I mostly remember he was trying to figure out why 1 worked differently. He was doing some fractional exponents stuff and then some algebra, which had fractions canceling out numbers as well as cases where he’s ending up at the law of identity (which he didn’t know the name of but I had to explain best I could) when he was canceling out variables and ending up with 1… and he was annoyed at all the different purposes of 1.
He’s an arithmetic guy: throw anything at him and he’ll get it figured out in his head but not be able to say why always. But algebra is a different animal and he’s been finding it annoying, especially how it seems to change how the number 1 works.
For the canceling = 1 .(a/a=1), I went over inversion a bit and he actually informed me (I forgot) that division is not communicative nor associative so i SUSPECT his real beef is with division…
but there was his trusty 1 betraying him, being used with strange results that don’t seem to follow the same pattern as other numbers do. [what’s 5 to the 1st power? 5, so it’s as if it did nothing at all].
So, I had to go over how 1 can also mean “whole”, complete, in the same way that 100% = 1.
That got a lightbulb after a bit of explaining but I gotta say that his rant (of what I remember) was spot on.
1 is used for an awful lot of stuff that has little to do with 100% and other things where that 100% meaning is all it does.
What I mean by ending up at the law of identity is a presumption on my part. I assume that since a=a, and a/a=1 that 1 = 100%, whole, so what’s happening is that while you lose the uniqueness of the variable, you’re replacing it with the knowledge of a completeness. Man that came out better in my head
I’d love to get your thoughts and/or find more reading on this oddness about 1. It could be as simple as “different axioms for different kinds of math”, much as how “chemistry math” is far different than counting rocks or cutting up a pizza… but still, I’d love to know more stuff on this.
Numbers are agnostic, 1 especially so. I don’t think there is an intrinsic value of 1 beyond unity/wholeness. It _is_ unit. But of what? Anything. Means “it is as it is”. “It’s it”. Or just “an it”.
You can’t just have a value of 1. You need to answer: “A value of 1 _what_?”.
Most basic numerical system consists of “1”, which is really 2. You have 1 and then the absence of 1.
But the absence of 1 can be implied or not. If it’s not, the numerical system really is just “1”
It is. But the others all have a relationship with 1. 1 needs nothing but itself.
You could go that far. I was just thinking of how many combinations you could go in.
b) 1, absence of 1
c) 1, absence of 1, superimposed (1, absence of 1)
d) 1, absence of 1, superimposed (1, absence of 1), choice of (1, absence of 1)
e) d) 1, absence of 1, superimposed (1, absence of 1), choice of (1, absence of 1), neither (1 nor absence of 1)
I think you could have a system with 1 and only 1.Only relationship would be with itself. Law of identity really.
I think you’re correct. I think if any unit can be broken up into constituents then it’s not really a unit but a convenience. A “carrying case”.