−1

Last updated • 3 min readFrom Wikipedia, The Free Encyclopedia
−2−1 0
−1 0 1 2 3 4 5 6 7 8 9
Cardinal −1, minus one, negative one
Ordinal −1st (negative first)
Divisors 1
Arabic ١
Chinese numeral 负一,负弌,负壹
Bengali
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.

Contents

In mathematics

Algebraic properties

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 = 1x + (−1)x = (1 + (−1))x = 0x = 0.

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

0x = (0 + 0)x = 0x + 0x.

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

0, 1, -1, i, and -i in the complex or Cartesian plane ImaginaryUnit5.svg
0, 1, −1, i , and −i in the complex or Cartesian plane

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

The reciprocal function f(x) = x where for every x except 0, f(x) represents its multiplicative inverse Geogebra f(x)=1/x 20211118.svg
The reciprocal function f(x) = x where for every x except 0, f(x) represents its multiplicative inverse

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.

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]

xq(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

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.

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

Related Research Articles

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<span class="mw-page-title-main">Imaginary unit</span> Principal square root of −1

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References

Notes

  1. For example, sin−1(x) is a notation for the arcsine function.

Sources

  1. 1 2 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. 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.