Additive inverse

In mathematics, the additive inverse of an element $x$ , denoted $-x$ ,^[1] is the element that when added to $x$ , yields the additive identity, 0 (zero).^[2] In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero element.

In elementary mathematics, the additive inverse is often referred to as the opposite number,^[3]^[4] or its negative.^[5] The unary operation of arithmetic negation^[6] is closely related to subtraction^[7] and is important in solving algebraic equations.^[8] Not all sets where addition is defined have an additive inverse, such as the natural numbers.^[9]

^ Gallian, Joseph A. (2017). Contemporary abstract algebra (9th ed.). Boston, MA: Cengage Learning. p. 52. ISBN 978-1-305-65796-0.
^ Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 169–170. ISBN 978-1-292-02496-7.
^ Mazur, Izabela (March 26, 2021). "2.5 Properties of Real Numbers -- Introductory Algebra". Retrieved August 4, 2024.
^ "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". learninglab.si.edu. Retrieved 2024-08-04.
^ Weisstein, Eric W. "Negative". MathWorld. Retrieved 2025-01-04.
^ Kinard, James T.; Kozulin, Alex (2008-06-02). Rigorous Mathematical Thinking: Conceptual Formation in the Mathematics Classroom. Cambridge University Press. ISBN 978-1-139-47239-5.
^ Brown, Christopher. "SI242: divisibility". www.usna.edu. Retrieved 2024-08-04.
^ "2.2.5: Properties of Equality with Decimals". K12 LibreTexts. 2020-07-21. Retrieved 2024-08-04.
^ Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 37–39. ISBN 978-1-292-02496-7.

[1] Gallian, Joseph A. (2017). Contemporary abstract algebra (9th ed.). Boston, MA: Cengage Learning. p. 52. ISBN 978-1-305-65796-0.

[2] Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 169–170. ISBN 978-1-292-02496-7.

[3] Mazur, Izabela (March 26, 2021). "2.5 Properties of Real Numbers -- Introductory Algebra". Retrieved August 4, 2024.

[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". learninglab.si.edu. Retrieved 2024-08-04.

[5] Weisstein, Eric W. "Negative". MathWorld. Retrieved 2025-01-04.

[6] Kinard, James T.; Kozulin, Alex (2008-06-02). Rigorous Mathematical Thinking: Conceptual Formation in the Mathematics Classroom. Cambridge University Press. ISBN 978-1-139-47239-5.

[7] Brown, Christopher. "SI242: divisibility". www.usna.edu. Retrieved 2024-08-04.

[:0-8] "2.2.5: Properties of Equality with Decimals". K12 LibreTexts. 2020-07-21. Retrieved 2024-08-04.

[:1-9] Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 37–39. ISBN 978-1-292-02496-7.

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Additive inverse

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