Why don't we define “imaginary” numbers for every “impossibility”? (2012)
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Why don't we define “imaginary” numbers for every “impossibility”? (2012)
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Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#2we do, it's called the algebraic numbers!
every polynomial with algebraic coefficients has 'n' solutions (counted with multiplicity)!
so e.g. x^121 + sqrt(7)x^9 + fithroot(22)x^7 + (1+i)x^3 + 22/7 = 0 has 121 solutions. and they're all algebraic numbers: nothing weird like pi in there.
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#3we do, it's called the algebraic numbers! every polynomial with algebraic coefficients has 'n' solutions (counted with multiplicity)! so e.g. x^121 + sqrt(7)x^9 + fithroot(22)x^7 + (1+i)x^3 + 22/7 = 0 has 121 solutions. and they're all algebraic numbers: nothing weird like pi in there.
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.
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#4One interesting case of this is the concept of dual numbers [1], where you have the symbol \epsilon !=0 but (\epsilon)^2 = 0.
It seems contradictory, but the resulting theory is very useful for automatic differentiation [2] and for mechanics (dual quaternions) [3].
[1]: https://en.m.wikipedia.org/wiki/Dual_number
[2]: https://book.sciml.ai/notes/08-Forward-Mode_Automatic_Differ...
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#5[dead]
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#6One interesting case of this is the concept of dual numbers [1], where you have the symbol \epsilon !=0 but (\epsilon)^2 = 0. It seems contradictory, but the resulting theory is very useful for automatic differentiation [2] and for mechanics (dual quaternions) [3]. [1]: https://en.m.wikipedia.org/wiki/Dual_number [2]: https://book.sciml.ai/notes/08-Forward-Mode_Automatic_Differ... [3]: https://en.m.wikipedia.org/wiki…
If anyone’s interested, I wrote up an example application of dual numbers in Ruby: https://tomstu.art/automatic-differentiation-in-ruby
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#7we do, it's called the algebraic numbers! every polynomial with algebraic coefficients has 'n' solutions (counted with multiplicity)! so e.g. x^121 + sqrt(7)x^9 + fithroot(22)x^7 + (1+i)x^3 + 22/7 = 0 has 121 solutions. and they're all algebraic numbers: nothing weird like pi in there.
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 understanding of a topic.
Most questions asked by beginners in an area are “stupid” and few as insightful as this one. I’ve taught mathematics at a community college for 20 years and I would be delighted to have been asked this. Usually questions are mundane like, “Why did you add x to both sides?”. Here the person is trying to understand what mathematicians do, what the basis of expanding a number system really involves. This is a fantastic question.
Peoples’ curiosity ought not be labeled as stupid.
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#8Well, 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 division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting.
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#9[deleted]
Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)
#10Well, 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…
"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?