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!
Imaginary numbers are numbers that can be written in the form of a+bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. These numbers are used to represent solutions to equations that do not have real solutions.
Yes, imaginary numbers can be divided just like real numbers. However, division by 0 is undefined for both real and imaginary numbers.
Yes, divisibility applies to imaginary numbers just like it does to real numbers. A number is divisible by another number if the remainder is 0 when the first number is divided by the second. This concept applies to both real and imaginary numbers.
The divisibility rule for imaginary numbers is the same as the rule for real numbers. For example, a number is divisible by 2 if the last digit is even, and a number is divisible by 3 if the sum of its digits is divisible by 3.
Yes, there are a few unique properties of divisibility for imaginary numbers. For example, if a+bi is divisible by c+di, then the ratio between a and c will be the same as the ratio between b and d. Additionally, if two imaginary numbers are divisible by the same number, then their sum and difference will also be divisible by that number.