Numbers

Building Numbers from Small Building Blocks: Any counting number, except 1, can be made by adding two or more small counting numbers. But only a few counting numbers can be formed by multiplying two or more smaller counting numbers.

Prime and composite numbers: We can make 36 by multiplying 9 and 4. Or we can make it from 6 and 6. or from 18 and 2; Or even by multiplying 2 × 2 × 3 × 3. Numbers like 10 and 36 and 49 that can be formed as a product of smaller counting numbers are called composite numbers.

Some numbers cannot be formed from such small pieces. For example, there is only one way to make 7 by multiplying 7 and using only counting numbers. So we’re not really building it with smaller building blocks. We need to start with it. Such numbers are called prime numbers.

Informally, primes are numbers that cannot be formed by multiplying other numbers. It captures the idea well, but it’s not a good enough definition, because it has too many flaws. The number 7 can be formed as a product of other numbers: for example, it is 2 × 3\frac{1}{2}. To get the idea that “7 is not divisible by 2,” we must make it clear that we are restricting numbers to include only counting numbers: 1, 2, 3….

A formal definition

A prime number is a positive integer that has exactly two distinct whole number factors (or divisors), namely 1 and the number itself.

Clarifying two common confusions

Two common confusions:

• The number 1 is not prime.
• The number 2 is prime. (It is the only even prime.)

The number 1 is not prime. Why not?

Well, the definition rules it out. It says “two distinct whole-number factors” and the only way to write 1 as a product of whole numbers is 1 × 1, in which the factors are the same as each other, that is, not distinct. Even the informal idea rules it out: it cannot be built by multiplying other (whole) numbers.

But why rule it out?! Students sometimes argue that 1 “behaves” like all the other primes: it cannot be “broken apart.” And part of the informal notion of prime — we cannot compose 1 except by using it, so it must be a building block — seems to make it prime. Why not include it?

Mathematics is not arbitrary. To understand why it is useful to exclude 1, consider the question “How many different ways can 12 be written as a product using only prime numbers?” Here are several ways to write 12 as a product but they don’t restrict themselves to prime numbers.

3 × 4

4 × 3

1 × 12

1 × 1 × 12

2 × 6

1 × 1 × 1 × 2 × 6