Can complex numbers (imaginary numbers) be classified as "prime" or "composite"?

Answer 1

Just to add to the previous answer... again the answer is yes.

Here are some concrete examples / definitions.

First of all, let's only consider number of the form a + bi where a and b are integers. These are the "Gaussian integers". We call a nonzero integer prime if it is only divisible by itself and one (and the negatives of these two numbers). So, for instance, 5 is only (evenly) divisible by 5, 1, -5, and -1.

But now let's enlarge our world of numbers to include the Gaussian integers. Then, we see that 5 is evenly divisible by 2 + i, because you can check that 5 = (2 + i)*(2 - i).

So, this is already pretty interesting. Now, we could ask if 2 + i is prime? In other words, can we factor 2 + i into the produce of two Gaussian integers neither of which is one or minus one. Actually, we should be more careful still. Any Gaussian integer can be factored thusly:

a + bi = i*(b -ai)

So, we should further insist that we only look for factorizations such that neither of the factors is 1, -1, i, or -i. (These special numbers are called the "units" of the "ring of Gaussian integers"; they are special in the sense that they have a multiplicative inverse.)

So, we could try to write 2 + i = (a + bi)*(c + di) = (ac - bd) + (ad + bc)i in such a way that neither a+bi nor c+di equals 1, -1, i, or -i. This already this seems like a daunting problem. (You might try to solve this as given. You'll probably need to consider several different cases.)

So how do you decide if a number like 2 + i is prime or not? Well, basically you figure this out just like you would for integers. We know 5 is prime because it is not evenly divisible by the positive integers strictly between 1 and 5:

2 divided into 5? We see that 5 = 2*2 + 1 (remainder 1)
3 divided into 5? 5 = 3*1 + 2
4 divided into 5? 5 = 4*1 + 1

To decide if 2 + i is prime we need an ordering of the Gaussian integers and we need a recipe for long division. Luckily both of this problems have nice solutions. The desciption of these is somewhat involved (and you should check out the links provided in the previous post), but here's the short version:

We really only need to check if 2 + i is divisible by Gaussian integers with smaller modulus; the modulus of a + bi is defined to be sqrt(a^2 + b^2). (If you visualize the Gaussian integers as the integer lattice in the plane, then you can see the possible set of divisors as those integer lattice points lying inside the circle with radius sqrt(modulus(2 + i)) centered at the origin.

Long division is a little trickier. When you ask what is 64 divided by 7, you see that 7 goes into 64 nine times with remainder 1:

64 = 7*9 + 1.

How did we know that it goes in 9 times and not 8 or 10? Well,
64 - 7*8 = 64 - 56 is greater than or equal to 7; and
64 - 7*10 is negative. Using our idea of modulus, we can rank a remainder as being to big or too small. The details are somewhat involved, but you might try, for instance, trying to figure out how to divide 8 - 9i by 2 + 3i so that the remainder has modulus less that that of 2 + 3i.

Answer 2

I would think that the idea of prime and composite can be applied to imaginary numbers. If you can't factor anything out of an imaginary number, I would say it's prime such as:
3i or 4i +7 as these numbers only have factors of 1 and the imaginary number.

On the other hand, a composite imaginary number would look like
4i + 6 which is the same as 2(2i + 3), which obviously has a factor of 2 in it, therefore making it composite.

Perhaps this is more about an imaginary number being simplified than being prime or composite...?

Answer 3

dukebdevi... is right, all the other answers I saw are wrong, or limited. The concept of prime and composite numbers is relevant only to integers. Even if a complex number x+iy has x, y chosen from the integers, that doesn't make it an integer.

Believe dukebdevi...!!

NvestR332... raises an interesting issue: Could we innovate by defining this concept for complex numbers? I don't know whether he's right in saying 4 + 7i has only itself and 1 as factors:

(a+bi)(c+di) = (ac-bd) + (ad+bc)i
and although I haven't time to try it now, I'd be surprised if there aren't integer solutions for
ac-bd = 4
ad+bc=7

Answer 4

prime numbers are defined as having exactly two integer factors, 1 and the number itself, since the imaginary number is not an integer, it can't be prime

it also can't be composite since a number of the form a + bi can't be divided by an integer and result in an integer answer

Answer 5

What Obama is saying is that, if he gets angry enough, he will send Assad a very angry letter with strong wording. He doesn't want to have to do that, as that would take him away from his golf.

Answer 6

This is actually a fascinating subject, and the answer is "yes". See links given. It's odd that it's not a much discussed subject, but it does have a long history.

Answer 7

From expeirence, I don't think you can.
Because a complex number can be negative and possitive, it is very confusing and most teachers / instructors do not teach that. It could be done, but it is highly unrecommended. I

Answer 8

sure, they're still numbers