A scalar (0-vector) DISTINGUISHES A POINT IN COORDINATE-SPACE; a vector (1-vector), as A SEGMENT OF A LINE, DISTINGUISHES A LINE IN COORDINATE-SPACE; a bivector (2-vector), as A PLANE-SECTION -- DISTINGUISHES A PLANE IN COORDINATE-SPACE; a trivector (3-vector) -- as A 3D-SECTION -- DISINGUISHES 3D IN COORDINATE-SPACE; etc.

Going beyond what Gauss, Wessel, and Argand distinguished as the vector i = √¯1, we construct the BIVECTOR, i.

Consider i º d₁Äd₂ = d₁ · d₂ + d₁Ùd₂ = 0 + d₁Ùd₂, that is, i º d₁Ù d₂.[1]
Then i · i = (d₁ Ùd₂) · (d₁Ùd₂) = ¯(d₂Ùd₁) · (d₁Ùd ₂) = ¯(d₂Ùd₁ · d₁Ùd₂) = ¯(d₂d₂) = ¯(d₂ Ùd₂) = ¯(1) = ¯1. That is, i · i = ¯1. [2]

So, we have our "imaginary" BIVECTOR, i, which as a BIVECTOR, is a SEGMENT OF A PLANE, DISTINGUISHING ITS PLANE WITH IN COORDINATE-SPACE AS THE i-PLANE (the PLANE implied when Gauss, Wessel, Argand conceived i = √-1) as a vector).

And, like any BIVECTOR (or MULTIVECTOR), i is an OPERATOR -- for example, SENDING A VECTOR INTO A VECTOR: d₁ Äi = d₁ · (d₁ Ùd₂) = 1 · d₂ = d₂. That is, d₁i = d₂. Thus, ALGEBRAICALLY -- as a RIGHT-OPERATOR -- i sends d₁ into d₂; GEOMETRICALLY, it ROTATES d₁ COUNTERCLOCKWISE THROUGH 90° or /2. [3]

On the other hand, d₂Äi = d₂ · (d₁Ùd₂) = ¯d₂ · (d₂Ù d₁) = ¯1(d₁) = ¯d₁. Again, as a RIGHT-OPERATOR, i ALGEBRAICALLY SENDS d₂ INTO ¯1d₁; GEOMETRICALLY, ROTATES IT CLOCKWISE THROUGH THROUGH 90° or -/2. [3a]

ASSIGNMENT: WHAT HAPPENS WHEN i ACTS AS A LEFT-OPERATOR FOR THE SAME TWO OPERANDS?

Thus, the great POWER and significance of the scalar, i = √-1, is only shown in using BIVECTOR i as a ROTATING OPERATOR. And, in general, the great power of multivectors is that a multivector can act as an OPERATOR upon an appropriate multivector as OPERAND.

Any RELATION which is 1-1 (hence, a FUNCTION) such that its OUTPUT IS A SUBSET OF ITS INPUT is an OPERATOR. Thus, an OPERATOR has DOUBLE CLOSURE. If the requirement were merely that "OUTPUT SET EQUALS INPUT SET", then this would provide THE CLOSURE THAT A DEFINITELY MATHEMATICAL STRUCTURE REQUIRES. But the "SUBSET" requirement "closes it within that closure".

Furthermore, American mathematician, Norbert Wiener (1894-1964), taught German physicist, Max Born (1882-1970), about OPERATORS, shifting QUANTUM THEORY FROM PERIODIC TO NONPERIODIC PROCESSES. If mathematicians and mathematical physicists had followed up on the work of Clifford, the OPERATOR-capability of MULTIVECTORS might have been widely known and physics might have advanced earlier. Not many contemporary mathematicians talk about this OPERATOR-capability!

Actually, "the imaginary" is present in Maxwell's Electromagnetic Equations, entering via rotation. Gurtler and Hestenes have also found that "spin" is in Schrödinger's allegedly "spinless" equation, in its "the imaginary". Apparently, "imaginaries" and "spinors" are absent only in a "rotationless universe".)

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HYPERBOLIC FUNCTIONS

From MULTIPRODUCT we find the VECTOR FORM of Euler's equation,:
aÄb = a · b + a Ù b = cos q + i sin q = e^iq. [4]
Then, bÄa = b · a + b Ù a = cos q - i sin q = e¯^iq. [4a]

Subtracting [4a] from [4] (so that their equal inner products cancel out), we have: aÙb — bÙa = aÙb + aÙb = 2aÙb = e^iq — e^iq → aÙb = (e^iq — e^iq)/2 = sinh q. [4b]

On the other hand, aÙb + bÙa = e^iq + e¯^iq → 1/2(aÙb + bÙa) = 1/2(e^iq + e¯^iq) = cosh q. [4c]

And the rest of HYPERBOLIC FUNCTION THEORY follows. (In the rotation operator, R, REPLACING COSINE AND SINE, respectively, by COSH AND SINH RESULTS IN ANOTHER ROTATION OPERATOR, from which THE BASIC EQUATIONS OF SPECIAL RELATIVITY MAY BE DERIVED. This is in "The Pauli Algebra", equivalent to QUATERNIONS.)

Other FUNCTIONS involving "complex exponentials", such as BESSEL FUNCTIONS can also be derived.

