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Imaginary numbers are math cope for when you're too cool to just use two numbers.
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I never got why they didn't just introduce tuples in maths
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They did, linear algebra and vector calculus are a thing, but complex numbers have certain properties that you don’t get with vectors and that are quite useful and worth studying.
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One definition of the complex numbers is the set of tuples (x, y) in R^(2) with the operations of addition: (a,b) + (c,d) = (a+c, b+d) and multiplication: (a,b) * (c,d) = (ac - bd, ad + bc). Then defining i := (0,1) and identifying (x, 0) with the real number x, we can write (a,b) = a + bi.
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Ok, that's actually quite interesting
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Yup, you'll notice the only thing distinguishing C from R^(2) is that multiplication. That one definition has extremely broad implications.
For fun, another definition is in terms of 2x2 matrices with real entries. The identity matrix
1 0 0 1is identified with the real number 1, and the matrix
0 1 -1 0is identified with i. Given this setup, the normal definitions of matrix addition and multiplication define the complex numbers.
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