Prime and Composite Numbers

Background

There are different ways we can classify groups of numbers. One such way is to look at if a whole number is the product of two smaller whole numbers or not.

Question

Classify the following as prime, composite, or other: 0, 1, 2, 3, 4, 5, 6

Answer

0 and 1 are other (see below for more detail). 2, 3, and 5 are prime. 4 and 6 are composite.

Analysis

Mathematicians like to divide and classify things into different groups, to show different properties of different numbers. One of the ways we can divide whole numbers is into the group of them that are the product of two smaller whole numbers (Composite Numbers) and those that aren't (typically these are Prime Numbers) but there are two exceptions that we'll get to in a minute.

Ok - first we need to nail down some definitions so that we can be precise in what we're talking about.

A Prime Number is one that is:

• a Whole Number, and
• is only the product of itself and 1 (the two numbers have to be different!)

A Composite Number is one that is:

• a Whole Number, and
• is the product of more than simply 1 and itself
• (I should note that sometimes that first requirement is dropped. Some people consider negative numbers to be composite numbers)

Let's look at the number 2. It's a whole number - that's the first requirement. And we can't multiply by a smaller whole number (0 and 1 being the only choices) and get to 2 (you can see that on the multiplication table). 2, therefore, is prime.

Now let's look at 4. It too is a whole number. But 4 is not only the product of itself and 1 (4 x 1 = 4), but we can also multiply 2 x 2 = 4. 4, therefore, is composite (in other words, other numbers aside from 1 and itself can be multiplied together to arrive at 4).

And we can work through the remaining numbers in the list:

3 is prime (1 x 3 = 3)
5 is prime (1 x 5 = 5)
6 is composite (1 x 6 = 2 x 3 = 6)

So what about 0 and 1?

Well... they aren't composite - they aren't the product of two smaller whole numbers.

And they aren't prime.

With the number 0, no matter the number we multiply 0 by, we still get 0. That means it isn't prime. And so if it isn't prime and it isn't composite, what is it?

The answer is super math-y, but the class of numbers 0 falls into is called zero-divisors - which simply means that if you multiply a whole number by it, you'll get 0. (Believe it or not, there is a field of study into this - check out the wikipedia page ending in Zero_divisor to take a look and have your head spin).

With the number 1, we can't multiply two different whole numbers to get a product of 1, meaning it too doesn't meet the definition of being prime. And so if it isn't composite and it isn't prime, what is it?

The answer again is super math-y, but 1 ends up in a different class of number, called units. Units are numbers where both itself and it's reciprocal (or in other words, you can take the number and divide by 1, and also divide 1 by the number) are whole numbers.

Vocabulary used:

• Whole Numbers - The set of numbers that start at 0 and increase by 1 (0, 1, 2, 3,...)
• Product - the result of a multiplication operation. For example, 8 is the product of 4 and 2

For more information check out these links (comment to add your favourite link):

https://www.mathsisfun.com/prime-composite-number.html
http://mathforum.org/library/drmath/view/57036.html
https://en.wikipedia.org/wiki/Zero_divisor

Where might you have come from?

Fact-orials Index

Numbers:

• The Number 1
• Counting/Natural Numbers
• The Number 0
• Whole Numbers
• Integers

Operations:

• Multiplication

Where might we go?

Operations:

• Multiplication Table

Associated Operations:

• Remainders and Modulo
• Factors and Factoring