Now, let x be ANY VECTOR IN THE i-PLANE: x = d₁ + d₂. And consider:
d₁Äx = d₁Ä(d₁ + d₂) = 1 · x₁ + x₂(d₁Ùd₂) = x₁ + x₂i º z. [5]

"Golly!" said The Rabbit. We've "a complex number form with real and imaginary parts". Furthermore, IT HAS MULTIPRODUCT FORM: SCALAR PLUS BIVECTOR! It's CONJUGATE is z^† = x₁ — x₂i. [5A]

We'll show that [5] ([5A]) is a SPINOR, THE SIMPLEST KIND OF SPINOR. This is a STRUCTURE thought only 20th Century. But it was present in Renaissance ALGEBRA, when "imaginary roots" were inserted to find "real roots" of NUMALGEBRAIC EQUATIONS!

However, for the nonce, we'll note only that

• I = d₁Ùd₂Ùd₃ DOES FOR 3-D WHAT i DOES FOR 2-D. ( ¯I is EQUIVALENT to the powerful HODGE OPERATOR of EXTERIOR ALGEBRA.)
• So, REAL AND COMPLEX ALGEBRA AND ANLYSIS become UNIFIED IN MULTIVECTOR THEORY!
• And MULTIVECTOR CONSTRUCTION of I means that "complex variable theory" can be EXTENDED to 3-D (see Brackx) in a way not possible in MAINSTREAM MATHEMATICS!

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QUATERNIONS

Please recall that Irish mathematician William Rowan Hamilton (1805-1865) created the word "vector" in recastingcomplexnumbers as vectors of real components, with a peculiar product and TWO BASIS UNITS: [1, 0] and [0, 1] (the latter equivalent to i = √-1). Noting that his vectors MODEL ROTATION in 2-D, Hamilton then attempted VECTORS to MODEL ROTATION in 3-D.

Since a 2 UNIT-SYSTEM MODELS ROTATION in 2-D, Hamilton variously sought a 3-D M0DEL on 3 BASIS UNITS, with no success. The story (legend?) goes that, one day while strolling with his family outside Dublin, Ireland, Hamilton thought of using FOUR BASIS UNITS. Taking out a pocket knife, Hamilton carved the EQUATIONS in the wood of a bridge they'd come to -- where one can see them today. This was the SYSTEM OF QUATERNIONS, which split mathematicians into opposing camps.

You can use QUATERNIONS to provide the SPINORS that Dirac found in his QUANTUM THEORY OF THE ELECTRON. But QUATERNIONS have always been considered "strange" by those favoring MATRICES. (The British physicist, Peter Tait (1831-1901) favored the matrices of British mathematician, Arthur Cayley (1821-1895) and described STANDARD vector algebra as "a sort of hermaphrodite monster, compounded of the notations of Hamilton and Grassmann.")

The FOUR BASIC UNITS OF QUATERNIONS are 1, i, j, k such that i² = j² = k² = ¯1, and ij = k, jk = i, ki = j, while ji = ¯k, kj = ¯i, ik = ¯j, etc.

(Elsewhere, I show what you'll find nowhere else! That UNITS WHOSE SQUARES ARE NEGATIVE, such as 1, i, j, k, BUT ALLOW SIMPLE SUBTACTION-RULES SEPARATE OUT SUBTRACTION-MODULS WITHIN THEIR SYSTEM, REQUIRING RECOGNITION AS UNIT VECTORS.

1. THE COMPLEX NUMBERS FORM A BIMODUL (my definition!): TWO INDEPENDENT SUBTRUCTURES EACH FORMING A MODUL -- CLOSED UNDER SUBTRACTION.
2. QUATERNIONS FORM A QUATROMODUL (my definition and label!).
3. OCTONIONS FORM AN OCTOMODUL (in my terminology).
4. And, in general, any ARITHMETIC OF CLIFFORD NUMBERS, C_n, n = 1,2,..., is (in my terminolgy) a 2ⁿ-MODUL.
5. Thus, THE ARITHMETIC OF CLIFFORD NUMBERS -- a.k.a. Clifford Algebra or Geometric Algebra or Multivector Theory -- might be called "The 2ⁿ-MODUL SYSTEM". This ENCAPSULATES "the whole secret of it". And opens up possibilities for research.)

   It's easy to TRANSLATE MULTIVECTORS INTO QUATERNIONS (REVERSING ORDER OF BASIC VECTOR UNITS to achieve NEGATIVE SELFPRODUCTS, and JUXTAPOSITION replacing VERTICAL, Ä, for convenience of writing): Let i = d₂d₁; j = d₃d₂; k = d₁d₃; then (by single permuting) ii = d₂d₁d₂d₁ = ¯d₁d₂d₂d₁ = ¯d₁d₁ = ¯1; jj = d₃d₂d₃d₂ = ¯d₂d₃d₃d₂ = ¯d₂d₂ = ¯1; kk = d₁d₃d₁d₃ = ¯d₃d₁d₁d₃ = ¯d₃d₃ = ¯1; (and by double permuting) ij = d₂d₁d₃d₂ = d₁d₂d₂d₃ = d₁d₃ = k; jk = d₃d₂d₁d₃ = d₂d₃d₃d₁ = d₁d₁ = i; ki = d₁d₃d₂d₁ = d₃d₁d₁d₂ = d₃d₂ = j; (again, single permuting) ij = ¯ji; etc.

   More properties from MULTIPRODUCT?