# Pairing function

In mathematics a **pairing function** is a process to uniquely encode two natural numbers into a single natural number.

Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. In theoretical computer science they are used to encode a function defined on a vector of natural numbers *f*:**N**^{k} → **N** into a new function *g*:**N** → **N**.

## Definition

A **pairing function** is a primitive recursive bijection

## Cantor pairing function

The **Cantor pairing function** is a pairing function

defined by

When we apply the pairing function to and we often denote the resulting number as

This definition can be inductively generalized to the **Cantor tuple function**

as

### Inverting the Cantor pairing function

Suppose we are given *z* with

and we want to find *x* and *y*. It is helpful to define some intermediate values in the calculation:

where *t* is the triangle number of *w*. If we solve the quadratic equation

for *w* as a function of *t*, we get

which is a strictly increasing and continuous function when *t* is non-negative real. Since

we get that

and thus

where is the floor function.
So to calculate *x* and *y* from *z*, we do:

Since the Cantor pairing function is invertible, it must be one-to-one and onto.

## References

- Steven Pigeon, "Pairing function",
*MathWorld*.