Cambridge Studies in Advanced Mathematics. Clifford Algebras and the Classical Groups (PDF). Reston, VA: National Council of Teachers of Mathematics. Historical Topics for the Mathematical Classroom. In software development, −1 is a common initial value for integers and is also used to show that a variable contains no useful information. (sequence A356136 in the OEIS).Ī non-integer or empty element is often represented by 0 as well. Īs an example, the number of regular convex polytopes in n-dimensional space is, Integer sequences commonly use −1 to represent an uncountable set, in place of " ∞" as a value resulting from a given index. More specifically, because the polynomial is not continuous, it is not a unit in F. Which is not possible, and therefore, F is not a field. If it did have an inverse q( x), then there would be x q( x) = 1 ⇒ deg ( x) + deg ( q( x)) = deg (1) ⇒ 1 + deg ( q( x)) = 0 ⇒ deg ( q( x)) = −1 In a polynomial domain F over any field F, the polynomial x has no inverse. When a subset of the codomain is specified inside the function f, its inverse will yield an inverse image, or preimage, of that subset under the function.Įxponentiation to negative integers can be further extended to invertible elements of a ring by defining x −1 as the multiplicative inverse of x in this context, these elements are considered units. Where f is bijective specifying an output codomain of every y ∈ Y from every input domain x ∈ X, there will be This definition is then applied to negative integers, preserving the exponential law x a x b = x ( a + b) for real numbers a and b.Ī −1 superscript in f −1( x) takes the inverse function of f( x), where ( f( x)) −1 specifically denotes a pointwise reciprocal. Inverse and invertible elements The reciprocal function f( x) = x −1 where for every x except 0, f( x) represents its multiplicative inverseĮxponentiation 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 the algebra of quaternions – where the fundamental theorem does not apply – which contains the complex numbers, the equation x 2 = −1 has infinitely many solutions. The only other complex number whose square is −1 is − i because there are exactly two square roots of any non‐zero complex number, which follows from the fundamental theorem of algebra. : p.48 Square roots of −1 Īlthough there are no real square roots of −1, the complex number i satisfies i 2 = −1, and as such can be considered as a square root of −1. The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers. But now adding 1 to both sides of this last equation implies The third equality follows from the fact that 1 is a multiplicative identity. 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. As a consequence, a product of two negative numbers is positive.įor an algebraic proof of this result, start with the equation So (−1) ⋅ x is the additive inverse of x, i.e. 0, 1, −1, i, and − i in the complex or cartesian plane Here we have used the fact that any number x times 0 equals 0, which follows by cancellation from the equationĠ ⋅ x = (0 + 0) ⋅ x = 0 ⋅ x + 0 ⋅ x. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity: Multiplying a number by −1 is equivalent to changing the sign of the number – that is, for any x we have (−1) ⋅ x = − x.
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