# n-tuple: Simple Definition, Examples

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 (a1, a2, … a1). 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 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.

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