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1 year agoI’d imagine you want something defined recursively like multiplication
- ( 0x = 0 )
- ( xy = x(y-1)+ x ) ( y > 0 ).
So it needs to be
- ( x^0 = c ) (c is some constant)
- ( x^y = xx^{y-1} ) (( y > 0 ) (to see why, replace multiplication with exponentiation and addition with multiplication). So what could ( c ) be? Well, the recursive exponentiation definition we want refers to ( x^0 ) in ( x^1 ). ( x^1 ) must be ( x ) by the thing we wish to capture in the formalism (multiplication repeated a single time). So the proposed formalism has ( x = x^1 = xx^0 = xc ). So ( cx = x ) hence ( c = 1 ), the multiplicative identity. Anything else would leave exponentiation to a zeroth power undefined, require a special case for a zeroth power and make the base definition that of ( x^1 ), or violate the intuition that exponentiation is repeated multiplication.
On an unrelated note, it’d be nice if Lemmy had Mathjax. I just wrote all this on mobile with that assumption, and I’m not rewriting now that I know better.
I completely agree with you about motivation in isolation. I’ve been doing a bit more this past week, but I need to keep pushing myself to stay focused on the same project after a day or so. It was easier in college because I had more external motivation. I did have the idea recently that I could learn a bit of graphics and get a bit more motivation out of what I code. I’ll probably stick with that for a couple of months because it is a fairly versatile skill to know how to tell the GPU to do things. Additionally, thank you for letting me know about the Out in Tech group. It sounds like it would be helpful.