← −2 −1 0 →
−1 0 1 2 3 4 5 6 7 8 9 →
Cardinal−1, minus one, negative one
Ordinal−1st (negative first)
Divisors1
Arabic−١
Chinese numeral负一，负弌，负壹
Bengali−১
Binary (byte)
S&M: 1000000012
2sC: 111111112
Hex (byte)
S&M: 0x10116
2sC: 0xFF16

In mathematics, −1 (negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0.

Algebraic properties

Multiplication

Multiplying a number by −1 is equivalent to changing the sign of the number – that is, for any x we have (−1) ⋅ x = −x. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity:

x + (−1) ⋅ x = 1 ⋅ x + (−1) ⋅ x = (1 + (−1)) ⋅ x = 0 ⋅ x = 0.

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

0 ⋅ x = (0 + 0) ⋅ x = 0 ⋅ x + 0 ⋅ x.
0, 1, −1, i, and −i in the complex or cartesian plane

In other words,

x + (−1) ⋅ x = 0,

so (−1) ⋅ x is the additive inverse of x, i.e. (−1) ⋅ x = −x, as was to be shown.

Square of −1

The square 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

0 = −1 ⋅ 0 = −1 ⋅ [1 + (−1)].

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

0 = −1 ⋅ [1 + (−1)] = −1 ⋅ 1 + (−1) ⋅ (−1) = −1 + (−1) ⋅ (−1).

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

(−1) ⋅ (−1) = 1.

The above arguments hold in any ring, a concept of abstract algebra generalizing integers and real numbers.

Square roots of −1

Although there are no real square roots of −1, the complex number i satisfies i2 = −1, and as such can be considered as a square root of −1.[1] 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. In the algebra of quaternions – where the fundamental theorem does not apply – which contains the complex numbers, the equation x2 = −1 has infinitely many solutions.

Exponentiation to negative integers

Exponentiation of a non‐zero real number can be extended to negative integers. We make the definition that x−1 = 1/x, meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition is then extended to negative integers, preserving the exponential law xaxb = x(a + b) for real numbers a and b.

Exponentiation to negative integers can be extended to invertible elements of a ring, by defining x−1 as the multiplicative inverse of x.

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 of the function. For example, sin−1(x) is a notation for the arcsine function, and in general f −1(x) denotes the inverse function of f(x),. When a subset of the codomain is specified inside the function, it instead denotes the preimage of that subset under the function.

Uses

Sequences

Integer sequences commonly use −1 to represent an uncountable set, in place of "∞" as a value resulting from a given index.[2]

As an example, the number of regular convex polytopes in n-dimensional space is,

{1, 1, −1, 5, 6, 3, 3, ...} for n = {0, 1, 2, ...} (sequence A060296 in the OEIS).

−1 can also be used as a null value, from an index that yields an empty set ∅ or non-integer where the general expression describing the sequence is not satisfied, or met.[2]

For instance, the smallest k > 1 such that in the interval 1...k there are as many integers that have exactly twice n divisors as there are prime numbers is,

{2, 27, −1, 665, −1, 57675, −1, 57230, −1} for n = {1, 2, ..., 9} (sequence A356136 in the OEIS).

A non-integer or empty element is often represented by 0 as well.

Computing

In software development, −1 is a common initial value for integers and is also used to show that a variable contains no useful information.

See also

References

  1. ↑ Bauer, Cameron (2007). Algebra for Athletes (2nd ed.). Nova Science Publishers. p. 273. ISBN 978-1-60021-925-2.
  2. 1 2 See searches with "−1 if no such number exists" or "−1 if the number is infinite" in the OEIS for an assortment of relevant sequences.
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