Abnormal complex number

Binary number is a redirect to this article. For binary numbers in the binary system, see dual system .

The abnormally complex numbers are an extension of the real numbers , which differs from that of the complex numbers in that the product of their non-real unit with itself is not equal to −1 but equal to +1. ${\displaystyle \mathbb {R} }$

definition

The abnormally complex numbers (English split-complex numbers or hyperbolic numbers; for reasons see below ) form a two-dimensional hypercomplex algebra over the field of real numbers. Like the algebra of complex numbers , this algebra is generated by two basic elements, the 1 and a non-real unit, which is referred to here as to distinguish it from the imaginary unit of complex numbers . Every abnormally complex number can therefore be clearly identified as ${\displaystyle \mathbb {R} }$ ${\displaystyle i}$ ${\displaystyle j}$

${\displaystyle z=a+b\cdot j}$

with , i.e. as a linear combination of 1 and . The definition of a general multiplication for abnormally complex numbers is completed by a definition for the square of the non-real unit, namely by ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle j}$

${\displaystyle j^{2}=1,}$ so ${\displaystyle (1+j)(1-j)=0,}$

whereby, of course, must be observed. It is also analogous to the complex numbers to having conjugated number ${\textstyle j\neq \pm 1}$ ${\displaystyle z=a+b\cdot j}$ ${\displaystyle a,b\in \mathbb {R} }$

${\displaystyle {\bar {z}}:=a-b\cdot j}$

Are defined.

properties

Like all hyper -complex algebras , the abnormally complex numbers also satisfy the right-hand and left-hand distributive law . Like the complex numbers , they are also commutative and associative, and inevitably because there is only one basic element other than 1, namely ${\displaystyle j:}$

${\displaystyle 1\cdot j=j\cdot 1=j}$

${\displaystyle 1\cdot (1\cdot j)=1\cdot j=(1\cdot 1)\cdot j=j}$

${\displaystyle 1\cdot (j\cdot j)=1\cdot 1=1=j\cdot j=(1\cdot j)\cdot j}$

The abnormally complex numbers thus form a commutative ring with one element , which - in contrast to  - is not a body, but a main ideal ring with two nontrivial ideals , the real-number multiples of and those of , i.e. the diagonals of the number plane running through the origin . They are the main ideals since they are each generated by a single element. They are both zero divisors , because 0 is the product of any element of one ideal with any element of the other: ${\displaystyle \mathbb {C} }$ ${\displaystyle 1+j}$ ${\displaystyle 1-j}$

${\displaystyle a(1+j)\cdot b(1-j)=ab(1^{2}-j^{2})=ab(1-1)=ab\cdot 0}$

A norm or an amount is not defined for abnormally complex numbers, but there are two properties that are "passed on" during multiplication, such as the norm for complex numbers or the determinant for matrices (in the sense of "norm / determinant of the Product equals product of the norms / determinants of the factors "):

1. The sum of the real and non-real parts (because when multiplied it behaves like 1) ${\displaystyle j}$
2. The product of a number (as above) with its conjugate: ${\displaystyle z}$

${\displaystyle z{\bar {z}}=(a+b\cdot j)(a-b\cdot j)=a^{2}-b^{2}\cdot j^{2}=a^{2}-b^{2},}$

which always results in a real number. This is

1. negative for ${\displaystyle |a|<|b|}$
2. equal to zero for ${\displaystyle |a|=|b|}$
3. positive for ${\displaystyle |a|>|b|.}$

Just as all complex numbers of the same amount lie on a circle, all abnormally complex numbers whose product with their conjugate has a fixed value lie on a hyperbola; therefore they are also called “hyperbolic numbers” in English. Thus the abnormally complex numbers follow a Minkowski metric such as time (= real axis) and spatial direction (= non-real axis) in the special theory of relativity . In describing the classic real Minkowski plane , the abnormally complex numbers play an analogous role as the complex numbers in the description of the classic real Möbius plane .

See also

• Dual number

literature

• IL Kantor, AS Solodownikow: Hypercomplex numbers. BG Teubner, Leipzig 1978.
• Walter Benz : Lectures on the geometry of algebras. Geometries by Möbius, Laguerre-Lie, Minkowski in uniform and basic geometric treatment. Springer, 1973, ISBN 9783642886706 , pp. 43-47.

supporting documents

1. ^ Walter Benz : Lectures on the geometry of algebras. Springer, 1973, ISBN 9783642886706 , p. 45.