>>13685900>Isn't that just a different way of writing the exact same thing?no.
the value of a square root is a number that when multiplied by itself gives the value under the square root. but this does not uniquely identify a number because multiple numbers can fit this definition. if you take i := sqrt(-1), then i * i = -1, but since -i * -i = -1 there is an ambiguity in the original definition for i; "do i take +i or -i to be the imaginary unit?". defining i^2 := -1 bakes this choice into the definition.
also, it's easy to fuck up using i = sqrt(-1) because some common manipulations that work on the reals lead to errors on complex quantities. for instance
i * i = sqrt(-1) * sqrt(-1) = sqrt(-1 * -1) = sqrt(1) = 1
which doesn't make sense
>>13685907>antisymmetric mapssuppose you want a multilinear function that maps the edges of an n-dimensional parallelpiped to a number that represent its volume. we want it to be multilinear so when we scale an edge, the volume gets scaled by the same amount. moreover, if any two edges are the same, then the volume should be 0 because the structure would be flat. the two conditions that the function is
1) multilinear
2) 0 when any two edges are equal
defines the class of antisymmetric functions, which get their name because of the property that when any two edges are exchanged the sign flips.
moreover, if you require that
3) the function maps the standard basis to
you've uniquely defined the determinate, which is a "signed volume" or "oriented volume". the determinate is useful because it appears in the analytic formula for a matrix inverse via Cramer's rule.
read this link to fill in the blanks:
https://sites.millersville.edu/bikenaga/linear-algebra/det-unique/det-unique.htmlwhy is this cool? this allows you to understand the properties of determinates and how to compute them from first principles, which is hard to do by brute force. also, these antisymmetric maps are key to generalizing calculus to n-dimensions
