The discussion explores the concept of extending complex numbers beyond two dimensions, specifically considering whether there are mathematical structures that can represent three or more dimensions. Participants examine the relationship between complex numbers, quaternions, and higher-dimensional extensions.
Participants express varying views on the dimensionality of complex numbers, with some supporting the idea of quaternions as a four-dimensional extension while others focus on the potential for three-dimensional representations. The discussion remains unresolved regarding the existence and utility of these higher-dimensional extensions.
Some limitations include the lack of consensus on definitions of dimensionality in this context and the historical usage of these mathematical structures, which may affect their acceptance and application.
thetexan said:Complex numbers ##a+bi## can be thought of as a second dimension extension of the real number line.
Is there a third dimension version of this? Are there even more complex numbers that not only extend into the y-axis but also the z axis?
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Ssnow said:giving rules for theaddictionaddition and multiplication
thetexan said:So is it correct to say that the single dimension real numbers can be considered a subset of the two dimensional complex number set where the real numbers are complex numbers with a 0 imaginary component?
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