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It made sense to worry that carbon dioxide did influence temperature.But by 2003 new data came in and it was clear that . Temperatures appear to control carbon, and while it’s possible that carbon also influences temperature these ice cores don’t show much evidence of that.Okay, so maybe imaginary and complex numbers make sense to introduce and lead to a reasonable theory, but how could they possibly be useful?After all, in the real world we have real numbers of things (e.g.Imaginary numbers, like real numbers, are simply ideas without any physical existence.They are both very useful (though with real numbers, it is much more obvious why that is so).However, as was pointed out, this hinges very much on your definition of “exist”.Although the idea of numbers is very natural (you might almost say “obvious”), and probably would have been invented by almost any highly intelligent beings that may have happened to inhabit our planet, they are indeed still simply just an idea that we created.
One way to think about this is that Taylor Series provide a method for locally approximating a function using polynomials, and the region in which such approximation succeeds depends in a crucial way on the function’s behavior when it’s thought of a function of complex rather than real numbers.But, if we are feeling creative, we might assume for a moment that there is a (special) number that satisfies this equation, (we’ll call it the number ), and then see if we can derive its properties.By definition, of course, our number satisfies , but we can ask other questions about this bizarre number, such as “what is for any positive integer n?Mathematician: Imaginary numbers arise quite naturally when you start asking certain basic mathematical questions.Probably the best example is the following: Once we know how to multiply and add, we might ask ourselves “are there any numbers x that satisfy the equation ? If we try some more equations of this form, we might at some point ask the similar question, are there any numbers x that satisfy the equation ?