Decimal Insights on Limits, Continuity, Convergence
Decimal and decimal-free error-control perspectives of continuity, limits
and Cauchy sequences are given below. These perspectives is followed by
comments on math education.
Continuity and Unlimited Error Control
Limits and continuity in calculus may be described geometrically, that
is, intuitively and informally, or more precisely in terms of say
epsilons and deltas. The roles of epsilon and delta below are played by E
> 0 and D > 0.
Imagine for instance we want to compute a function f(x) at the point x =
A accurately. So we can ask the error control question how close must x
be to A in order for f(x) to agree with f(A) to say k-decimal places. The
answer might be that x must agree with A to m decimal places. In some
computational problems, this answer for a specified number k of decimal
places may be all that is needed. But in other situations, we want in
practice or in principle, unlimited error control. Here we may want to
say for any k, there is an m such f(x) will agree with f(A) to k decimals
if x agrees with A to m decimals. Unlimited error control offers
motivation and a perspective on the discussion of continuity.
Now will say that f(x) is continuous at x = A if for each whole
number k, there is a number m such the limit f(a) and the value of f(x)
will agree to k-decimals whenever the number x agrees with the value of
a to m decimal places. Continuity here represents the concept of
unlimited error control in decimal computations.
More generally, we can ask (following Cauchy), given an error control
target E > 0, how close must x be to A for the difference of f(x) and
f(A) to be less than E in magnitude? The answer follows by obtaining a
number D with the property that if |x-A| < d then |f(x) - f(A)| <
E.
Without reference to decimals we can say that f(x) is continuous at x = A
if for every error control tolerance E > 0, there is a number D > 0
such that whenever |x-A| < d then |f(x)- f(A)| < E. Here continuity
at x=A corresponds to the idea of unlimited error control at x=A.
This second concept is decimal free. It is traditional to use epsilons
and deltas in place of E> 0 and D > 0.
Limits
We will say that a number L is the limit of a function f(x) as x
approaches A if one of the following conditions hold:
-
(Decimal Perspective): For
every whole number k, there exist a whole number m such f(x) will agree
will L to k decimal places if x agrees with A to m decimal
places.
-
(Decimal Free Perspective):
For every positive number E > 0, there exists a positive number
D> 0 such that if |x-A| < d then |f(x)- f(A)| < E.
Both conditions are equivalent. Each
implies the other. Why or how depends on how you think of (or represent)
the real numbers. For most people, assuming that real numbers are
represented by signed decimal expansions (infinite or finite) is
sufficient. Modern mathematics has alternate decimal (or base) -free
representations of real numbers.
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