This will be new and surprising to most feathermerchants because the Standard presentation of THE REAL NUMBER System is "exogamous" (not-All-in-the-family), explaining THIS ARITHMETIC PROBLEM (need for "real numbers") IN TERMS OF GEOMETRY ("the diagonal of a square") -- making logarithm appear as "something the cat drug in". Rather, REAL NUMBERS can be explained "endogamously", that is, ARITHMETICALLY, as THE NEED TO MAKE TOTAL ONE OF THE INVERSES (LOGARITHM) OF EXPONENTIATION! (I learned this in 1948, when my father-in-law gave me his copy of "Fine's Algebra", named for the math prof to whom Fine Hall at Princeton U. is dedicated.)

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For any rational number, r, The REPEATED PRODUCT r · r · r · ... · r is TEEEDJUS! So humans invented a SHORTCUT, DEFINED RECURSIVELY: b⁰ = 1; b^S(p) = b^p · p, where b, p are rational numbers.

Obviously, THE NATURALS, INTEGERS, and RATIONALS ARE CLOSED UNDER EXPONENTIATION.

Does exponentiation have "an inverse"? Well, exponentiation is DOUBLY WELL-DEFINED:

1. b^p = b^q iff p = q.

2. b^p = c^p iff b = c.

So, like ADDITION and MULTIPLICATION, EXPONENTIATION HAS TWO INVERSES. But the two inverses for ADDITION and MULTIPLICATION merge, because these operations are COMMUTATIVE. However, EXPONENTIATION IS NOT COMMUTATIVE. A single exception proves this: 2³ = 8, while 3² = 9; their nonequivalence proves NONCOMMUTATIVITY OF EXPONENTIATION.

Hence, EXPONENTIATION HAS TWO INVERSES: Given b^p = E ("exponential", avoiding "e" of "natural logarithms"), that is, "base b to power p equals exponent E":

1. log_bE = p iff b^p = E.

2. (E)^1/p = b iff b^p = E.

So, EXPONENTIATION has two (nonmerging) INVERSES: LOGARITHM and ROOT EXTRACTION. Both are PARTIAL in the RATIONAL NUMBER SYSTEM:

1. log_ba is irrational ("outside" RATIONAL NUMBERS) whenever either of a, b contains a PRIME FACTOR the other lacks. Well, "that's just about everything!".

2. (-1)^1/2 is irreal ("outside" RATIONAL and REAL NUMBERS).

TOTALIZING LOGARITHM yields THE REAL NUMBER SYSTEM. TOTALIZING ROOT EXTRACTION yields THE COMPLEX NUMBER SYSTEM. (And launches "THE ARITHMETIC OF CLIFFORD NUMBERS"!)

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1. We've seen that ANY RATIONAL NUMBER CAN BE REPRESENTED AS A DECIMAL NUMBER. Example: 1/3 = 0.|3|, where |n| means that n is REPEATED INFINITELY, that is, PERIODICITY beginning at some point in the DECIMAL NUMBER.

2. We've also seen that any such "periodic decimal" can be represented as a GEOMETRIC SUM of INFINITE LENGTH. Thus, 1/3 Þ 0.33333.... Þ 3/10 + 3/100 + 3/1000 + ... + 3/10ⁿ + ....

3. We've also seen that such INFINITE SUMS "equal IN THE LIMIT the number that is the RATIONAL form".

Question: Do other INFINITE SUMS HAVE LIMITS? If so, we may use these to FILL IN THE GAPS FOR THE VALUES OF LOGARITHMS -- hence, MAKE LOGARITHM TOTAL.

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The problem cannot be solved FINITARILY -- as the DIFFERENCE and QUOTIENT problems have been solved. We must solve the "log" (EXPONENTIAL INVERSE) problem TRANSFINITARILY by a new OPERATION: LIMIT. (I say "transfinitarily" since "infinite" means "approaching it", whereas "transfinite" means "using the infinitely attained result in finite doings".)

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The LIMIT OPERATION allows us to DEFINE CAUCHY SEQUENCES FOR REPEATING AND NONREPEATING RATIONAL SEQUENCES -- extending to IRRATIONAL NUMBERS.

DEFINITION: A metric is a nonnegative function g(x,y) describing the "Distance'' (difference between coordinate values) between neighboring points for a given Set. A metric satisfies the Triangle Inequality, g(x,y) + g(y,z) ³ g(x,z), and is symmetric: g(x,y) = g(y,x).

DEFINITION: A Cauchy sequence, {a₁, a₂, a₃, ...}, is a sequence such that its metric, d(a_m,a_n) satisfies lim_{min(m,n)→¥} d(a_m,a_n) = 0.

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Cauchy sequences are IMPLICIT in every "nonending" decimal number you encounter. In particular, the "square root of two" which the first or one of the first irrationals encountered.

When we see a number listed thus, √2 = 1.41421..., we can rewrite this as a Cauchy sequence of rational numbers: 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ....

We can show that √2 is "pinched" between LOWER and UPPER BOUNDS:

1 < √2 < 2 -- since (by SQUARING) 1 < 2 < 4
1.4 < √2 < 1.5 -- since (by SQUARING) 1.96 < 2 < 2.25
1.41 < √2 < 1.42 -- since (by SQUARING) 1.9881 < 2 < 2.0164
1.414 < √2 < 1.415 -- since (by SQUARING) 1.99941 < 2 < 2.0022
1.4142 < √2 < 1.4143 -- since (by SQUARING) 1.99996 < 2 < 2.00024
1.41421 < √2 < 1.41422 -- since (by SQUARING) 1.99999 < 2 < 2.00002
.....

Please note that the (left) SEQUENCE OF LOWER BOUNDS IS INCREASING CONTINUOUSLY IN VALUE, while CORRESPONDINGLY the (right) SEQUENCE OF UPPER BOUNDS IS DECREASING CONTINUOUSLY IN VALUE. The real number, √2, is being "pinched more and more tightly" by these BOUNDS. You can carry the DECIMAL NUMBER OUT TO ANY NUMBER OF POSITIONS, EQUIVALENTLY EXTEND THE CAUCHY SEQUENCE TO ANY NUMBER OF TERMS, EQUIVALENTLY EXTEND THE SEQUENCE OF "PINCHES" TO ANY NUMBER. You can imagine that, "eventually" or IN THE LIMIT, ONLY ONE NUMBER IS DESIGNATED OR "PINCHED". This defines the given REAL NUMBER -- here, √2.

And this fits my definition of a CANTITONE -- a CONTINUOUS ANTITONIC PROCESS.

We use these results to create "infinite vectors" as REPRESENTATION OF LOGARITHMS -- achieving our desired inverse.