Number Systems - Classifications

Number Class		Ordered short for Total Order 𝑥<𝑦, 𝑥>𝑦, or 𝑥=𝑦 for all 𝑥,𝑦 in number class	Commutative Commutative 𝑥 + 𝑦 = 𝑦 + 𝑥 for all 𝑥,𝑦 in number class	Associative Associative 𝑥 + (𝑦 + 𝑧) = (𝑥 + 𝑦) + 𝑧 for all 𝑥,𝑦,𝑧 in number class	Dimension	# of Solutions to an n^th Degree Polynomial	Representation 𝛼,𝑏,𝜒,𝛿 are Reals 𝑖,𝑗,𝑘,𝑒_𝑖 are the Basis 𝑤,𝑥,𝑦,𝑧 are Complex	Description
Real Numbers (ℝ)		✔	✔	✔	1	at most n	𝛼
Complex Numbers (ℂ)		❌	✔	✔	2	exactly n	𝛼 + 𝛽𝑖
H y p e r c o m p l e x	Quaternions (ℚ)	❌	❌	✔	4	possibly ∞	𝛼 + 𝛽𝑖 + 𝜒𝑗 + 𝛿𝑘
	Octonions (𝕆)	❌	❌	❌	8	possibly ∞	𝛼𝑒₀ + 𝛽𝑒₁ + ... + 𝛾𝑒₆ + 𝜂𝑒₇
	Sedenions (𝕊)	❌	❌	❌	16	possibly ∞	𝛼𝑒₀ + 𝛽𝑒₁ + ... + 𝛾𝑒₁₄ + 𝜂𝑒₁₅
	Tessarines Bicomplex Numbers					n²	(𝛼 + 𝛽𝑖, 𝜒 + 𝛿𝑖) 𝑤 + 𝑥𝑖	are the tensor product of the complex numbers and the split-complex numbers
	Coquaternions Split-Quaternions							the algebra of split-quaternions is isomorphic to the ring of the 2×2 real matrices
	Biquaternions						𝑤 + 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘

Subsets of Real Numbers

Counting Numbers Natural Numbers (ℕ)	are positive integers {1, 2, 3, ...} other definitions include 0 (i.e. non-negative integers {0, 1, 2, 3, ...})
Whole Numbers	are natural numbers including 0
Integers (ℤ)	are positive and negative counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}
Rational Numbers (ℚ)	numbers that can be expressed as a ratio of an integer to a non-zero integer all integers are rational, but the converse is not true; there are rational numbers that are not integers	1/2
Irrational Numbers	are all real numbers that are not rational	√2, 𝑒, 𝜋

Pure Imaginary Numbers (𝕀)	numbers that equal the product of a real number and the square root of −1 the number 0 is both real and imaginary	𝑖
Algebraic Numbers (𝔸)	is a real or complex number that is a root of a non-zero polynomial	9 + 5𝑖
Transcendental Numbers	is a real or complex number that is not an algebraic number	𝑒, 𝜋

$p$ -adic numbers	various number systems constructed using limits of rational numbers, according to notions of "limit" different from the one used to construct the real numbers
Computable Numbers	are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm
Finitely Describable Numbers	is a number that can be unambiguously defined by a finite string over a finite alphabet