[Submitted on 16 Aug 2000]
Authors:Silviu Olariu (National Institute of Physics and Nuclear Engineering, Tandem Laboratory, Magurele, Bucharest, Romania)
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Abstract: A system of commutative hypercomplex numbers of the form w=x+hy+kz are introduced in 3 dimensions, the variables x, y and z being real numbers. The multiplication rules for the complex units h, k are h^2=k, k^2=h, hk=1. The operations of addition and multiplication of the tricomplex numbers introduced in this paper have a simple geometric interpretation based on the modulus d, amplitude \rho, polar angle \theta and azimuthal angle \phi. Exponential and trigonometric forms are obtained for the tricomplex numbers, depending on the variables d, \rho, \theta and \phi. The tricomplex functions defined by series of powers are analytic, and the partial derivatives of the components of the tricomplex functions are closely related. The integrals of tricomplex functions are independent of path in regions where the functions are regular. The fact that the exponential form of the tricomplex numbers contains the cyclic variable \phi leads to the concepts of pole and residue for integrals of tricomplex functions on closed paths. The polynomials of tricomplex variables can be written as products of linear or quadratic factors.
Submission history
From: Silviu Olariu [view email]
[v1] Wed, 16 Aug 2000 17:40:58 UTC (80 KB)
FAQs
The answer is no. The only sets of numbers which satisfy all the usual rules of elementary algebra (that is satisfy the field axioms) have dimension one or two. We can define division of complex numbers but we cannot define division of three dimensional vectors.
Are complex functions 3 dimensional? ›
Every complex number has both a real part and an imaginary part, so one complex variable is two-dimensional and a BBC pair of complex variables is four-dimensional.
How many dimensions does a complex number have? ›
Complex numbers in the coordinate plane. Each complex number x + yi corresponds to a number pair (x, y) in the plane, so we may say that the complex numbers form a two-dimensional collection. The two coordinates of the pair (x, y) are called the real part and the imaginary part of the complex number.
Can numbers be three dimensional? ›
No, but there is a four-dimensional analogue, the Quaternion . Where a complex number is represented as a+bi a + b i , an quaternion is represented as a+bi+cj+dk a + b i + c j + d k , where i2=j2=k2=ijk=−1 i 2 = j 2 = k 2 = i j k = − 1 .
Do we exist in 3 dimensions? ›
We live in a world of three dimensions.
We move through space , either left or right, forward or backward, up or down. Everything around us, from the houses we live in to the objects we use in everyday life, has three dimensions: height, length, and width.
Are there really more than 3 dimensions? ›
Current versions of string theory require 10 dimensions total, while an even more hypothetical über-string theory known as M-theory requires 11. But when we look around the universe, we only ever see the usual three spatial dimensions plus the dimension of time.
Is the complex plane 4d? ›
The complex plane is a 2 dimensional plane which enables us to visualise complex numbers in a Cartesian coordinate system by using the x-axis to represent the real part and the y-axis to represent the imaginary part. The complex plane is also commonly known as the Argand diagram.
What are the 3 dimensions in 3D? ›
Dimensions in 3D Space
All 3D models will have three measurements to them: height, width, and depth.
What is not three-dimensional? ›
A solid shape that has length, breadth and height (depth) is known as a three-dimensional shape. So, cube, cone and cylinder are three dimensional, while the parallelogram is a two-dimensional shape.
Are there infinite complex numbers? ›
When talking about z=∞, we are referring to something called complex infinity, which can be regarded as a complex number with infinite magnitude and undefined argument.
The short answer is this: Yes, the complex numbers are closed for polynomials using the algebraic operations of addition, multiplication, subtraction, and division. Yes, there are larger domains than the complex numbers, such as the quaternions and octonians.
What are 8 dimensional complex numbers? ›
Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
Why are there no 3-dimensional complex numbers? ›
The answer is that such a system is impossible to exist, at least if we want it to have certain 'nice' properties. Hamilton couldn't find a way to divide his 3D numbers and it was later proven that this is exactly the point where a possible three dimensional extension of the complex numbers breaks down.
Do we see in 3 dimensions? ›
We are 3D creatures, living in a 3D world but our eyes can show us only two dimensions. The depth that we all think we can see is merely a trick that our brains have learned; a byproduct of evolution putting our eyes on the front of our faces.
What do the 3 numbers in dimensions mean? ›
Length, width, and height are the three dimensions of an object. They are measured in units such as inches or centimeters.
Is the complex plane 2d or 3D? ›
The complex plane is a two dimensional real vector space (using the natural identification (x,y)=x+iy).
What are the three forms of complex numbers? ›
Complex numbers have three primary forms: the general form, z=a+ib; the polar form, z=r(cosθ+isinθ); and the exponential form, z=rexp(iθ).
Is there a set beyond complex numbers? ›
The quaternions are an extension of the complex numbers, and the octonions are a further extension. In a sense you pay a price for making these extensions, though. The reals are ordered; the complex numbers are not ordered as a field because to be an ordered field the square of any value can't be negative.
Can 3 be a complex number? ›
That fact is that integers are also complex numbers! For example, 3 is a complex number because it can be written as 3+0i. You can also have negative complex numbers!
