An **n-tuple** (also called an *ordered n-tuple* or just a *list*) is an ordered sequence of natural numbers (whole, non-negative numbers that we use to count), where n ≧ 1.

N-tuples are often denoted by an expression like (a_{1}, a_{2}, … a_{1}). For example:

- 1-tuple (
*monad*or*singleton*): (7) - 2-tuple (
*pair*or*twin*): (3, 4) - 3-tuple (
*triple*or*triad*): (3, 3, 7) - 4-tuple (
*quadruplet*): (3, 6, 7, 8) - 5-tuple (
*quintuplet*): (1, 3, 6, 7, 8) - 6-tuple: (
*sextuplet*):(4, 5, 7, 9, 11, 13)

A **zero n-tuple** (or empty function) has all-zero entries:

**0** = (0, 0,…0).

Entries for tuples don’t have to be distinct (they can all be the same).

## Addition of n-Tuples

Addition of tuples is easy. You just add the first components, then the second, and so on. For example, let’s say you bought 2 bags of rice and 3 cans of beans (2, 3), then went to a second store and bought 1 more bag of rice and 2 cans of beans (1, 2). The addition would be:

(2, 3)+

(1, 2)

= 3, 5

For this to work, your tuples must be the same size.

## N-Tuples in Linear Algebra

If you’ve ever taken linear algebra, this type of addition (called *component addition)* might look familiar, with good reason: In matrices, an n-tuple is synonymous with “l x n matrix” (Rosen, 2013).

Tuples can be row matrices, like [0, 2, 3] or column matrices, such as

An n-tuple is commonly used to represent a vector. For example, the 2-tuple [3, 4] could represent a vector in 2D space where the tail of the vector is at the origin and head is at [3, 4]. Similarly, a 3-tuple like [-1, 4, 5] could represent a vector in 3D Space.

## References

Cowin, S. & Doty, S. (2007). Tissue Mechanics. Springer, New York.

Davis, M. (2013). Computability and Unsolvability. Dover Publications.

Dr. Math. (1999). Definition of an N-tuple – Math Forum – Ask Dr. Math. Retrieved July 24, 2020 from: http://mathforum.org/library/drmath/view/55500.html

Rosen, G. (2013). Formulations of Classical and Quantum Dynamical Theory. Academic Press.