Introduction:

When we say that integers are closed under addition, it means that adding any two integers will yield a result that is also an integer. This fundamental property is crucial in understanding the structure and behavior of integers within mathematics. In this article, we will delve into the details of what it means for integers to be closed under addition and explore the underlying mathematical concepts that support this property.

What Does It Mean to Say That Integers Are Closed Under Addition?

Integers, in their simplest form, are composed of a natural number (absolute value) and a sign (positive or negative) or zero, which is neither positive nor negative. When we add two integers, the result will always be another integer. For example, if a and b are integers, then a b is also an integer.

This property can be demonstrated through different methods, such as defining addition in cases based on the signs of the integers involved. However, a more general and abstract method defines integers as equivalence classes of ordered pairs of natural numbers under an equivalence relation.

Standard Definition of Addition of Integers

Addition of integers can be defined through a series of cases based on the signs of the integers involved. For instance:

If both a and b are positive, then a b a b. If both a and b are negative, then a b a b. If a and b have different signs, the result is the difference between a and b with the sign of the term whose absolute value is larger.

For example, 6 (-4) 2, because the absolute value of the negative term is larger. This definition, while useful for concrete problems, can be cumbersome for proofs. Therefore, a more abstract and commonly used method defines integers as equivalence classes of ordered pairs of natural numbers under an equivalence relation.

Equivalence Classes and Integers

The integers can be defined as the equivalence classes of ordered pairs of natural numbers under the equivalence relation:

a – b ~ c – d if and only if a – d b – c.

This means that the integer n is the equivalence class of n – 0, and -n is the equivalence class of 0 – n. This way, natural numbers are identified with their corresponding equivalence classes.

Component-wise Addition of Equivalence Classes

Component-wise addition of ordered pairs (a, b) and (c, d) is defined as:

(a, b) (c, d) (a c, b d)

A straightforward computation shows that the equivalence class of the result depends only on the equivalence classes of the summands. Thus, this operation defines an addition of equivalence classes that are integers.

This definition is equivalent to the case-based definition, which simplifies the proof and makes it more accessible.

Abstract Construction of Integers

The construction of integers as equivalence classes of ordered pairs can be generalized to embed any commutative semigroup with a cancellation property into a group. Here, the semigroup is the natural numbers, and the group is the additive group of integers.

The rational numbers can be constructed similarly by taking the nonzero integers with multiplication as the semigroup, and the direct sum as the group operation.

The Grothendieck group is an even more generalized construction that applies to any commutative semigroup, potentially resulting in a non-injective semigroup homomorphism. Originally, the Grothendieck group was specifically applied to the equivalence classes of objects in an abelian category with the direct sum as the semigroup operation.

Conclusion:

Understanding the closure property of integers under addition is essential for grasping the fundamental concepts in mathematics. Whether through concrete examples, case-based definitions, or abstract constructions, the property that the sum of any two integers is always an integer is a cornerstone of arithmetic and number theory.