything which went earlier. For example finding the roots of a polynomial often represents a problem where complex solutions aren’t applicable, and “no solution” is the more meaningful result. One kind of mathematics may be bigger and more complete than another, but that doesn’t make it better or more true. It just depends on what you need from it.
You could make the same argument for things like mathematics before the discovery about imaginary numbers.
Ehh imaginary numbers added to the scope of mathematics it didn’t take away anything other than no’s.
“hey look, i got your no’s!”
No, it changed things like “how many roots does x² + 2x + 2 have” from “none” to “two”.
The answer to that question didn’t change, what changed is how you might interpret the question.
If I asked “what are the REAL roots of x² + 2x + 2” the answer is still “none”. And prior to imaginary numbers being widely used, that is how the question would have been understood.
Mathematics involves making choices about what set of rules we’re working with. If you don’t allow the concept of negative numbers, the equation “x+1=0” has no solution. If you give me an apple, then I have no apples, how many apples did I have before? The question describes an impossible situation, and that’s a perfectly valid way to view it.
Different sets of rules can change what’s possible but don’t invalidate conclusions based on other sets of rules. We just need to specify what set of rules we’re working with.
My entire point is that before they weren’t saying “real” versus “imaginary”. You’re proving my point. In the other fields mentioned you could make the same argument about the interpretation changing but the book still being useful.
The other fields are attempting to describe reality. While Newtonian physics is useful, as an approximation, it’s also quite clearly wrong. You can imagine a universe which follows those rules but it’s not this universe, and that’s why it’s wrong. Mathematics doesn’t care about this universe, so you can pick whatever rules you want. Imaginary numbers are not “more accurate”, they don’t invalidate any previous understanding. They are an imaginary concept with interesting properties. For mathematics, that’s enough.
No. Imaginary numbers have the worst name. Like the Schrodinger’s Cat thought experiment it was something meant to mock the concept originally but stuck once real applications were found. Imaginary and complex numbers describe very real processes in nature and are not just some weird artifact of trying to get the square root of a negative number.
Here is an interesting video on the topic that also covers some of the applications used to describe things in nature. https://youtu.be/cUzklzVXJwo
If you prefer text here is an article listing some. https://www.geeksforgeeks.org/maths/applications-of-imaginary-numbers-in-real-life/
I agree, because really all numbers are imaginary. Numbers are also wonderfully useful for describing nature, and it’s amazing how what might start as a quest for completeness and elegance ends up reflecting something about the real world. Each extension on our use of numbers is an augmentation, an extended toolkit to solve different problems, but doesn’t negate anything which went earlier. For example finding the roots of a polynomial often represents a problem where complex solutions aren’t applicable, and “no solution” is the more meaningful result. One kind of mathematics may be bigger and more complete than another, but that doesn’t make it better or more true. It just depends on what you need from it.