The discussion centers around the concept of divisibility as it applies to imaginary and complex numbers, exploring whether divisibility is a valid concept outside of integers. Participants examine examples involving imaginary numbers and rational numbers, questioning the definitions and usefulness of divisibility in various mathematical contexts.
Participants express differing views on the applicability and usefulness of divisibility for imaginary and complex numbers. There is no consensus on whether divisibility is a valid concept in these contexts, and the discussion remains unresolved.
The discussion highlights the ambiguity surrounding the definition of divisibility, particularly in relation to non-integer numbers, and the varying interpretations of what it means for one number to be divisible by another.
Yes, "m is divisible by n" means there exist x such that m= nx and so "0 is divisible by 0" because 0= 0*0 (with x= 0). I just want to make it clear to others that you are not saying 0 can be divided by 0.Hurkyl said:The imaginary numbers aren't closed under multiplication: I assume you meant the complex numbers.
Divisibility makes sense in any algebraic structure with a multiplication operation... but it's not very useful when most things are invertible.
e.g. in the complex numbers, every number is divisible by every nonzero number. (and zero is divisible by zero)
The same is true for the real numbers and the rational numbers: every rational number is divisible by every nonzero rational number (and zero is divisible by zero).
Just to emphasize that -- in the rational numbers, 3 is divisible by 2.
It's in the integers that 3 is not divisible by 2. Divisibility is useful for the integers because it has very few invertible elements (among its other nice properties). Another useful ring is the Gaussian integers Z: the set of all complex numbers of the form m+ni where m and n are integers. Divisibility is useful there too, and it has pretty much all of the nice properties that one wants out of it.
glass.shards said:Thanks for the reply, Hurkyl!
Yes, what I meant was "evenly" divisible in terms of integers. Despite it not being "useful" to consider divisibility for non-integers, I am curious if it is actually valid at all to say that 5i is 'evenly' divisible by 5.
Is divisibility defined only for integers, or can it be applied to the above example? I think what confuses me most is the 'true definition' of division. Any further insight would be much appreciated!