- 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.
We've also seen that any such "periodic decimal" can be represented as a GEOMETRIC SUM of INFINTE LENGTH. Thus, 1/3 Þ 0.33333.... Þ 3/10 + 3/100 + 3/1000 + ... + 3/10n + ....
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.
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 "hovering near there", whereas "transfinite" means "using the infinite result in finite or infintary doings".)
We can use these results to create "infinite vectors" as REPRESENTATION OF LOGARITHMS -- achieving our desired inverse.
The DEFAULT DEFINITION OF LIMIT is from For Auld Lang Syne, I'll state this classical defintion (you
can close your eyes and plug up your ears). Fortunately, modern mathematics
provides a more math-friendly definition, which I'll soon display. (Also,
my graph for LIMIT is as pedestrian
as climbing stairs, or finding a sock in a drawer.)
Let Sn = s1, s2,
s3, ..., denote a sequence -- as in,
.3,
.33, .333, .3333, ..., or 3/10, 33/100, 333/1000,
3333/10000, ...., the sequence representing the RATIONAL 1/3.
DEFINITION: The (real) number L is the
limit of sequence Sn iff, for every
e > 0, there exists a
d > 0 such that
|sn - L| < e if
n > d.
The sense of this: Designate any limit L's NEIGHBORHOOD (of the form
|sn - L|), and I can specify a NONZERO d
(indexing an element in sequence Sn) such that every sequent beyond that position
n resides in that NEIGHBORHOOD, |sn - L|.
LIMIT enables Achilles to catch up with the Tortoise (in ZENO'S PARADOX), even
though Tortoise has ahead start of ONE UNIT. Achilles can run HALF OF THIS
UNIT; HALF OF THE HALF; HALF OF THE HALF OF THE HALF; ad infinitum.
That is, 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, ....
We can obtain PARTIAL SUMS of this sequence: 1/2,
3/4, 7/8, ..., (2n — 1)/2n = 1 — 1/2n,
.....
But Zeno knew that you can't SUM THIS BY A FINITARY OPERATION,
hence, ARGUED THAT ARITHMETIC CANNOT EXPLAIN MOTION -- STYMYING THE MECHANICAL
INDUSTRIAL REVOLUTION AND PROLONGING SLAVERY FOR 2000 years!
We can show that the LIMIT of this sequence is L = 1.
What NEIGHBORHOOD do you wish? The "one-millionth NEIGHBORHOOD"? That is,
|sn
- L| < e = 1/1000000. Well, for
n
= 20, s20 = 1 — 1/1048576, so
|1
- 1/1048576 — 1| = 1/1048576| < 1/1000000. And ACHILLES CLOSES
IN ON TORTOISE!
Thus, lim(n->¥) (1 — 1/rn)
= 1 — 0 = 1.
The epsilon-delta definition can be restated by use of the metric
concept, an extension of the familiar "Pythagorean Theorem".
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).
We can now define a general theory of sequences, which will include all those for RATIONAL NUMBERS, with REPEATING subsequences, as well as CONVERGENT SEQUENCES without REPEATERS to REPRESENT ANY REAL NUMBER.
DEFINITION: A Cauchy sequence, {a1,
a2, a3, ...}, is a sequence such that its metric, d(am,an) satisfies limmin(m,n)→¥ d(am,an) = 0.
(A Cauchy sequence does not necessarily CONVERGE for all RATIONALS, but will
CONVERGE for all REAL NUMBERS.) However, before seeing how LIMIT makes TOTAL THE LOGARITHMIC INVERSE OF EXPONENTIATION via REAL NUMBERS, let's study "periodic and nonperiodic decimal numbers", in preparation for real numbers (including irrationals). By adjoining, to the FINITARY OPERATIONS (addition, subtaction, multiplication, etc.)
the TRANSFINITARY OPERATION OF LIMIT, the LOGARITHMIC INVERSE OF EXPONENTIATION CAN BE MADE
TOTAL, as we see by returning to its modeling.
Augustin
Cauchy (1789-1857), with "epsilons and deltas and the momes rath outgrabe!",
which puzzles or scares many students. I agree. If I saw an Epsilon walking
hand in hand with a Delta down the sidewalk, I'd shuffle off to Buffalo.