Imaginary Numbers Debate
Discussions focus on the history, poor naming, mathematical validity, and fundamental nature of imaginary and complex numbers, often comparing them to real and negative numbers while defending their importance in math and physics.
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Why aren't real and imaginary numbers both invented?
Complex/imaginary numbers are just badly named for historical reasons, they represent an objectively central concept in math and physics, and can be derived from axioms of what we expect from a well-behaved number field. For reals, we have: (A) expected properties of addition and multiplication, (B) total order and other order-related nice properties (Dedekind-complete). Any mathematical structure satisfying (A,B) will be equivalent to real numbers. Now if we extend it to get (C) algebraic
imaginary numbers to me are more real than real numbers, they are the numbers of geometry - i like them being first class numbers
"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." - Gauss
If math is not real, it should be complex then.
Every time you're tempted to freak out because complex numbers, reflect that you should also be freaking out about negative numbers. Those don't exist either.
It's analogous to the real and imaginary parts of a complex number. Does it makes sense to add a real number and a purely imaginary number? Aren't they different kinds of things? Yes, and yes!
"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." — Gauss
complex/negative numbers don't actually exist in contrast to natural numbers that don't actually exist in a different way
That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question. - Gauss