−1
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|---|---|---|---|---|---|
| Cardinal | −1, minus one, negative one | ||||
| Ordinal | −1st (negative first) | ||||
| Divisors | 1 | ||||
| Arabic | −١ | ||||
| Chinese numeral | 负一，负弌，负壹 | ||||
| Bengali | −১ | ||||
| Binary (byte) |
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| Hex (byte) |
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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.
In mathematics
[edit]Algebraic properties
[edit]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.
The square 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
- 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
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
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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 y ∈ Y from every input domain x ∈ X, 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.
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 constant, it is not a unit in F.
In other uses
[edit]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 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,
A non-integer or empty element is often represented by 0.
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]
See also
[edit]
References
[edit]Notes
[edit]Sources
[edit]- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.