Numeric types¶

Disco has four types which represent numbers:

Natural numbers, written N, ℕ, Nat, or Natural. These represent the counting numbers 0, 1, 2, … which can be added and multiplied.
Disco> :type 5 5 : ℕ
Integers, written Z, ℤ, Int, or Integer, allow negative numbers such as -5. They extend the natural numbers with subtraction.
Disco> :type -5 -5 : ℤ
Fractional numbers, written F, 𝔽, Frac, or Fractional, allow fractions like 2/3. They extend the natural numbers with division.
Disco> :type 2/3 2 / 3 : 𝔽
Rational numbers, written Q, ℚ, or Rational, allow both negative and fractional numbers, such as -2/3.
Disco> :type -2/3 -2 / 3 : ℚ

We can arrange the four numeric types in a diamond shape, like this:

Each type is a subset, or subtype, of the type or types above it. For example, the fact that \(\mathbb{N}\) is below \(\mathbb{Z}\) means that every natural number is also an integer.

The values of every numeric type can be added and multiplied.
The arrow labelled \(x-y\) indicates that going up and to the left in the diamond (i.e. from \(\mathbb{N}\) to Z or F to Q) corresponds to adding the ability to do subtraction. That is, values of types on the upper left of the diamond (\(\mathbb{Z}\) and \(\mathbb{Q}\)) can also be subtracted.
Going up and to the right corresponds to adding the ability to do division; that is, values of the types on the upper right of the diamond (\(\mathbb{F}\) and \(\mathbb{Q}\)) can also be divided.
To move down and to the right (i.e. from \(\mathbb{Z}\) to \(\mathbb{N}\), or from \(\mathbb{Q}\) to \(\mathbb{F}\)), you can use absolute value.
To move down and to the left (i.e. from \(\mathbb{F}\) to \(\mathbb{N}\), or from \(\mathbb{Q}\) to \(\mathbb{Z}\)), you can take the floor or ceiling.