# −1

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.

 ← −2 −1 0 →
Cardinal−1, minus one, negative one
Ordinal−1st (negative first)
Divisors1
Arabic١
Chinese numeral负一，负弌，负壹
Bengali
Binary (byte)
 S&M: 1000000012 2sC: 111111112
Hex (byte)
 S&M: 0x10116 2sC: 0xFF16

## 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.

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.[1]: p.48

### 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.[2] 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.[3][4]

## Inverse and invertible elements

Exponentiation 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:

x−1 = 1/x.

This definition is then applied to negative integers, preserving the exponential law xaxb = x(a + b) for real numbers a and b.

A −1 superscript in f −1(x) takes the inverse function of f(x), where ( f(x))−1 specifically denotes a pointwise reciprocal.[a] Where f is bijective specifying an output codomain of every yY from every input domain xX, there will be

f −1( f(x)) = x,  and f −1( f(y)) = y.

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.

### Rings

Exponentiation 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.[1]: p.49

In a polynomial domain F [x] over any field F, the polynomial x has no inverse. If it did have an inverse q(x), then there would be[5]

x q(x) = 1 ⇒ deg (x) + deg (q(x)) = deg (1)
⇒ 1 + deg (q(x)) = 0
⇒ deg (q(x)) = −1

which is not possible, and therefore, F [x] is not a field. More specifically, because the polynomial is not continuous, it is not a unit in F.

## Uses

### Sequences

Integer sequences commonly use −1 to represent an uncountable set, in place of "" as a value resulting from a given index.[6]

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.[6]

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.[citation needed]

## References

1. ^ For example, sin−1(x) is a notation for the arcsine function.
1. ^ a b Nathanson, Melvyn B. (2000). "Chapter 2: Congruences". Elementary Methods in Number Theory. Graduate Texts in Mathematics. Vol. 195. New York: Springer. pp. xviii, 1−514. doi:10.1007/978-0-387-22738-2_2. ISBN 978-0-387-98912-9. MR 1732941. OCLC 42061097.
2. ^ Bauer, Cameron (2007). "Chapter 13: Complex Numbers". Algebra for Athletes (2nd ed.). Hauppauge: Nova Science Publishers. p. 273. ISBN 978-1-60021-925-2. OCLC 957126114.
3. ^ Perlis, Sam (1971). "Capsule 77: Quaternions". Historical Topics in Algebra. Historical Topics for the Mathematical Classroom. Vol. 31. Reston, VA: National Council of Teachers of Mathematics. p. 39. ISBN 9780873530583. OCLC 195566.
4. ^ Porteous, Ian R. (1995). "Chapter 8: Quaternions". Clifford Algebras and the Classical Groups (PDF). Cambridge Studies in Advanced Mathematics. Vol. 50. Cambridge: Cambridge University Press. p. 60. doi:10.1017/CBO9780511470912.009. ISBN 9780521551779. MR 1369094. OCLC 32348823.
5. ^ Czapor, Stephen R.; Geddes, Keith O.; Labahn, George (1992). "Chapter 2: Algebra of Polynomials, Rational Functions, and Power Series". Algorithms for Computer Algebra (1st ed.). Boston: Kluwer Academic Publishers. pp. 41, 42. doi:10.1007/b102438. ISBN 978-0-7923-9259-0. OCLC 26212117. S2CID 964280. Zbl 0805.68072 – via Springer.
6. ^ a b 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.