As addition is based upon the idea of combining groups of things to yield a larger group, subtraction, the opposite of addition, is based on the idea of removing objects from a group, thereby reducing its size. These intuitive ideas of combining and reducing were undoubtedly engrained in daily life for even the earliest humans. When numbers are used to represent the quantity of objects composing groups, subtraction becomes an operation upon those numbers. Indeed, subtraction is one of four fundamental arithmetic operations (the others being addition, multiplication, and division). The symbol that denotes subtraction is "-", read "minus". For instance, "A minus B" means that quantity B is to be subtracted from quantity A, where "A" is the minuend and "B" is the subtrahend. The word minus comes from the Latin word "minus", which means "less".
By calling subtraction an operation, it is meant that any number subtracted from another number yields a third unique number. For instance, "7 - 2" represents "5", unique among the whole numbers {1, 2, 3, 4, 5,...}. However, many of the arithmetic laws that govern the operation of addition do not hold for subtraction. For example, the commutative law of addition "a + b = b + a" is true for whole numbers, rational numbers, real numbers, etc., but does not hold for subtraction (e.g., "5 - 7 7 - 5"). Within arithmetic the only fundamental addition laws that are valid for subtraction are the distributive and monotonic laws: the distributive law for subtraction of products states that for numbers a, b and c, "a ⋅ (b - c) = (a ⋅ b) - (a ⋅ c)", while the monotonic law for subtraction states that "if a < b then (a - c) < (b - c)". So as an arithmetic operation, subtraction does not generally possess the symmetry that addition possesses. However, as shall be shown, subtraction can be modified to conform to the addition laws.
In the historical development of number systems subtraction played an important part. For people using number systems that lacked terms for zero and/or negative numbers, subtraction leads to confounding mathematical situations. For example, subtraction of whole numbers leads to expressions like "5 - 5" and "7 - 9". Without zero or negative numbers, expressions such as these are impossible. For some ancient peoples the answer was to expand their number system to include zero and negative numbers. Thus the set of integers {..., -3, -2, -1, 0, 1, 2, 3,...} became an extension of the whole numbers. By including negatives and zero in the number system, the operation of subtraction can be recast as the addition of positive and negative numbers. As was pointed out earlier, "5 - 7" and "7 - 5" represent two distinct numbers (i.e., subtraction is not commutative). However, if one views subtraction as the addition of positives and negatives, then the subtraction of "7 from 5" becomes equivalent to the addition of "5 and (-7)". As a result, the commutative law holds (i.e., 5 + (-7) = (-7) + 5). The pertinent algebraic law is the law of additive inverse, which states that for any number "a" there exists another unique number "-a" such that: "a + (-a) = 0". This law holds for the set of integers, rationals, reals, etc. It does not hold for the whole or natural numbers, since neither system contains negatives.
By treating subtraction of one number from another (a - b) as simply the addition of the number "a" with the additive inverse of "b" (that is, a + (-b)), the additive laws of arithmetic may be applied. These laws are valid on the set of integers, rationals, reals, etc.; and subtraction of quantities beyond the real numbers is now considered.
Subtraction on vectors is performed by defining the negative of a vector as being equal to a vector of equal magnitude but opposite direction. The corresponding components of the two vectors are subtracted to form the resultant vector. Specifically, subtracting the vector "b = bxi + byj from the vector "a = axi + ayj" yields a vector "r", such that "r = a - b = a + (-b) = (ax - bx) i + (ay - by) j", where i and j are unit vectors in the x-direction and y-direction (respectively) of the x-y plane, ax and ay are scalar components of vector a, and bx and by are scalar components of vector b.
The subtraction of functions is accomplished by subtracting common terms. For example, the function g(x) = x3 - 2x2 + x + 3 can be subtracted from the function f(x) = x2 + 3x - 4 to result in f(x) - g(x) = (x2 + 3x - 4) - (x3 - 2x2 + x + 3) = (-x3 + 3x2 + 2x - 7).
Subtraction on matrices is valid only if each of the matrices to be subtracted is of the same order (contain the same number of rows and columns). Subtraction of matrices is performed by subtracting the elements of a particular matrix from the corresponding elements of other matrices. Therefore, the resulting matrix is of the same order as the subtracted matrices.
Subtraction can be applied to complex numbers, which are composed of a real part followed by an imaginary part. It is defined as the subtracting of complex quantities in which the individual real and imaginary parts are separately subtracted. Specifically, subtracting "y = c + id" from "x = a + ib" results in "x - y = (a - c) + i(b - d)". For example, subtracting "y = 1 + i(6)" from "x = 2 + i(3)" yields: "x - y = (2 + i(3)) - (1 + i(6)) = (2 - 1) + i(3 - 6) = 1 + i(-3)".
As is the case with addition, subtraction is founded upon concepts found in the branch of mathematics known as set theory. Under set theory, the arithmetic operation of subtraction on the natural (or whole) numbers can be recast as the subtraction (or difference) of sets, denoted symbolically for sets "A" and "B" as "A \ B". The various laws for subtraction between whole numbers can then be derived.
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