# N-Tuples

An **N-tuple** is just a sequence of `N` real numbers:
(`x`_{1}, `x`_{2}, ...,
`x`_{N}).

For example, here is a 12-tuple which stores how many days each
month has in a non-leap year:
(31, 28, 31, 30, 31, 30, 31,
31, 30, 31, 30, 31).

Special cases of tuples include 1-tuples or **singletons** like
(3), 2-tuples or **pairs** like (7, 11.3), and 3-tuples
**triples** like (23, 47, 0.03). In the 5-tuple
(8, 6.5, 10, 9, 5) we
would call 8 the first **component**, 6.5 the second component,
etc.
The set of all N-tuples is refered to as
**R**^{N}, where the **R**
stands for ``Real Number''.
[You could also talk about
**Q**^{N}, the set of N-tuples of
rational numbers (``Quotients''), or
**Z**^{N} the set of N-tuples of
integers (``Zahlen'' means ``Integer'' in German).] In these pages,
we will be working mostly with **R**^{1} = **R**,
**R**^{2} and **R**^{3}.

Tuples are useful whenever you want to gather a set of data together
and consider it as a single unit. You might want to combine the result
of a measurement with the time the measurement was taken into a pair,
or combine all the 30 answers to a question in a psychological profile
into a 30-tuple.

It is really easy to do arithmetic with tuples of the same size: you
just do the operation on the first components of both tuples to get
the first component of the answer, then repeat on the second
componenets, etc. For example, if I sold 3 apples and 4 oranges on
Monday, and 2 apples and 7 oranges on Tuesday, I could calculate my
combined sales by adding the two 2-tuples:
(3, 4) + (2, 7) = (5, 11). This is
called component-wise addition.

One can also operate on a tuple by a single real number. To convert a
5-tuple of temperatures measured in Celsius to a 5-tuple of temperatures
measured in Kelvin, I might want to add 273.15 to each component:

Similarly, to find the
average of three triplets, I would add them up component-wise and
then divide by three:

All by themselves, tuples may not represent a great deal of
mathematical interest. Soon, however, we will use
tuples to represent more interesting objects: points
and vectors.

### Exercises:

- What is the fourth component of
(31, 28, 31, 30, 31, 30, 31,
31, 30, 31, 30, 31)?

- Calculate
(1, 2, 3) - (4, 8, 2)
using component-wise subtraction.

Solutions to Exercises |
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watko@athena.mit.edu
Last modified November 19, 1998