minus one

−1

−1 is the additive inverse of 1 in mathematics , that is, when it is added to 1, one obtains the neutral element of addition  0 . It is a negative integer that is greater than minus two (−2) and less than zero .

Minus one has some properties that are similar but slightly different from positive one.

−1 is related to Euler's identity since   e ^iπ  = −1.  

In computer science is a common initial value -1 for such Integer - variables whose values typically are not negative, thereby indicating that the variable (still) does not contain any meaningful information.

Algebraic properties

Multiplying a number by −1 is equivalent to changing the sign. This can be shown by means of the distributive law and the axiom that 1 is the neutral element of the multiplication : For a real number  x holds

${\displaystyle x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0\cdot x=0}$

taking advantage of the fact that a real number x times 0 is equal to 0, which results from the shortening of the following equation

${\displaystyle 0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x\,}$

In other words

${\displaystyle x+(-1)\cdot x=0}$,

so (−1) x is the additive inverse of x or - x .

Squaring −1

The square of −1 (that is, −1 times −1) is equal to 1. In the sequence, a product of two negative numbers is positive.

To prove this algebraically, one starts with the equation

${\displaystyle 0=-1\cdot 0=-1\cdot [1+(-1)]}$

The first equation follows from the above result. The second follows from the definition of −1 as the additive inverse of 1: It is exactly the number that results in 0 when it is added to 1. By applying the distributive law one sees

${\displaystyle 0=-1\cdot [1+(-1)]=-1\cdot 1+(-1)\cdot (-1)=-1+(-1)\cdot (-1)}$.

The second equation follows from the fact that 1 is the neutral element of the multiplication. Adding 1 on either side of the last equation follows

${\displaystyle (-1)\cdot (-1)=1}$.

The above implications are also valid in any ring that generalizes the abstract algebra of whole and real numbers.

Square root of −1

The complex number i satisfies   i ² = −1 and is therefore regarded as the square root of −1. The only other complex number x that satisfies the equation   x ² = −1 is −i. In the quaternion algebra, which contains the complex plane, the equation   x ² = −1 has   an infinite number of solutions .

Powers of negative integers

The power of real numbers without zero can be expanded to negative exponents . It is defined x ⁻¹  = 1 / x ; that is, if a number is raised to the power of −1, one gets its reciprocal value . If this definition is extended to negative integers, the exponential law for real numbers a , b not equal to 0 is retained:   x ^a x ^b  =  x ^{( a  +  b )}

Powers with negative exponents can be extended to the invertible elements of a ring by defining x ⁻¹ as the inverse element of the multiplication by  x .

Binary representation in the computer

There are a number of different representations of −1 and negative integers in general on computer systems. The most commonly used is the two's complement of its positive form. Minus one has the same representation in two's complement as the positive integer 2 ⁿ  - 1, where n is the number of binary digits in the representation (the number of bits in the data type). For example, 11111111 ₂ ( binary ) or FF ₁₆ ( hex ) for n  = 8 represents the number −1 in two's complement, but 255 in the standard representation.

See also

• Menelaus theorem

Individual evidence

1. ↑ Jayant V. Deshpande: Mathematical analysis and applications , ISBN 1842651897
2. ↑ mathforum.org