A set in which all of the elements are numbers.

### Notations

The different sets of numbers are identified by letters :

- \(\mathbb{W}\) refers to the set of whole numbers;
- \(\mathbb{Z}\) refers to the set of integers;
- \(\mathbb{D}\) refers to the set of decimal numbers;
- \(\mathbb{Q}\) refers to the set of rational numbers;
- \(\mathbb{Q}\)‘ refers to the set of irrational numbers;
- \(\mathbb{R}\) refers to the set of real numbers;
- \(\mathbb{\overline{Q}}\) refers to the set of algebraic numbers;
- \(\mathbb{C}\) refers to the set of complex numbers.

### Historical Note

We may well wonder where the choice of the letters to describe the different sets of numbers comes from. Here are a few explanations :

- In his work on the axiomatization of the set of non-zero integers, the Italian mathematician Giuseppe Peano used the capital letter N, for the Italian word
*naturale*, which later became \(\mathbb{N}\), to indicate the set of non-zero whole numbers. He also used the letter \(\mathbb{Q}\), for the first letter of the Italian word*quotient*which means “quotient”, to refer to the set of rational numbers. - In the works of the Bourbaki group (around 1970), they used the letter \(\mathbb{D}\) to indicate the set of decimal numbers and \(\mathbb{Z}\) to refer to the set of integers (for the German word
*zahlen*which means “to count”. This may also be an idea from the German mathematician Dedekind. - The choice of the letter \(\mathbb{R}\) to refer to the set of real numbers comes from Julius Wilhelm Dedekind (1831-1916), a German mathematician, who introduced it in his texts.
- Dedekind also discussed complex numbers and algebraic numbers in his texts, but the choices of the letters \(\mathbb{C}\) for complex numbers and \(\mathbb{\overline{Q}}\) for algebraic numbers are more recent although we find them in many texts.