Number Class

Ordered Total Order

𝑥<𝑦, 𝑥>𝑦, or 𝑥=𝑦 for all 𝑥,𝑦 in number class

short for

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

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 numberthat 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