Why Study Prime and Composite Numbers?

Date: 01/25/2001 at 21:43:23
From: Kim Howell
Subject: What is the purpose of studying Prime and Composite numbers?

My daughter is in Grade 6. She is learning about prime and composite 
numbers but my husband and I wonder why this is taught in school at 
all. Who uses this in the real world? Why does someone need to know 
whether a number is a prime number or not?

Date: 01/26/2001 at 00:12:55
From: Doctor Ian
Subject: Re: What is the purpose of studying Prime and Composite 
numbers?

Hi Kim,

Every time you send a credit card number over the Internet, it gets 
encrypted by your browser, and the encryption algorithm is based on 
the theory of prime numbers. At some point, electronic money will 
become as common as paper money, and _that_ will also be based on the 
theory of prime numbers. And what's used more in the real world than 
money? 

The importance of prime numbers is that any integer can be decomposed 
into a product of primes. For example, if you want to know how many 
different pairs of numbers can be multiplied to get 360, you can start 
trying to write them down,

  1 * 360
  2 * 180
  3 * 120
  4 *  90
  5 *  72
  6 *  60

checking every single number up to 180, and hope that you don't miss 
any; or you can decompose 360 into its prime factors, 

  360 = 2 * 2 * 2 * 3 * 3 * 5

with the assurance that every factor of 360 will be a product of a 
subset of these prime factors.  

This kind of analysis is extremely convenient when working with 
fractions (since prime factorization tells you which common 
denominators are available for any two fractions), when factoring 
polynomials... when doing just about anything where integers are 
involved, really. 

Think of it this way.  You don't need to learn to multiply, since you 
can always use repeated addition to solve any multiplication problem, 
right? If you want to know what 398 times 4612 is, you can just start 
adding:

  398   (1) 
  398   (2)
 ----
  796
  398   (3)
 ----
 1194
  398   (4)
 ----
  etc.

Knowing about multiplication saves you time. That's all it does... but 
that's a lot!  

Mostly, prime numbers are good for quickly transforming a situation 
with zillions of possible outcomes into an equivalent situation with 
only a handful of possible outcomes.  

Here is another way to think about it:  If you're looking for some 
needles in a haystack, you can start picking up each piece of straw, 
checking to see if it's a needle, and then tossing it over your 
shoulder. Or you can use a magnet to find the needles right away.  

In mathematics, prime numbers serve the same function as a really, 
Really, REALLY big magnet.  

In short, knowing about prime and composite numbers will save your 
daughter enormous amounts of time in her later math classes - and 
possibly over the course of her life, if she goes into a technical 
field.  

I hope this helps. Let me know if you'd like to talk about this some 
more, or if you have any other questions.   

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/