[applied_heat]

I or j don’t need to originate from Sqrt(-1) ?

[anon291]

No not at all. I is just something that behaves as if it is equivalent to negative one (that is, the additive inverse of the multiplicative identity) after combining it with itself in some way. We commonly call this multiplication. If such a thing comes with another operation called addition that behaves similarly to addition and multiplication (i.e. form a ring), then they will behave like i. Geometrically, multiplication by I can be seen as a 90deg rotation of a 2d vector. Complex numbers are simply 2-d coordinates (or rather, they are isomorphic to 2-d coordinates). Nothing special really. Easy to measure with a protractor and ruler.

In general there are many algebraic rings with an element that, when multiplied by itself, produces the additive inverse of the multiplicative identity.

[gus_massa]

In math, officially i is the "root" of x^2+1=0 or to be more precise, C is R[x]/x^2+1, i.e. you take all the polynomials in x and pretend that the polynomials A and B they are equivalent when A-B is a multiple of x^2+1.

There is also a construction with matices instead of polynomials.

And perhaps others. Each of them are useful in some cases.

[anon291]

X*X + 1 = 0 is a fundamental statement on an algebraic rings behavior with the additive and multiplicative identities and the additive and multiplicative group operations. Namely, it says that the ring contains an element that when multiplied by itself is equal to the additive inverse of the multiplicative identity . Plenty of rings have such an element. You can complete any ring with such an element and call it whatever you want. The use of the term imaginary for it is incredibly unfortunate. There's nothing strange or mystical about it. It's very real. In fact the rational complex numbers are more real than the non complex real numbers

[itishappy]

Easier to see without the square root:

    i^2 = -1

What action when applied twice results in a sign change?

"A 90 turn" is one answer. There are probably others.

[kgwgk]

“A -90 turn” would be another.