Axiom of Closure

One of the early axioms adopted by mathematicians must have been the Axiom of Closure, commonly called Closure Property.

This axiom states that a set of numbers should be closed for all the four basic operations & exponentiation.

This means that the result of operating on numbers in a set, using the four operators should result in a number which also belongs to that set.

Though this looks like a constraint, the closure property helped discovering new types of numbers which progressively increased the scope of these sets.

A Brief History of Different Numbers

Let us see some examples in brief.

The detailed history of the development of various types of numbers can be found on the chapter on Development of Number Systems

When Natural Numbers & operations were studied, it was found that sometimes, subtraction & division did not result in Natural numbers.

4 – 4 was 0, 3 – 5 was -2 and 3 ÷ 5 was 3/5.

This led to the inclusion of 0 with Natural numbers and the set was called Whole Numbers

It also led to the inclusion of Negative Numbers with Whole numbers and the set was called Integers

It further led to the inclusion of Fractions with Integers and the set was called Rational numbers

The operation of taking roots of Rational numbers led to numbers like √2 which were not rational numbers.

These were called Irrational numbers

The inclusion of Irrational numbers with Rational numbers led to the set of Real numbers

Taking roots of negative numbers resulted in numbers which were not Real numbers.

Such numbers were called Imaginary numbers and represented as “xi” where x was any real number and i was √(-1)

Imaginary numbers were included with Real numbers and called Complex numbers

It was found that the set of Complex numbers was closed with respect to the four major operations.

Hence starting with Natural numbers, the closure property led us to the closed set of Complex numbers.

Other aspects of Closure

There are other examples of the closure property.

The Set of even numbers is closed only for additions. Set of odd numbers is not, since odd + odd = even.

Set of positive numbers & the set of negative numbers differ only in the closure property. Positive numbers are closed for multiplication. Negative numbers are not, since – X - = +

Closure is a sign of completeness and invariance. It is, perhaps, one of the most beautiful and useful properties in all of mathematics.

Closure of an operation acting on a set, implies we don't need to search for larger sets to contain the results of any series or number of uses of the operation on arguments from the set. The final outcome will always be a member of the set.

This allows us to abstract the actions and objects of the operation of a closed set.

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 like vectors & matrices.