mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, −1 (negative one or minus one) is the

additive inverse In mathematics, the additive inverse of an element , denoted , is the element that when added to , yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero el ...

of 1, that is, the number that when added to 1 gives the

additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element in the set, yields . One of the most familiar additive identities is the number 0 from elementary ma ...

element, 0. It is the negative

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

greater than negative two (−2) and less than 0.

In mathematics

Algebraic properties

Multiplying a number by −1 is equivalent to changing the sign of the number – that is, for any we have . This can be proved using the

distributive law In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary ...

and the axiom that 1 is the

multiplicative identity In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

: :. Here we have used the fact that any number times 0 equals 0, which follows by cancellation from the equation :. In other words, :, so is the additive inverse of , i.e. , as was to be shown. The

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

of −1 (that is −1 multiplied by −1) equals 1. As a consequence, a product of two negative numbers is positive. For an algebraic proof of this result, start with the equation :. The first equality follows from the above result, and the second follows from the definition of −1 as additive inverse of 1: it is precisely that number which when added to 1 gives 0. Now, using the distributive law, it can be seen that :. The third equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies :. The above arguments hold in any ring, a concept of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

generalizing integers and

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

Although there are no real square roots of −1, the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

satisfies , and as such can be considered as a

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

of −1. The only other complex number whose square is −1 is − because there are exactly two square roots of any non‐zero complex number, which follows from the fundamental theorem of algebra. In the algebra of

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

s – where the fundamental theorem does not apply – which contains the complex numbers, the equation has infinitely many solutions.

Inverse and invertible elements

Exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...

of a non‐zero real number can be extended to negative integers, where raising a number to the power −1 has the same effect as taking its

multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...

: :. This definition is then applied to negative integers, preserving the exponential law for real numbers and . A −1

superscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, wh ...

in takes the

inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...

of , where specifically denotes a

pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some Function (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...

reciprocal. Where is

bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

specifying an output

codomain In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...

of every from every input domain , there will be : and . When a subset of the codomain is specified inside the function , its inverse will yield an

inverse image In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each ...

, or preimage, of that subset under the function. Exponentiation to negative integers can be further extended to invertible elements of a ring by defining as the multiplicative inverse of ; in this context, these elements are considered units. In a polynomial domain over any field , the polynomial has no inverse. If it did have an inverse , then there would be : : : which is not possible, and therefore, is not a field. More specifically, because the polynomial is not constant, it is not a unit in .

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