In mathematics, the additive inverse of a number a (sometimes called the opposite of a)[1] is the number that, when added to a, yields zero. The operation taking a number to its additive inverse is known as sign change[2] or negation.[3] For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself.
The additive inverse of a is denoted by unary minus: −a (see also § Relation to subtraction below).[4] For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0.
Similarly, the additive inverse of a − b is −(a − b) which can be simplified to b − a. The additive inverse of 2x − 3 is 3 − 2x, because 2x − 3 + 3 − 2x = 0.[5]
The additive inverse is defined as its inverse element under the binary operation of addition (see also § Formal definition below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect: −(−x) = x.
YouTube Encyclopedic
-
1/3Views:135 957100 76216 535
-
Additive Inverse 127-3.3
-
Math Shorts Episode 1 - Additive Inverse
-
Maths - What is additive inverse of an integer - English
Transcription
This video is provided as supplementary material for courses taught at Howard Community College, and in this video I want to talk about the additive inverse of numbers. Additive inverse goes something like this: Let's say I start with a number, let's say 5. Now the additive inverse of a number is going to be the number I add to it to end up with a zero. So I started out with a 5, that's a positive 5, and I wanna add something to it so that the result will be zero. Well if I add negative 5... positive 5 plus negative 5 equals zero. So one way of thinking of this would just be to say you take the opposite of the number you starting with, and you can add those together and get a zero. If you start with a negative number, for instance if you started with negative 4, I would take the opposite of that negative 4, which is a positive 4, when I add negative 4 and positive 4 together, I get zero. And that's basically the concept. Now you can have an additive inverse for an expression, it doesn't have to be just a number. So if I had an expression like 'x minus 2' and I wanted to find its additive inverse, I could multiply it by making negative 1 and that would give me the opposite of 'x minus 2'. So I'm going to distribute negative 1 across 'x minus 2'. Negative 1 times x is negative x and negative 1 times negative 2 is positive 2. So the additive inverse of 'x minus 2' is going to be 'negative x plus 2.' And that's the basic idea. Take care, I'll see you next time.
Common examples
For a number (and more generally in any ring), the additive inverse can be calculated using multiplication by −1; that is, −n = −1 × n. Examples of rings of numbers are integers, rational numbers, real numbers, and complex numbers.
Relation to subtraction
Additive inverse is closely related to subtraction, which can be viewed as an addition of the opposite:
- a − b = a + (−b).
Conversely, additive inverse can be thought of as subtraction from zero:
- −a = 0 − a.
Hence, unary minus sign notation can be seen as a shorthand for subtraction (with the "0" symbol omitted), although in a correct typography, there should be no space after unary "−".
Other properties
In addition to the identities listed above, negation has the following algebraic properties:
- −(−a) = a, it is an Involution operation
- −(a + b) = (−a) + (−b)
- −(a − b) = b − a
- a − (−b) = a + b
- (−a) × b = a × (−b) = −(a × b)
- (−a) × (−b) = a × b
- notably, (−a)2 = a2
Formal definition
The notation + is usually reserved for commutative binary operations (operations where x + y = y + x for all x, y). If such an operation admits an identity element o (such that x + o ( = o + x ) = x for all x), then this element is unique (o′ = o′ + o = o). For a given x, if there exists x′ such that x + x′ ( = x′ + x ) = o, then x′ is called an additive inverse of x.
If + is associative, i.e., (x + y) + z = x + (y + z) for all x, y, z, then an additive inverse is unique. To see this, let x′ and x″ each be additive inverses of x; then
- x′ = x′ + o = x′ + (x + x″) = (x′ + x) + x″ = o + x″ = x″.
For example, since addition of real numbers is associative, each real number has a unique additive inverse.
Other examples
All the following examples are in fact abelian groups:
- Complex numbers: −(a + bi) = (−a) + (−b)i. On the complex plane, this operation rotates a complex number 180 degrees around the origin (see the image above).
- Addition of real- and complex-valued functions: here, the additive inverse of a function f is the function −f defined by (−f )(x) = − f (x), for all x, such that f + (−f ) = o, the zero function (o(x) = 0 for all x).
- More generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group):
- Sequences, matrices and nets are also special kinds of functions.
- In a vector space, the additive inverse −v is often called the opposite vector of v; it has the same magnitude as the original and opposite direction. Additive inversion corresponds to scalar multiplication by −1. For Euclidean space, it is point reflection in the origin. Vectors in exactly opposite directions, but not necessarily the same magnitude, are sometimes referred to as antiparallel vectors.
- vector space-valued functions (not necessarily linear),
- In modular arithmetic, the modular additive inverse of x is also defined: it is the number a such that a + x ≡ 0 (mod n). This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3 + x ≡ 0 (mod 11).
Non-examples
Natural numbers, cardinal numbers and ordinal numbers do not have additive inverses within their respective sets. Thus one can say, for example, that natural numbers do have additive inverses, but because these additive inverses are not themselves natural numbers, the set of natural numbers is not closed under taking additive inverses.
See also
- −1
- Absolute value (related through the identity |−x| = |x|).
- Additive identity
- Group (mathematics)
- Monoid
- Inverse function
- Involution (mathematics)
- Multiplicative inverse
- Reflection (mathematics)
- Reflection symmetry
- Semigroup
Notes and references
- ^ Tussy, Alan; Gustafson, R. (2012), Elementary Algebra (5th ed.), Cengage Learning, p. 40, ISBN 9781133710790.
- ^ Brase, Corrinne Pellillo; Brase, Charles Henry (1976). Basic Algebra for College Students. Houghton Mifflin. p. 54. ISBN 978-0-395-20656-0.
...to take the additive inverse of the member, we change the sign of the number.
- ^ The term "negation" bears a reference to negative numbers, which can be misleading, because the additive inverse of a negative number is positive.
- ^ Weisstein, Eric W. "Additive Inverse". mathworld.wolfram.com. Retrieved 2020-08-27.
- ^ "Additive Inverse". www.learnalberta.ca. Retrieved 2020-08-27.