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Natural numbers

Natural numbers are the numbers we meet most often in everyday life. They are the positive integers, that is, the numbers 1, 2, 3, 4, …

Notation

The natural numbers form a set containing the positive integers 1, 2, 3, 4, … We usually write this set with the letter N with a double first stroke, like this: ℕ. It comes from the English word “naturals”. In this article the natural numbers are 1, 2, 3, … without zero; some authors (and Wikipedia) count zero as a natural number too.

Sometimes we assume that the set of natural numbers also contains zero. When we need to tell the two apart, we usually write the plain ℕ for the set without zero, and if we want zero as well, we add a zero to the subscript: ℕ0, the natural numbers together with zero. We also often use a plus sign, ℕ+, to stress that we mean the natural numbers without zero.

We use natural numbers mainly to say how many of something there are (“we have three chairs at home”, “there are thirty benches in the park”, …) and to give an order (“the first person on the Moon”, “Canada is the second largest country in the world”, …).

Properties

Natural numbers have some interesting properties:

  1. The set of natural numbers is infinite, but it is countable: we can list all of them in a sequence.
  • Natural numbers are closed under addition and multiplication. This means that if we add or multiply any two natural numbers, we get a natural number again.
  • They are not closed under subtraction, because if we subtract a larger number from a smaller one, we get a negative number.
  • Similarly, they are not closed under division either: for example, 7/2 is not a natural number.

Division with remainders

In the previous section we saw that the natural numbers are not closed under division. However, we can define division with remainders, which you all certainly know. If we work out 7/2, we get 3.5. If we use division with remainders, we get an answer of 3 remainder 1. So if we multiply 3 · 2 and add the remainder, we get 7 back: 3 · 2 + 1 = 7.

The natural numbers are not closed under this operation either, but the natural numbers including zero are. So both the quotient and the remainder will lie in the set ℕ0.

The definition of division with remainders looks like this:

a=bq+r;a,q,r∈N0,b∈N,r<b

In the definition we divided a÷b. The number r is called the remainder and q the quotient. What do the conditions mean? We assume that we are working with the natural numbers including zero, but since we cannot divide by zero, we choose b from the set without zero.

For example, to work out 19 ÷ 5 we would take a = 19 and b = 5. The decomposition would look like this:

19=5⋅3+4

So q = 3 is the quotient and r = 4 is the remainder.