Is i = √(-1) prime or composite... or else?

Answer 1

You would have to indicate in what field (e.g., complex integers of the form m + ni where m and n are real integers).

Whatever the field, a number is a prime if it can not be divided by another number (except itself, its additive inverse or a unit).

A unit is a number that has a multiplicative inverse in the field. In real integers, 1 and -1 are "units". They are not considered primes.

The multiplicative inverse of 1 is 1 (because 1*1=1) and 1 belongs to the integers, therefore 1 is a unit.

In the complex integers, the inverse of i is -i (because i * -i = 1), and -i is a member of the complex integers (because it can be written as 0 + (-1)i ).

Therefore, i is a unit. It cannot be a prime (just like 1 is not a prime in real integers).

---

2 is not a prime in the complex integers. It can be written as (1+i)(1-i), neither of which is a unit.

(1+i) is a prime because it is not divisible by any complex integer m + ni, other than itself (and its additive inverse) or a unit.


---

An additive inverse is simply 'minus' the number.

In real integers, -7 is the additive inverse of 7.
7 is prime even though it is divisible by -7; however that division yields a unit as the only other possible factor:

7 = (-7)*(-1)

That is why we had to drop the old definition of "not divisible by numbers other than itself or 1", because -7 is not 7 and -1 is not 1.

Answer 2

Speaking technically, there are two concepts called 'prime' and 'irreducible' which are distinct, but I will not make this distinction here.

I am not going to be too technical in my discussion that follows. If you are interested in a technical discussion, I suggest that you read a book on abstract algebra.

Let me first review what it means to be prime in the integers.

In general, what it means for a number p to be prime is two things:
(1) It is ``indivisible enough'' that it cannot be written as the product of smaller primes.
(2) It is ``large enough'' that, as a factor, it counts significantly when multiplied by other things

In the integers, 1 is not prime because it fails (2). You can factor 7 as 1 x 7; you can factor it as 1 x 1 x 7; you can factor it as 1 x 1 x 1 x 7; and so on. The number 1 is somehow not substantial enough to actually do anything if we take it as a factor. (The number 1 is what is technically referred to as a `unit.' It turns out that 1 is not composite, either, because it cannot be written as the product of primes.)

2 is prime because it cannot be written as the product of smaller primes. Same with 3, 5, 7, and so on.

12 is not prime because we can split it up into 2 x 2 x 3, the product of smaller primes. So 12 is composite.

Now, let me move to the Gaussian integers--that is, the set of all numbers of the form a + bi, where a and b are integers. (Such numbers include 2 + 5i, 8 - 7i, -3 + 2i, i, 3i, 2, 0, and so on.)

Within the Gaussian integers, it turns out that i is not prime, nor is it composite. Recall that i^4 = 1. So we can write

2 = 2 x 1
2 = 2 x 1 x i x i x i x i
2 = 2 x 1 x i x i x i x i x i x i x i x i

See what's happening? The number i, like 1, is somehow ``insubstantial'' and doesn't add any substance when we use it as a factor. So i is a unit.

It turns out that 1, -1, i and -i are the only units in the Gaussian integers.

There are, however, some Gaussian integers which are prime. The number 1 + i is an example. You can't write 1 + i as the product of ``smaller'' primes. (Here ``smaller'' means ``having a smaller magnitude,'' where ``magnitude'' is defined as the complex absolute value, or modulus.)

Some numbers which are prime as integers are no longer prime in the Gaussian integers. For example,

(1 + 2i)(1 - 2i) = 5.

So 5 is not prime in the Gaussian integers.

Answer 3

The imaginary number is in the same pool as 0 and 1. neither prime nor composite. Traditional Prime numbers are all Real Integers, thus excluding imaginary numbers. Composite numbers are generally also discussed in the domain of Real Integers.

Answer 4

Only natural numbers greater than 1 can be prime or composite.

Answer 5

It depends on how you look at it.
These terms prime and composite etc are subject to definitions. For example, 1 is not a prime number. Why? Just because.

In the complex plane, i = 0 + 1*i
So you could say the coefficient of the imaginary part is 1 hence it is not prime.

But i can also be taken as exp(i*n*pi) where n = odd

So i = exp*(i*n*pi/3) * exp(i*n*2*pi/3)

I say it is not prime.

Answer 6

Only natural numbers are prime or composite.

Answer 7

In the complex plane exist the so called Gaussian Primes and the units are 1,-1,i and -i so even in this case sqrt(-1) is not prime

Answer 8

i is a complex number.