can a negative number be prime? how about an aboslute number, can it be prime?

Answer 1

>>>Here's to answer your first question:

Answer One: No.

By the usual definition of prime for integers, negative integers can not be prime.

By this definition, primes are integers greater than one with no positive divisors besides one and itself. Negative numbers are excluded. In fact, they are given no thought.
Answer Two: Yes.

Now suppose we want to bring in the negative numbers: then -a divides b when every a does, so we treat them as essentially the same divisor. This happens because -1 divides 1, which in turn divides everything.

Numbers that divide one are called units. Two numbers a and b for which a is a unit times b are called associates. So the divisors a and -a of b above are associates.

In the same way, -3 and 3 are associates, and in a sense represent the same prime.

So yes, negative integers can be prime (when viewed this way). In fact the integer -p is prime whenever p, but since they are associates, we really do not have any new primes. Let's illustrate this with another example.

The Gaussian integers are the complex numbers a+bi where a and b are both integers. (Here i is the square root of -1). There are four units (integers that divide one) in this number system: 1, -1, i, and -i. So each prime has four associates.

It is possible to create a system in which each primes has infinitely many associates.
Answer Three: It doesn't matter

In more general number fields the confusion above disappears. That is because most of these fields are not principal ideal domains and primes then are represented by ideals, not individual elements. Looked at this way (-3), the set of all multiples of -3, is the same ideal as (3), the set of multiples of 3.

-3 and 3 then generate exactly the same prime ideal.

>>>And here's to answer your second question:

Did you mean: can the absolute value of a number be prime?

In this case, I don't think it really matters since absolute values just show the magnitude of something whether small, big, or in-between in size.

But since technically the absolute value of a number is positive, I guess the absolute value of a number could be prime.

Answer 2

Yes, a negative can be prime (in Z, the integers), because -1 is a unit in Z.

Of course, some definitions are in order here. In any ring (or semi-ring)--see
http://en.wikipedia.org/wiki/Ring_%28mathematics%29
and
http://en.wikipedia.org/wiki/Semi-ring
respectively--a "unit" is a number with a multiplicative inverse. For example, in N (the natural numbers), only 1 is a unit. In Z, the units are 1 and -1. In Q (the rationals) and R (the reals), EVERY non-0 number is a unit. (0 is never a unit.)

A prime element (number) in a ring is any number a with the following property: if a divides bc, and neither b nor c is a unit, then a divides b or a divides c. However, in a Unique Factorization Domain (which all these rings we're talking about are), primes are the same as irreducibles--elements which cannot be expressed as products of two non-primes, which is the definition of "prime" we usually think about.

So, with all that machinery, in fact it is the case that in ANY ring with a prime p, -p is also a prime because there's always a -1 in a ring, and -1 is always a unit.

That's a pretty technical answer, but there ya go!
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As to the other... I don't know what you mean by an "absolute number," nor can I find a definition--and I know where to look!

Answer 3

By definition, a prime number is a *natural number* which has exactly two distinct natural number divisors: 1 and itself.

Natural numbers are the set of *positive integers* {1, 2, 3, ...}

Negative numbers aren't in the set of positive integers, so they can't be prime.

However, an absolute number (assuming it is an integer and not zero) can result in a prime number.

| -2 | = 2 which is prime
| -3 | = 3 which is prime
| -5 | = 5 which is prime
etc.

Answer 4

Negatives can't be prime

Answer 5

No, because a prime number can only be divided by one and iteself. A negative number can also be divided by -1.

Answer 6

sorry no clue