Hypercomplex numbers (a+bi with a,b hyperreal)

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In summary, the field of "hypercomplex numbers" (C; a+bi with a,b hyperreal numbers) can satisfy the transfer principle and all arithmetical facts with complex numbers are also true with hypercomplex numbers. It is possible to construct a nonstandard complex analysis using hypercomplex numbers and obtain identical results to standard complex analysis. This approach is not limited to real numbers and encompasses all of analysis. However, it should be noted that the term "hypercomplex numbers" is commonly used with a different meaning and "complex numbers in nonstandard analysis" may be a more appropriate phrasing.
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lolgarithms
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Does the field of "hypercomplex numbers" (*C; a+bi with a,b hyperreal numbers) satisfy the transfer principle? Are all of the arithmetical facts with complex numbers true with the hypercomplex numbers?

One might be able to construct a nonstandard complex analysis with the hypercomplex numbers, with identical results as as standard complex analysis, with the standard part function st(z) = st(Re(z)) + i st(Im(z)) used to define complex derivatives and integrals.
 
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Yes, there is no problem doing what you want. I learned nonstandard analysis from Robinson's book, which is still a classic. From that approach, you see how it is not limited to real numbers, but encompasses all of analysis.

P.S. The term "hypercomplex numbers" is used more commonly with a completely different meaning. For what you want, maybe say "complex numbers in nonstandard analysis".
 

1. What are hypercomplex numbers?

Hypercomplex numbers are a generalization of complex numbers, which are numbers of the form a+bi, where a and b are real numbers and i is the imaginary unit. Hypercomplex numbers also include other types of units, such as j and k, which behave similarly to i but have different properties.

2. How are hypercomplex numbers different from complex numbers?

Hypercomplex numbers extend the idea of complex numbers to include additional units, such as j and k. This means that hypercomplex numbers have more dimensions than the two dimensions of a traditional complex plane. They also have different properties and behaviors compared to complex numbers.

3. What are hyperreal numbers?

Hyperreal numbers are a type of non-standard number that extends the real numbers to include infinitesimal and infinite quantities. They are used in the construction of hypercomplex numbers, which can have both real and hyperreal components.

4. How are hypercomplex numbers used in science?

Hypercomplex numbers have many applications in science, including in physics, engineering, and computer graphics. They are particularly useful for representing rotations and transformations in three-dimensional space, as well as in quantum mechanics and relativity.

5. Are there any real-world examples of hypercomplex numbers?

Yes, hypercomplex numbers have many real-world applications. For example, they are used in computer graphics to represent 3D rotations and transformations, in quantum mechanics to describe the spin of particles, and in electrical engineering to model electromagnetic fields. They are also used in the study of fractals and chaos theory.

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