In formal definitions of arithmetics, division can be defined via multiplication: as a simplified example with real numbers, because a ÷ 2 is the same as a × 0.5, this means that if your axioms support multiplication you’ll get division out of them for free (and this’ll work for integers too, the definition is just a bit more involved.)
Mathematicians also subtract by adding, with the same logic as with division.
a/b is the unique solution x to a = bx, if a solution exists. This definition is used for integers, rationals, real and complex numbers.
Defining a/b as a * (1/b) makes sense if you’re learning arithmetic, but logically it’s more contrived as you then need to define 1/b as the unique solution x to bx = 1, if one exists, which is essentially the first definition.
if your axioms support multiplication you’ll get division out of them for free
this is true… except when it isn’t.
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist
if your axioms support multiplication you’ll get division out of them for free*
*certain terms and conditions may apply. Limited availability in some structures, North Korea, and Iran. Known to the state of California to cause cancer or reproductive toxicity
And mathematicians divide by multiplying!
In formal definitions of arithmetics, division can be defined via multiplication: as a simplified example with real numbers, because a ÷ 2 is the same as a × 0.5, this means that if your axioms support multiplication you’ll get division out of them for free (and this’ll work for integers too, the definition is just a bit more involved.)
Mathematicians also subtract by adding, with the same logic as with division.
a/b is the unique solution x to a = bx, if a solution exists. This definition is used for integers, rationals, real and complex numbers.
Defining a/b as a * (1/b) makes sense if you’re learning arithmetic, but logically it’s more contrived as you then need to define 1/b as the unique solution x to bx = 1, if one exists, which is essentially the first definition.
That’s me, a degree-holding full time computer scientist, just learning arithmetic I guess.
Bonus question: what even is subtraction? I’m 99% sure it doesn’t exist since I’ve never used it, I only ever use addition.
It’s just addition wearing a trench coat, fake beard and glasses
The example was just to illustrate the idea not to define division exactly like that
this is true… except when it isn’t.
https://en.wikipedia.org/wiki/Ring_(mathematics)
Yeah I should maybe just have written