What are imaginary numbers?
Can someone please also do this for:<p>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.<p>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)?<p>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).<p>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.