mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, −1 (also known as negative one or minus one) is the

additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (op ...

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 ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elemen ...

element, 0. It is the negative

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

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

Algebraic properties

Multiplying a number by −1 is equivalent to changing the

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or ...

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 generalizes 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 arithmetic, ...

and the axiom that 1 is the

multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

: :. Here we have used the fact that any number times 0 equals 0, which follows by cancellation from the equation :. ImaginaryUnit5

In other words, :, so is the additive inverse of , i.e. , as was to be shown.

Square of −1

The

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

of −1, i.e. −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, we see 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. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...

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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

Square roots of −1

Although there are no

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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 fo ...

satisfies , and as such can be considered as a

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

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 The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...

. 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. Hamilton defined a quater ...

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

Exponentiation to negative integers

Exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...

of a non‐zero real number can be extended to negative integers. We make the definition that , meaning that we define raising a number to the power −1 to have the same effect as taking its

reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...

. This definition is then extended to negative integers, preserving the exponential law for real numbers and . Exponentiation to negative integers can be extended to invertible elements of a ring, by defining as the multiplicative inverse of . A −1 that appears as a superscript of a function does not mean taking the (pointwise) reciprocal of that function, but rather 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 X ...

of the function. For example, is a notation for the

arcsine In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...

function, and in general denotes the inverse function of ,. When a subset of the

codomain In mathematics, the codomain or set of destination of a function is the 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 refer to either th ...

is specified inside the function, it instead denotes the

preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...

of that subset under the function.

Uses

* In

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, −1 is a common initial value for integers and is also used to show that a variable contains no useful information. * −1 bears relation to

Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...

since .

References