Additive inverse

Common examples

When working with integers, rational numbers, real numbers, and complex numbers, the additive inverse of any number can be found by multiplying it by −1.[[Image:NegativeI2Root.svg|thumb|right|These complex numbers, two of eight values of [[root of unity|]], are mutually opposite]]

The concept can also be extended to algebraic expressions, which is often used when balancing equations.

Relation to subtraction

The additive inverse is closely related to subtraction, which can be viewed as an addition using the inverse: :. Conversely, the additive inverse can be thought of as subtraction from zero: :. This connection lead to the minus sign being used for both opposite magnitudes and subtraction as far back as the 17th century. While this notation is standard today, it was met with opposition at the time, as some mathematicians felt it could be unclear and lead to errors.

Formal definition

Given an algebraic structure defined under addition (S, +) with an additive identity e \in S, an element x \in S has an additive inverse y if and only if y \in S, x + y = e, and y + x = e.

Addition is typically only used to refer to a commutative operation, but for some systems of numbers, such as floating point, it might not be associative.{{cite journal

The definition requires closure, that the additive element y be found in S. However, despite being able to add the natural numbers together, the set of natural numbers does not include the additive inverse values. This is because the additive inverse of a natural number (e.g., -3 for 3) is not a natural number; it is an integer. Therefore, the natural numbers in set S do have additive inverses and their associated inverses are negative numbers.

Further examples

• In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction.
• In modular arithmetic, the modular additive inverse of x is the number a such that and always exists. For example, the inverse of 3 modulo 11 is 8, as .
• In a Boolean ring, which has elements {0, 1} addition is often defined as the symmetric difference. So 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 0. Our additive identity is 0, and both elements are their own additive inverse as 0 + 0 = 0 and 1 + 1 = 0.

References

1. Gallian, Joseph A.. (2017). "Contemporary abstract algebra". Cengage Learning.
2. Fraleigh, John B.. (2014). "A first course in abstract algebra". Pearson.
3. Mazur, Izabela. (March 26, 2021). "2.5 Properties of Real Numbers -- Introductory Algebra".
4. "Standards::Understand p + q as the number located a distance {{!}}q{{!}} from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.".
5. (1982). "College Algebra". Elsevier.
6. (2008-06-02). "Rigorous Mathematical Thinking: Conceptual Formation in the Mathematics Classroom". Cambridge University Press.
7. Brown, Christopher. "SI242: divisibility".
8. (2020-07-21). "2.2.5: Properties of Equality with Decimals".
9. Fraleigh, John B.. (2014). "A first course in abstract algebra". Pearson.
10. Cajori, Florian. (2011). "A History of Mathematical Notations: two volume in one". Cosimo Classics.
11. Axler, Sheldon. (2024). "Vector Spaces". Springer International Publishing.
12. Gupta, Prakash C.. (2015). "Cryptography and network security". PHI Learning Private Limited.
13. (1989-03-01). "Boolean unification — The story so far". Journal of Symbolic Computation.