What are imaginary numbers?? Why can’t the square root of a negative be real?
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Suppose you had a real number, whose square is -1. Let us call this number i.
Now you have 3 options:
– i=0. Nope, then i*i=0 is not -1.
– i is positive, but then i*i is also positive, so not -1.
– i is negative, but then i*i is positive, also does not work.
Okay, so that does not work and we need something more than real numbers.
One idea is for any real number x, we can identify it with a matrix ((x,0), (0,x)), which just *scales* the plane with a factor x. Then multiplication, addition etc. for matrices just work. Additionally, we can look at the matrix that *rotates* everything by 90 degrees: ((0, -1), (1,0)). Now this does what we want, because rotating by 90 degrees twice gives you 180 degrees, which is the same as ((-1,0),(0,-1)).
So we now found a generalization (called extension) of the real numbers as matrices ((x,-y),(y,x)). For historical reasons, we call these “complex numbers” and write them as x+iy. That is it, nothing magical or “imaginary” about them, just matrices.
So why do we say “imaginary” and all that? Historically these came from trying to solve higher order polynomial equations like x^3 + a x +b =0. If a,b are “nice” you can write down a formula involving roots (of positive numbers) to find the solutions. However, these surprisingly sometimes also worked even if the roots supposedly did not make sense in calculations in between, e.g. calculations like 3 + sqrt{-2} -sqrt{-2}=3. People considered that rather strange or nonsensical at the time, but it worked, so they said “fine, we will use it since it works, but call it ‘imaginary’ because it seems weird”.
Complex numbers stayed controversial among mathematicians for some time, but people got used to them and nowadays they seem quite normal. However, the names stuck.
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