"Closure" is a property which a set either has or lacks with respect to a given operation. A set is closed with respect to that operation if the operation can always be completed with elements in the set.
For example, the set of even natural numbers, 2, 4, 6, 8,..., is closed with respect to addition because the sum of any two of them is another even natural number. It is not closed with respect to division because the quotients 6/2 and 4/8, for instance, cannot be computed without using odd numbers or fractions.
Knowing the operations for which a given set is closed helps one understand the nature of the set. Thus one knows that the set of natural numbers is less versatile than the set of integers because the latter is closed with respect to subtraction, but the former is not. Similarly one knows that the set of polynomials is much like the set of integers because both sets are closed under addition, multiplication, negation, and subtraction, but are not closed under division.
Particularly interesting examples of closure are the positive and negative numbers. In mathematical structure these two sets are indistinguishable except for one property, closure with respect to multiplication. Once one decides that the product of two positive numbers is positive, the other rules for multiplying and dividing various combinations of positive and negative numbers follow. Then, for example, the product of two negative numbers must be positive, and so on.
The lack of closure is one reason for enlarging a set. For example, without augmenting the set of rational numbers with the irrationals, one cannot solve an equation such as x2 = 2, which can arise from the use of the pythagorean theorem. Without extending the set of real numbers to include imaginary numbers, one cannot solve an equation such as x 2 + 1= 0, contrary to the fundamental theorem of algebra.
Closure can be associated with operations on single numbers as well as operations between two numbers. When the Pythagoreans discovered that the square root of 2 was not rational, they had discovered that the rationals were not closed with respect to taking roots.
Although closure is usually thought of as a property of sets of ordinary numbers, the concept can be applied to other kinds of mathematical elements. It can be applied to sets of rigid motions in the plane, to vectors, to matrices, and to other things. For example, one can say that the set of three-by-three matrices is closed with respect to addition.
Closure, or the lack of it, can be of practical concern, too. Inexpensive, four-function calculators rarely allow one to use negative numbers as inputs. Nevertheless, if one subtracts a larger number from a smaller number, the calculator will complete the operation and display the negative number which results. On the other hand, if one divides 1 by 3, the calculator will display 0.333333, which is close, but not exact. If an operation takes a calculator beyond the numbers it can use, the answer it displays will be wrong, perhaps significantly so.