Transformations of Complex, Dual, and Hyperbolic Numbers

This Demonstration compares the multiplicative transformation properties on the plane of complex, dual, and hyperbolic numbers. These number types share the form , where and are real numbers and has the defining characteristics: for complex numbers, for dual numbers, and for hyperbolic numbers. Their conjugates share the form and their modulus the form The figure shows, for the selected number type,
1. a transform point on its unit-modulus locus (in blue),
2. an object point in the plane (in red), and
3. the point resulting from their multiplicative transform on its trajectory (in brown).
In the complex plane, multiplying by a unit-modulus transform produces a rotation about the origin; in the dual plane, it produces a translation in the pure dual direction, and in the hyperbolic plane, a traversal of an hyperbola that passes through the object point.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Hyperbolic numbers (sometimes called split-complex numbers) are useful for measuring distances in the Lorentz space-time plane (see G. Sobczyk, "The Hyperbolic Number Plane," The College Mathematics Journal, 26(4), 1995 pp. 268–280).
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+