−1

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For the 1963 short story, see Minus One.
−2 −1 0
−1 0 1 2 3 4 5 6 7 8 9
Cardinal −1, minus one, negative one
Ordinal −1st (negative first)
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 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.

Negative one bears relation to Euler's identity since e^{i \pi} = -1.\!

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

Negative one has some similar but slightly different properties to positive one.[1]

Algebraic properties[edit]

Multiplying a number by −1 is equivalent to changing the sign on the number. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity: for x real, we have

x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0 \cdot x=0

where we used the fact that any real x times 0 equals 0, implied by cancellation from the equation

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

In other words,

x+(-1)\cdot x=0 \,

so (−1) · x is the arithmetic inverse of x, or −x.

Square of −1[edit]

The square of −1, i.e. −1 multiplied by −1, equals 1. As a consequence, a product of two negative real numbers is positive.

For an algebraic proof of this result, start with the equation

0 =-1\cdot 0 =-1\cdot [1+(-1)]

The first equality follows from the above result. The second follows from the definition of −1 as additive inverse of 1: it is precisely that number that when added to 1 gives 0. Now, using the distributive law, we see that

0 =-1\cdot [1+(-1)]=-1\cdot1+(-1)\cdot(-1)=-1+(-1)\cdot(-1)

The second equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies

(-1) \cdot (-1) = 1

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

Square roots of −1[edit]

The complex number i satisfies i2 = −1, and as such can be considered as a square root of −1. The only other complex number x satisfying the equation x2 = −1 is −i.[2] In the algebra of quaternions, containing the complex plane, the equation x2 = −1 has an infinity of solutions.

Exponentiation to negative integers[edit]

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 then extended to negative integers preserves the exponential law xaxb = x(a + b) for a,b real numbers.

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

−1 that appears next to functions or matrices does not mean raising them to the power −1 but their inverse functions or inverse matrices. For example, f−1(x) is the inverse of f(x), or sin−1(x) is a notation of arcsine function.

Computer representation[edit]

Most computer systems represent negative integers using two's complement. In such systems, −1 is represented using a bit pattern of all ones. For example, an 8-bit signed integer using two's complement would represent −1 as the bitstring "11111111", or "FF" in hexadecimal (base 16). If interpreted as an unsigned integer, the same bitstring of n ones represents 2n − 1, the largest possible value that n bits can hold. For example, the 8-bit string "11111111" above represents 28 − 1 = 255.

References[edit]

  1. ^ Mathematical analysis and applications By Jayant V. Deshpande, ISBN 1-84265-189-7
  2. ^ "Ask Dr. Math". Math Forum. Retrieved 2012-10-14.