| 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
Imaginary and Complex Numbers
The Cayley-Dickson Series
| Name | dim | Property lost |
| Real | 1 | n/a |
| Complex | 2 | conjugate(x) = x |
| Quaternion | 4 | commutivity |
| Sedonion | 8 | alternativity |
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 a3) 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.
The smaller numbers (reals, complex, quaternion, octonion and sedonion) all have power-associative multiplication. The bitredeconions do not.