4DLab - Critical Thinking  

How to use the Transcomplex Calculator

Preliminaries

For a complete description of the transcomplex numbers system download the book Foundations of the Transcomplex Numbers System in this link.

For a short introduction of the multiplication rules of the  transcomplex numbers read the short comparison between the quaternions and the transcomplexs  in this link.

To download the Transcomplex Numbers Calculator follow this link.

Using the Transcomplex Calculator

This calculator is a specialized scientific calculator programmed to make multiplications only, but for 2-dimensional complex numbers or to multiply 4-dimensional transcomplex numbers. Therefore, you can use it to multiply any kind of number.

First of all, keep in mind that every kind of number, like the real numbers, the imaginary numbers, and the common complex numbers are a subclasses of the transcomplex numbers. For that reason the Transcomplex Numbers Calculator is also useful for multiplying any kind of numbers.

The calculator has three horizontal panels: two for the transcomplex factors, and one for the result of the multiplication.

Transcomplex numbers are 4-dimensional complex numbers. For that reason each of the three panels of the the calculator has four small windows.

The two multiplicands are labeled as Transcomplex 1 and Transcomplex 2. The result is given in the panel labeled Transcomplex Product.

Above the boxes for the factor entries there are the unitary numbers for each one of the four entries of the transcomplex number. They are:

• 1 is the unitary basis of the real numbers.
• 1^~ is the unitary basis of the image-real numbers.
• i is the unitary basis of the imaginary numbers.
• i^~ is the unitary basis of the image-imaginary numbers.

Thus, the simple complex number 2 + 3i, which is also 2*1 + 3*i is also the transcomplex number 2*1 + 0*1^~ + 3*i + 0 *i^~. The notation of the transcomplexs is simplified when using the ordered pair notation. In the example just given, we have 2 + 3i = (2, 0, 3, 0). It is implicit that the order of the entries is the same order of the unitary basis.

There are many combinations of multiplications. What follows is just an example.

To multiply numerical values

First select the Numeric option. The 0/1 check box is available when using numerical values only. The use of this box is that double clicking on it changes an entry from 0 to 1 or vice versa.

• To multiply two real numbers: Enter each factor in the 1 unitary boxes and press the Multiply button.
• To multiply two imaginary numbers: Enter each factor in the i unitary boxes and press the Multiply button.
• To multiply two complex numbers: Enter the real part of in the 1 unitary box of each factor, enter the imaginary part in the i unitary box of each factor and press the Multiply button.
• To multiply a real number by an imaginary number: Enter the real factor into one of the 1 unitary box, enter the imaginary number in the i box of the other factor, press the Multiply button.

To multiply alphabetic values

First select the Alphabetic option. The 0/1 check box is not available when using alphabetic values. Talking of alphabetic values is the same as talking of alphabetic variables.

For alphabetic multiplication follow the same examples and rules as for numerical values.

For example, the complex number x + yi is the same as the transcomplex number (x, 0, y, 0), so to find the square of this number enter it twice and press the Multiply button. The result of this example is the complex number xx - yy + (xy +yx )i. The parentheses are added here to avoid confusion. In simplified notation: (x + yi)² = x² - y² +2xyi.