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Natural numbers

Natural numbers are the most common numbers we encounter in everyday life. These are positive integers, i.e. the numbers 1, 2, 3, 4, ...

Marking

Natural numbers is the set that contains the positive integers 1, 2, 3, 4, ... We usually denote this set by the letter N with the first leg doubled, as follows: ℕ. It is from the English "naturals".

We sometimes assume that the set of naturals also contains zero. If we need to distinguish this, we usually use the classic ℕ for the set without zero, and if we want zero as well, then we add zero to the index as follows: ℕ0. Often we then still use the plus sign to highlight that we are counting natural numbers without zero $\mathbb{N}^+$.

We use natural numbers mainly to determine the quantity of something ("we have three chairs at home", "there are thirty benches in the park", ...) and to determine the order ("the first man on the moon", "Canada is the second largest country in the world", ...).

Features

Natural numbers have some interesting properties:

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

Division with remainder

In the last chapter, we saw that natural numbers are not closed with respect to division. However, we can define the operation of division with remainder, which I'm sure you're all familiar with. If we divide 7/2, we get 3.5. If we use division with remainder, we get the result 3 and the remainder 1. So if we multiply 3 · 2 and add the remainder, we get back 7: 3 · 2 + 1 = 7.

While natural numbers are not then closed with respect to this operation, natural numbers including zero are. Thus both the result and the remainder will be contained in the set ℕ0.

The definition of division with remainder looks like this:

$$a=bq+r; \qquad a, q, r\in\mathbb{N}_0, b\in\mathbb{N}, r < b$$

In the definition we divided a:b, the number r is called the remainder after the division and q the result of the division. What do the terms mean? We assume that we are in the natural numbers including zero, but since we cannot divide by zero, we pick b from the set without zero.

So for the example, to calculate 19:5, a = 19 and b = 5 would be valid. The decomposition would look like this:

$$19=5\cdot3+4$$

So q = 3, that's the result after division, and r = 4, that's the remainder.