# Numbers Other Than Positive Reals

This page discusses numbers other than the types on my numbers page (which discusses positive reals) and my large numbers page (which discusses very large reals in general, and infinite quantities).

The page is arranged, as much as possible, by the history of "discovery" or invention of the different types of numbers.

## Zero

It took a long time for zero to be considered a numerical quantity in the same way as the other natural numbers (positive integers). The use of a digit in combination with other digits, as we do now, began independently in Babylonia (ca. 700 BC), Mesoamerica (ca 50 BC), and India (628 AD); the first two never spread beyond their culture of origin but the third took hold and continued through to the present day. Each of these cultures, as well as Greece and China, developed the concept of number as an abstraction at about the same time.

## Negative Numbers

Negative numbers seem to have been developed a little bit after zero. They are needed to make addition and subtraction "complete" (or technically, "closed under addition"): with negative numbers, you can add or subtract any two numbers and get a number as an answer, and solve equations involving any combination of numbers added and subtracted. As a specific example, the equation:

x + x + 2 = 1

has no solution using just positive numbers and zero, but once you add negative numbers, this equation does have a solution.

## Imaginary and Complex Numbers

These were developed sometime in the 16^{th} century in Europe.

In a similar manner to negative numbers, Complex numbers are needed to make multiplication and divison "complete". As a specific example, the equation:

x × x + 2 = 1

has no solution using just real numbers, but once you add complex numbers, this equation does have a solution. Using complex numbers, you can multiply or divide any two numbers and get another number, and find the roots of equations involving any numbers with addition, subtraction, multiplication and division.

Furthermore (and this is the really cool bit), due to Euler's formula and its extention to the generalized complex exponential function, we also get exponentiation and its two inverses, (logarithms and arbitrary radicals) "for free". In other words, there is no analogue to the above equation, for example

x^{x} + 2 = 1

does have a solution:

x = 2.64836... + 4.20934... i

and there are other solutions (based on the multiple branches of the
Lambert W function), but no additional solutions come from
expanding the field to quaternions or anything like that. Furthermore,
you can even add the lower hyper_{4} function
and its two inverses.

## The Cayley-Dickson Series

Name | dim | Property lost |

Real | 1 | n/a |

Complex | 2 | conjugate(x) = x |

Quaternion | 4 | commutivity |

Octonion | 8 | associativity |

Sedonion | 16 | alternativity |

Bitredeconion | 32 | power-associativity |

## Quaternions

An operation (.) is commutative if a(.)b = b(.)a. The smaller numbers (real and complex) all have commutative multiplication. The quaternions are no longer commutative, however they are still associative.

## Octonions

An operation (.) is associative if (a(.)b)(.)c = a(.)(b(.)c). The smaller numbers (reals, complex and quaternion) all have associative multiplication.

The octonions are no longer associative, however they are still alternative.

## Sedonions

An operation (.) is alternative if (a(.)a)(.)b = a(.)(a(.)b) and (b(.)a)(.)a = b(.)(a(.)a). The smaller numbers (reals, complex, quaternion and octonion) all have alternative multiplication.

The sedonions are no longer alternative, however they are still power-associative.

sedonions also have non-zero values b so that for all a, ab=0.

## Bitredeconions

In the bitredeconions, there is no well-defined integer exponent
operation. In order for integer exponents (like a^{3}) to be
well-defined, multiplication must be power-associative.

An operation (.) is power-associative if a value can be operated on multipe times with the operator, producing the same result no matter what order the parts are combined — for example, a(.)(a(.)(a(.)a)) = (a(.)(a(.)a))(.)a = (a(.)a)(.)(a(.)a). It is possible for an operator to be non-associative but still power-associative, because associativity requires equality for all values a and b and c whereas power-associativity only requires equality when combining multiple copies of a.

In ordinary numbers, the multiplication operator
a×b is power-associative, but the
exponentiation operator a{^}b is not, because
a{^}(b{^}c) is not always the same as (a{^}b){^}c. For
example, 2{^}(2{^}3), ordinarily written 2^{23} is 256, but
(2{^}2){^}3, ordinarily written (2^{2})^{3} is 64.

The smaller numbers (reals, complex, quaternion, octonion and sedonion) all have power-associative multiplication. The bitredeconions do not.

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