My point. The standard presentation of arithmetic explains geometrically the need for "real numbers": to provide a number measure for the diagonal of a unit square. Also, an algebraic explanation of "complex numbers": to solve the algebraic equation, x2 + 1 = 0. ONLY HERE do we RESTORE ARITHMETIC TO ARITHMETIC BY SHOWING THAT THE NONCOMMUTATIVE ARITHMETIC OPERATION OF EXPONENTIATION HAS TWO INVERSES: LOGAIRTHM AND ROOT-EXTRACTION. To remder each operatotion TOTAL requires, respectively, real numbers (with irrationals) and complex numbers.
For, the Latin word "redux" is from the infinitive "reducis", meaning "to restore". But in other associated files, we've seen that the term "restore" is used in describing NUMALGEBRA, wherein one WORKS BACKWARDS to UNRAVEL THE ALGEBRA INTO ARITHMETIC. And this IMPLIES AN INVERSE OPERATION FOR ANY GIVEN OPERATION TO BE UNRAVELED. (If the OPERATION IS ADDITION, we must SUBTRACT; and vice versa. If the OPERATION IS MULTIPLICATION, we must DIVIDE; and vice versa. If the OPERATION IS EXPONENTIATION, we must EXTRACT ROOTS.)
What's so important about this? One word, "group", THE ULTIMATE CLOSURE STRUCTURE. Given a SET, S with an OPERATION, O: If S is CLOSED under O; and O has an INVERSE O¯1 under which it is also CLOSED, THEN S is a GROUP! (The ULTIMATE CLOSURE STRUCTURE!)
Now, THE NATURAL (COUNTING) NUMBER SYSTEM IS CLOSED UNDER ALL THE PRIMARY OPERATIONS OF ARITHMETIC (ADDITION, MULTIPLICATION, EXPONENTIATION). But NOT ONE OF THESE HAS A TOTAL INVERSE (INVERSE UNDER WHICH NATURALS IS CLOSED). Hence, THE MAJOR PROBLEM IN ARITHMETIC IS MAKING TOTAL THE INVERSE OF EACH OF THESE PRIMARIES.
And ARRANGING CLOSURE FOR ALL THESE INVERSE OPERATIONS LEADS US FROM THE NATURAL NUMBER SYSTEM TO the INTEGER SYSTEM (CLOSED UNDER SUBTRACTION); the RATIONAL NUMBER SYSTEM (CLOSED UNDER NONZERO DIVISION); the REAL NUMBER SYSTEM (CLOSED UNDER LOGARITHM -- ONE OF THE TWO INVERSES OF EXPONENTIATION); the COMPLEX NUMBER SYSTEM (CLOSED UNDER ROOT EXTRACTION -- THE OTHER INVERSE OF EXPONENTIATION).
This extension encompasses over 25 fields of MAINSTREAM MATHEMATICS!
Soooooooooooooooooooooooooooooo! ALMOST ALL OF MATHEMATICS IS ALGEBRA!!!
First, we must GENERATE ARITHMETIC, then see it GO CLIFFORD.
WAKE UP! Do you know what happened in those last three lines? Instead of "completing the job" of INVERSES FOR ALL PRIMARIES, THE NATURE AND PURPOSE OF ARITHMETIC JUST GOT CHANGED! In fact, ALL OF MATHEMATICS JUST GOT CHANGED! MATHEMATICS IS REBORN WITH THESE CHANGES! REDUX BECOMES REBIRTH! Briefly, as the great British mathematician, William Kingdon Clifford
(1845-1879) recognized in 1884, the RATIONAL, REAL, and COMPLEX NUMBER SYSTEMS FORM THE BEGINNING OF AN UNENDING EXTENSION which, today, goes by the name of "Clifford Algebra" (a.k.a "Geometric Algebra", a.k.a. "Multivector Theory") -- but an almost unbelievable history. For our purposes, we choose the label of THE ARITHMETIC OF CLIFFORD NUMBERS, Ci , i = 0, 1, 2, ..., wherein C0 is THE RATIONAL NUMBER SYSTEM; C1 is THE REAL NUMBER SYSTEM; C2 is THE COMPLEX NUMBER SYSTEM.
WHY STUDY ALGEBRA?