Why Aren't There Negative Prime Numbers?
Date: 12/10/1999 at 00:12:32 From: Edgar Lopez Subject: Prime Numbers Can negative numbers be prime numbers? In my math book it says that a prime number is a whole number greater than 1 that has only two factors, 1 and itself. So why can't, for example, (-5) be prime? Its factors would be -5 and 1. I don't get it. Is it because 5 and (-1) are also factors of (-5)? I say no because (-1) and (-5) would then be factors of 5, and 5 wouldn't be prime. I don't understand and neither does my teacher. We tried to figure it out, and then she told me to ask you.
Date: 12/10/1999 at 19:24:14 From: Doctor Ian Subject: Re: Prime Numbers Hi Edgar, This is an excellent question, but in order to understand the answer, you have to get a feel for what mathematics is about. It's not about cranking numbers through equations and getting answers, and it's not about balancing your checkbook or figuring out how long it will take Bill and Janet to mow the lawn if they work together, and it's not about building bridges or telephones or sending spaceships to other planets. Those are all _uses_ of math, but mathematics itself is about searching for patterns. The most interesting patterns are the ones that hold for the largest classes of numbers. So something that is true for all numbers is more interesting than something that is true for just integers, or just prime numbers, or just numbers smaller than 10, or just the number 17. (Note that 'patterns' are sometimes called 'theorems,' and sometimes called 'properties'. For example, the prime number theorem, which says that any composite number can be broken into a product of primes, is one kind of pattern. The commutative property of addition, which says that you can add numbers in any order, is another kind of pattern.) Before anyone figured out the need for negative numbers, mathematicians had already discovered lots of patterns involving prime numbers. So when negative numbers came along, making some of them prime would have caused a lot of patterns (patterns that looked like "For any prime number, blah blah blah") to stop being true. That would have been annoying without serving any real purpose, and so the definition of primes was adjusted to exclude negative numbers. This, by the way, is why zero and one aren't prime numbers either, although you can make some good arguments for why they should be. If they were considered prime, then a lot of patterns that now look like "For any prime number, blah blah blah" would have to be changed to "For any prime number except one or zero, blah blah blah." That makes the patterns uglier, and harder to remember. ("Oh, wait - does this pattern for prime numbers also apply to zero?") Mathematics turns out to have so many practical uses that it's easy to forget that the mathematicians who are creating it, in many cases, really don't care whether the patterns they are finding have any more uses than a song, or a painting. I hope this helps. Thanks for asking an interesting question. Be sure to write back if you have others. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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