What are imaginary numbers?
Can someone please also do this for:1.  Matrices, especially matrix multiplication.  Unlike matrix addition, multiplication is defined in a very weird way.  I think I understand where it is coming from -- defining it that way allows representing and solving linear equations.  More insights, however, would help.2.  Dot and cross products.  E.g., the magnitude of dot product in 3D is a.b.cos(theta), while for cross product, it is a.b.sin(theta).  I never got sure what happens to other combinations like a new "vector" product whose magnitude is a.b."cos"(theta)?3.  How is a set and "belongs-to" operator defined?  Most books I have come across just assume these (and later define natural numbers and addition from them).4.  Why is 0.9999... considered to be "equal" to 1.  I understand them to be equal "under the limit", but not without.  This seems to be in my way of understanding Cantor's infinities.