I heard of Hypercomplex numbers like quaternions but whereever I search for them its unintelligible…
In: 5
(also answer for /u/JudgeAdvocateDevil )
There are no technical definitions of numbers. Mathematicians just went “hey this sounds similar enough to what we had called ‘number’ so far, so let’s call these things ‘numbers’ too”. So there are really no limitations or consistency in what are called numbers. Late 19th century-early 20th century have a lot of mathematicians devising new algebraic system and calling them ‘number’. Eventually people stopped that and start calling them “algebra” instead, but previously named ‘number’ retain their historical names.
There are very little commonalities between different systems of numbers. A common pattern is as follow: you have a system consisting of objects and 2 operations on them, called addition and multiplication. These operations needs to satisfy a few expected properties that we commonly know: addition is commutative and associative, multiplication is distributive over addition. It also needs to contains an older previously accepted system of number as a subset.
And…that’s it. Note that many other properties we commonly see are not required, such as multiplication being associative. There are also no “chain” of increasingly larger system of number, a lot of time these system are incompatible.
So let me list a few different flavors:
– Discrete extension of natural number into the infinite realm: ordinals, cardinals. These numbers allows you to perform induction into infinity or to count sets of infinite size.
– Extension of real number to includes infinity and infinitesimal: hyperreal, surreal. Surreal is biggest and including everything of this kind; it also has exponentiation. This allows you to measure comparable quantities that can be infinitely large.
– Extension of complex number to includes infinity and infinitesimal: hypercomplex, surcomplex. If you extend real numbers, you can extend complex as well.
– Composition algebra: complex, quaternion, octonion, split-complex, bicomplex, etc. They’re mostly defined algebraically without an intrinsic meaning to them, though some were used as an earlier form of Clifford algebra. The main property is that when you multiply 2 numbers, their “length” also get multiplied.
– Geometrically-motivated extension: Clifford algebra, Grassman number. These numbers correspond to geometric objects, higher dimensional version of vectors. If you think of vectors as something with a magnitude, an orientation, and a line, then these numbers replace line with plane, 3D spaces, and so on.
– Completion of rational number under other metric: p-adic number, real number. It’s proved that the only other sensible metric for rational number is p-adic metric, and filling “holes” in the rational number as defined by this metric creates p-adic numbers (metric means a way to measuring distance between numbers). They have applications in number theory, since they correspond to the problem to finding solutions in modular arithmetic.
– Analog in different system of logic: real number in intuitionistic math, fuzzy number. Different system of logic needs to construct numbers in a different ways.
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