What Is Addition?
There are many things that can be added: numbers, vectors, matrices, spaces, shapes, sets, functions, equations, strings, chains...
Mathematics originated with the desire and need to count and measure. But ever since the invention of numbers it began acquiring abstract features that characterize it nowadays. The number 1 is an abstraction corresponding to a single object, be it one cow, one fish, flower or molecule. With counting naturally comes operation of addition - passing from the current object to the next means adding one to the set of already counted objects. Ian Stewart defines Mathematics as the science of pattern
that detects and studies commonality in diverse phenomena. m + n means the result of first counting m and then n objects. Regardless of what was counted, the pattern emerged that claimed that first counting n objects and afterwards additional m will produce the same result:
Thus Mathematics went from the abstraction of a number to the abstraction of operation; addition being just one such operation. Operations apply to elements of arbitrary sets which, in turn, may be distinguished by the variety of operations (and their properties) that are defined for elements of a set. Addition is a binary operation that applies to two objects simultaneously and results in another element of the same set. Breeding might be looked at as another binary operation. Negation, i.e. changing sign, is a unary operation since it applies to a single element. A ternary operation applies to three elements at once, and so on.
Addition, as an abstract operation, has several properties.
In a set for whose elements addition is defined there exists a very special element (most often) called zero and denoted as 0, such thata + 0 = 0 + a = a,
for any element a of the given set.
What you added you must be able to take back so that for every element a there exists an element b such thata + b = b + a = 0.
This element is denoted as -a and is called the (additive) inverse of a.
Addition is required to be associative, i.e., producing the same result regardless of the sequence in which elements are added:(a + b) + c = a + (b + c)
- There is one more property, that of commutativity
a + b = b + a
which is often imposed on the operation of addition. But sometimes it's more natural and convenient to allow a non-commutative addition.
You won't be surprised to learn that mathematicians have given names to sets on which addition is defined. Such sets are called (additive) groups. If the addition is commutative, the group is said to be commutative or Abelian. They also found use to sets in which the inverse element -a does not always exist as in case of whole numbers (positive integers). Such sets are called semigroups. (Just in passing, if the operation is not even associative, the set is called groupoid, or magma, see wikipedia.Groupoid.)
- Ian Stewart, Nature's Numbers, BasicBooks, 1995
- Oystein Ore, Number Theory and Its History, Dover Publications, 1976