Does divisibility apply to imaginary numbers?

[glass.shards]

For example, is 5i "divisible" by 5? Or does divisibility only apply to integers?

On that note, is 5pi divisible by 5? Is 5/6 not divisible by 5?

Thanks in advance! =)

 

[Hurkyl]

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]

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!

 

[HallsofIvy]

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)

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.

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.

 

[CRGreathouse]

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!

In the complex numbers C, 5i is divisible by 5 (but also by 7 or pi + 3i).

In the Gaussian integers Z, 5i is divisible by 5 (but not by 7).

 

[WWGD]

As a quick comment to add, look up "Euclidean Rings". These are the structures
where a lot of the ideas of number theory in Z (integers) can be extended. This
is thanks to the existence of a gauge function. Once you have a gauge function
you can define a Euclidean algorithm, divisibility, etc.