Split-complex numbers and dual numbers

• ForMyThunder
In summary, the term "modulus" has different definitions depending on the number system being used. In the case of real, complex, and split-complex numbers, the modulus is defined as z times its conjugate. However, in the dual numbers, the modulus is simply equal to the real component of the number. Although the split-complex modulus is not a norm, it is still multiplicative and a quadratic form.
ForMyThunder
I recently had a conversation with a friend (who for some reason likes number theory) about these split-complex numbers and dual numbers. I'm more into topology, so I've never heard of them and he brought up that the modulus (I've only heard this term used for complex numbers) of the split-complex numbers can be negative. I made the guess that the modulus was a norm. He said it wasn't and I asked him what exactly a modulus was and he said it was just some real-valued function.

So, in general what is the definition of a "modulus" that coincides its definitions on the reals, complex, dual, and split complex numbers?

As I have come across the term in these contexts the modulus isn't just "some" real-valued function but rather the function
$$\|z\| = z\overline{z}$$
where $\overline{z}$ denotes the conjugate, so:
$$\overline{a} = a\qquad\textrm{for real a}$$
$$\overline{a+bi} = a-bi\qquad\textrm{for complex a+bi}$$
$$\overline{a+bj} = a-bj\qquad\textrm{for split complex a+bj}$$
$$\overline{a+b\epsilon} = a-b\epsilon\qquad\textrm{for dual }a+b\epsilon$$
So in the split complex case we have
$$\|a+bj\| = (a+bj)(a-bj) = a^2 -b^2$$
which is not a norm, but it is multiplicative and a quadratic form. In the dual case we have
$$\|a+b\epsilon\| = (a+b\epsilon)(a-b\epsilon) = a^2$$

1. What are split-complex numbers?

Split-complex numbers, also known as hyperbolic numbers, are a type of number system that extends the complex numbers by adding a second imaginary unit, typically denoted by j. These numbers have the form a + bj, where a and b are real numbers and j satisfies the equation j^2 = 1. Unlike complex numbers, split-complex numbers do not satisfy the commutative property of multiplication.

2. How are split-complex numbers used?

Split-complex numbers have applications in various fields of mathematics, physics, and engineering. They are used to represent rotations and translations in 2D and 3D spaces, as well as in the study of hyperbolic geometry. They are also used in special relativity, quantum mechanics, and signal processing.

3. What are dual numbers?

Dual numbers are another type of number system that extends the real numbers by adding an infinitesimal element, typically denoted by ε. These numbers have the form a + bε, where a and b are real numbers and ε satisfies the equation ε^2 = 0. Dual numbers are often used in differential calculus to represent and manipulate infinitesimal quantities.

4. How are dual numbers related to split-complex numbers?

Dual numbers can be seen as a special case of split-complex numbers, where the real part is represented by a and the imaginary part by bε. In other words, dual numbers have the form a + bj, where b is a real number multiplied by an infinitesimal ε. This relationship allows for the use of split-complex arithmetic in calculations involving dual numbers.

5. Can split-complex and dual numbers be visualized?

Yes, split-complex and dual numbers can be visualized on a number line or in a 2D plane. In the case of split-complex numbers, the real part is represented on the horizontal axis and the imaginary part on the vertical axis. Similarly, dual numbers can be visualized as a point on a number line, where the real part is represented by the horizontal distance and the infinitesimal part by the vertical distance from the origin.

• General Math
Replies
1
Views
188
• Linear and Abstract Algebra
Replies
1
Views
913
• General Math
Replies
13
Views
3K
• General Math
Replies
2
Views
387
• General Math
Replies
3
Views
2K
• Linear and Abstract Algebra
Replies
1
Views
750
• Classical Physics
Replies
13
Views
897
• General Math
Replies
7
Views
6K
• General Math
Replies
17
Views
2K
• Topology and Analysis
Replies
4
Views
744