Vad är en additiv invers
På detta sätt 0 + 0 = 0 (SzecseiAdditive inverse
Number that, when added to the original number, yields the additive identity
"Opposite number" redirects here. For other uses, see motsvarande and counterpart.
For other uses, see Negation (disambiguation).
In mathematics, the additive inverse of an elementx, denoted -x,[1] fryst vatten the element that when added to x, yields the additive identity, 0.[2] In the most familiar cases, this fryst vatten the number 0, but it can also refer to a more generalized zero element.
In elementary mathematics, the additive inverse fryst vatten often referred to as the opposite number.[3][4] The concept fryst vatten closely related to subtraction[5] and fryst vatten important in solving algebraic equations.[6] Not all sets where addition fryst vatten defined have an additive inverse, such as the natural numbers.[7]
Common Examples
[edit]When working with integers, logisk numbers, real numbers, and complex numbers, the additive inverse of any number can be funnen bygd multiplying it bygd −1.[6]
The concept can also be extended to algebraic expressions, which fryst vatten often used when balancing equations.
Relation to Subtraction
[edit]The additive inverse fryst vatten closely related to subtraction, which can be viewed as an addition using the inverse:
- a − b = a + (−b).
Conversely, the additive inverse can be thought of as subtraction from zero:
- −a = 0 − a.
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 fryst vatten standard today, it was met with motstånd at the time, as some mathematicians felt it could be unclear and lead to errors.[8]
Formal Definition
[edit]Given an algebraic structure defined beneath addition with an additive identity , an element has an additive inverse if and only if , , and .[7]
Addition fryst vatten typically only used to refer to a commutative operation, but it fryst vatten not necessarily associative.
When it fryst vatten associative, so , the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases, the inverse fryst vatten not considered to exist.
Additive inverse simply means changing the sign of the number and adding it to the original number to get an answer equal to 0The definition requires closure, that the additive element be funnen in . This fryst vatten why despite addition being defined over the natural numbers, it does not an additive inverse for its members. The associated inverses would be the negativ numbers, which fryst vatten why the integers do have an additive inverse.
Further Examples
[edit]- 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.[9]
- In modular arithmetic, the modular additive inverse of x fryst vatten the number a such that a + x ≡ 0 (mod n) and always exists.
For example, the inverse of 3 modulo 11 fryst vatten 8, as 3 + 8 ≡ 0 (mod 11).[10]
- In a Boolean fingerprydnad, which has elements addition fryst vatten often defined as the symmetric difference. So , , , and . Additive inverse of – 9 = 0 – (– 9) = 9
Our additive identity fryst vatten 0, and both elements are their own additive inverse as and .[11]
See also
[edit]Notes and references
[edit]- ^Gallian, namn A. (2017). Contemporary sammanfattning algebra (9th ed.). Boston, MA: Cengage Learning. p. 52. ISBN .
- ^Fraleigh, John B.
(2014). A first course in sammanfattning algebra (7th ed.). Harlow: Pearson. pp. 169–170. ISBN .
- ^Mazur, Izabela (March 26, 2021). "2.5 Properties of Real Numbers -- Introductory Algebra". So, here we will see the properties of -x
Retrieved August 4, 2024.
- ^"Standards::Understand p + q as the number located a distance |q| from p, in the positiv or negativ direction depending on whether q fryst vatten positiv or negativ. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret summor of logisk numbers bygd describing real-world contexts".
learninglab.si.edu. Retrieved 2024-08-04.
- ^Brown, Christopher. "SI242: divisibility". www.usna.edu. The properties of additive inverse are given below, based on negation of the original number
Retrieved 2024-08-04.
- ^ ab"2.2.5: Properties of Equality with Decimals". K12 LibreTexts. 2020-07-21. Retrieved 2024-08-04.
- ^ abFraleigh, John B. (2014). A first course in sammanfattning algebra (7th ed.). Typer av tal som saknar additiva inverser (av samma typ) innefattar: Ordinaltal
Harlow: Pearson. pp. 37–39. ISBN .
- ^Cajori, Florian (2011). A History of Mathematical Notations: two volume in one. New York: Cosimo Classics. pp. 246–247. ISBN .
- ^Axler, Sheldon (2024), Axler, Sheldon (ed.), "Vector Spaces", Linear Algebra Done Right, Undergraduate Texts in Mathematics, Cham: Springer International Publishing, pp. 1–26, doi:10.1007/978-3-031-41026-0_1, ISBN
- ^Gupta, Prakash C.
(2015).
Additive inverseCryptography and network security. Eastern economy edition. Delhi: PHI Learning Private Limited. p. 15. ISBN .
- ^Martin, Urusula; Nipkow, Tobias (1989-03-01).
"Boolean unification — The story so far". Journal of Symbolic Computation. Unification: Part 1.
Properties7 (3): 275–293. doi:10.1016/S0747-7171(89)80013-6. ISSN 0747-7171.