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:
y + y + 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 16th 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:
y × y + 2 = 1
(where "×" is ordinary multiplication) 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 exponential analogue to the above problems of closure of addition and multiplication — for example
yy + 2 = 1
does have a solution:
y = 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. And, you can even add the lower hyper4 function and its two inverses.
However, for the standard "hyper" function (which I call the higher hyper4 function, or just "tetration") it seems unlikely there is a unique definition for real arguments.
The Cayley-Dickson Series
Given a ring, i.e. a set of numbers with suitable definitions of "addition", "conjugate", and "multiplication" operations, one can construct another ring whose elements are two-element tuples of members of the original ring, and a set of definitions of how to add, conjugate, and multiply these tuples. For example, to extend the reals to the complex numbers, these definitions suffice:
given: the "real numbers" are a set of values for which we have
definitions of:
addition a+b,
an additive identity 0 (such that a+0 = a for all a),
a conjugacy operation a, and
multiplication a×b.
(We will use letters a, b, c, etc. to repesent elements
of the set of real numbers.)
the additive identity is the tuple (0,0)
addition : (a,b) + (c,d) = (a+b, c+d)
conjugacy : (a,b) = (a, -b)
multiplication : (a,b) × (c,d) = (ac-bd, ad+bc)
Given these definitions one can see that (0,1)×(0,1) = (-1,0), and (0,-1)×(0,-1) is also (-1,0); which is another way of saying that i2 = -i2 = -1. For another example, the conjugate of (-2,-1) is (-2,1), or in normal notation -2-i = -2+i.
Comparing to the Wikipedia page for Cayley-Dickson construction you can wee I am not using the raised-asterisk symbol * for conjugation, instead I am using an "overline" as in the complex conjugate article.
For a slightly longer example, we know that the product of a complex number with its complex conjugate has a complex component of zero (normally just called "a real number", though we are still treating it as something with an imaginary component). We can see this by deriving the product of two conjugates: (a,b)×(a,b) = (a,b)×(a,-b) = (aa-(-bb), ab-ba) = (a2+b2, 0) provided that the underlying multiplication operation is commutative.
The choice of definitions is important, and the above set of definitions, while they do generate something like the ring of complex numbers from the reals, do not suffice for going further.
However, the Cayley-Dickson construction, used to make the quaternions and higher algebras in the table below, uses something very similar. We merely need to involve conjugacy in our definition of multiplication, and be clear about order of multiplication.
multiplication : (a,b) × (c,d) = (ac-db, da+bc)
On the right-hand side of the definition, the ordering of the multiplications has been changed in a couple places, and in two of the multiplications we're using the conjugate of one of the elements. If a, b, etc. were actually real numbers, this wouldn't matter because in the reals conjugacy is the identity function and multiplication is commutative. But it does matter when we move on to higher examples, because (first of all) the conjugacy operation is not an identity starting with the complex numbers, and then (later) multiplication is not always commutative.
In this table, each row represents a ring that is constructed from the previous one by the same process (using the more sophisticated rule for constructing the multiplication operation):
| Name | dim | based on | Property lost |
| Real | 1 | n/a | n/a |
| Complex | 2 | Real | orderdness; conjugate(x) = x |
| Quaternion | 4 | Complex | commutativity |
| Octonion | 8 | Quaternion | associativity |
| Sedonion | 16 | Octonion | alternativity |
| Bitredeconion | 32 | Sedonion | 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 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.
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 223 is 256, but (2{^}2){^}3, ordinarily written (22)3 is 64.
The smaller numbers (reals, complex, quaternion, octonion and sedonion) all have power-associative multiplication. The bitredeconions do not.
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2014 Dec 18.
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