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I personally find Serge Lang's "Basic Mathematics" to be super useful in explaining the "why" of a particular formula in pre-college level math, as most books

by lsc 9y ago

I personally find Serge Lang's "Basic Mathematics" to be super useful in explaining the "why" of a particular formula in pre-college level math, as most books at my current level just seem to say "memorize this" He has an explanation of why x^0 is 1 and not 0 that doesn't involve calculus, for instance.

yorwba 9y ago
What would be an explanation that involves calculus? The only one I know is that x^(a+b) = x^a * x^b holds for positive a and b, and to uphold that pattern, it's necessary that x^0 = 1, x^-1 = 1/x and so on. For me, all of that was covered long before calculus.
- sethammons 9y ago
  I actually referred to a different pattern to help knowing what x^0 is above. Not from calc, but just a different way than yours. If you start with x^3, x^2, x^1, x^0, x^-1, x^-2, and expand the forms, you get xxx, xx, x, ... where the pattern is each step divides by x. So x/x = 1 is the next, and 1/x is the next, and 1/xx is the next, ...
- lsc 9y ago
  Interesting. When I took high school math as a kid, there weren't many explanations, (and for some reason, I wasn't up for looking them up/figuring them out myself- weird, 'cause I totally was up for that sort of thing in some other classes) - just x^(a+b) = x^a * x^b and x^0=1 -- memorize these facts, with the emphasis on memorization rather than on the connection between those facts. Most of what I remember is a lot of guess and check and factorization; something I automated using the rom basic on my Tandy model 100. These were the mid '90s, and looking back with a less resistant mind, I think they really were trying to get us to develop a mathematical intuition, and that if I had read the math text book the way I read the history or English lit textbooks (they were full of stories!) I would have done just fine. But I still can't read math books the way I read literature, and I haven't figured out why yet. I know my first decade of reading, I skipped a lot, like if I didn't understand something, I moved on and filled out the blanks from what I did understand as best I could. (These days when I'm reading literature I get all excited when I see a word or phrase I don't understand, and I immediately stop and research. (the world is awesome for that today. Once I was blocked on my way home, waiting for a family of skunks to move off of the trail. I was introduced to rimbaud by miller, so I bought and downloaded a bunch of his stuff to my kindle. By the time I was done reading, it was dark and the skunks started to approach me, so I took off (without incident) but I wish I could do that with math, and I'm not quite sure why I cant. I mean, reading things I half understand, and halfway getting something out of it.) the calculus explanation I heard from a friend that I was talking about this about was extrapolating from x^.5 x^.4, etc, etc, and as the exponent approaches 0 the answer approaches 1 for all non zero values of x - but this was maybe two years back and the friend in question was educated overseas, and I still am not educated enough to use calculus notation to write that down (or even really to understand if I understood him correctly.) so it could be garbled, or only tradition on the continent, or something like that.

by lsc 9y ago

yorwba 9y ago

What would be an explanation that involves calculus? The only one I know is that x^(a+b) = x^a * x^b holds for positive a and b, and to uphold that pattern, it's necessary that x^0 = 1, x^-1 = 1/x and so on. For me, all of that was covered long before calculus.

sethammons 9y ago
I actually referred to a different pattern to help knowing what x^0 is above. Not from calc, but just a different way than yours. If you start with x^3, x^2, x^1, x^0, x^-1, x^-2, and expand the forms, you get xxx, xx, x, ... where the pattern is each step divides by x. So x/x = 1 is the next, and 1/x is the next, and 1/xx is the next, ...
lsc 9y ago
Interesting. When I took high school math as a kid, there weren't many explanations, (and for some reason, I wasn't up for looking them up/figuring them out myself- weird, 'cause I totally was up for that sort of thing in some other classes) - just x^(a+b) = x^a * x^b and x^0=1 -- memorize these facts, with the emphasis on memorization rather than on the connection between those facts. Most of what I remember is a lot of guess and check and factorization; something I automated using the rom basic on my Tandy model 100. These were the mid '90s, and looking back with a less resistant mind, I think they really were trying to get us to develop a mathematical intuition, and that if I had read the math text book the way I read the history or English lit textbooks (they were full of stories!) I would have done just fine. But I still can't read math books the way I read literature, and I haven't figured out why yet. I know my first decade of reading, I skipped a lot, like if I didn't understand something, I moved on and filled out the blanks from what I did understand as best I could. (These days when I'm reading literature I get all excited when I see a word or phrase I don't understand, and I immediately stop and research. (the world is awesome for that today. Once I was blocked on my way home, waiting for a family of skunks to move off of the trail. I was introduced to rimbaud by miller, so I bought and downloaded a bunch of his stuff to my kindle. By the time I was done reading, it was dark and the skunks started to approach me, so I took off (without incident) but I wish I could do that with math, and I'm not quite sure why I cant. I mean, reading things I half understand, and halfway getting something out of it.) the calculus explanation I heard from a friend that I was talking about this about was extrapolating from x^.5 x^.4, etc, etc, and as the exponent approaches 0 the answer approaches 1 for all non zero values of x - but this was maybe two years back and the friend in question was educated overseas, and I still am not educated enough to use calculus notation to write that down (or even really to understand if I understood him correctly.) so it could be garbled, or only tradition on the continent, or something like that.