I "invent" it in the same kind of way that I "invent" that a certain cloud has the shape of a rabbit. Your "invention", on the other hand, is like inventing a new animal just so that it will fit the cloud. But it's not only the invention that matters - it is where it starts and what are its foundations. Your "invention" starts at the end and its foundations are hidden. You keep drawing those general analogies between things as if they mean anything, but they are so general that the only thing they do is hide the difference, instead of exposing the similarities.
By the same kind of reasoning we can justify an extremely large amount of bad expository articles on mathematics. There are clear conceptual differences between making people "see" multiplication as scaling, then asking a question about the possibility of extending it, out of which i arises naturally and it's clear to everybody how it fits in and in what sense i^2=-1 despite the seeming contradiction with everything we know about numbers and multiplication up to that point, and between your way of inventing this "number" because "if you think about it long enough, there are ways to make it work without contradictions". People don't understand it just as they don't understand what is the square root of an apple: how can you even take the square root of a fruit without extending the definition of what it means to take a square root? We are not dealing with mathematicians who have experience in abstract algebra where multiplication has this abstract meaning and we can just add new stuff to our set and see if it works. We are talking about people who's experience with multiplication and numbers comes from calculating the acceleration of a ball and counting coins. It shouldn't come as a surprise that one approach to teaching something works better than the other: Tim Gowers presents normal subgroups and quotient groups in the much better way [1] than the traditional approach, even though the same very broad analogies can be drawn between the two. VCA presents certain ideas about complex analysis in a much more intuitive and clear way than the traditional approach, even though all it does is place a much higher emphasis on geometry.
Why don't you grab a bunch of students with little prior knowledge of complex numbers, present them the complex numbers using those two ways, then ask them what seems more obvious and natural.
I don't care about what seems natural to you. I care about what produces better results for the majority of people. Your way of explaining complex numbers produces bad results.
You can either accept this, or you can keep ignoring it because you know better. There is no shortage of smart mathematicians with the wrong ideas about how to teach mathematics, and they all have plenty of excuses.
Comments
I "invent" it in the same kind of way that I "invent" that a certain cloud has the shape of a rabbit. Your "invention", on the other hand, is like inventing a new animal just so that it will fit the cloud. But it's not only the invention that matters - it is where it starts and what are its foundations. Your "invention" starts at the end and its foundations are hidden. You keep drawing those general analogies between things as if they mean anything, but they are so general that the only thing they do is hide the difference, instead of exposing the similarities.
By the same kind of reasoning we can justify an extremely large amount of bad expository articles on mathematics. There are clear conceptual differences between making people "see" multiplication as scaling, then asking a question about the possibility of extending it, out of which i arises naturally and it's clear to everybody how it fits in and in what sense i^2=-1 despite the seeming contradiction with everything we know about numbers and multiplication up to that point, and between your way of inventing this "number" because "if you think about it long enough, there are ways to make it work without contradictions". People don't understand it just as they don't understand what is the square root of an apple: how can you even take the square root of a fruit without extending the definition of what it means to take a square root? We are not dealing with mathematicians who have experience in abstract algebra where multiplication has this abstract meaning and we can just add new stuff to our set and see if it works. We are talking about people who's experience with multiplication and numbers comes from calculating the acceleration of a ball and counting coins. It shouldn't come as a surprise that one approach to teaching something works better than the other: Tim Gowers presents normal subgroups and quotient groups in the much better way [1] than the traditional approach, even though the same very broad analogies can be drawn between the two. VCA presents certain ideas about complex analysis in a much more intuitive and clear way than the traditional approach, even though all it does is place a much higher emphasis on geometry.
Why don't you grab a bunch of students with little prior knowledge of complex numbers, present them the complex numbers using those two ways, then ask them what seems more obvious and natural.
I don't care about what seems natural to you. I care about what produces better results for the majority of people. Your way of explaining complex numbers produces bad results. You can either accept this, or you can keep ignoring it because you know better. There is no shortage of smart mathematicians with the wrong ideas about how to teach mathematics, and they all have plenty of excuses.
[1] - http://gowers.wordpress.com/2011/11/20/normal-subgroups-and-...