Thursday, August 11, 2016

МF111-113 Real numbers and Cauchy sequences of rationals (English)

...It involves epsilons and N's, much as does the notion of a limit, and suffers from... issues: how to guarantee that we can find an infinite number of N's for an infinite number of epsilons (making the very generous assumption that the term `sequence' does not really have to be defined properly!)

Built on top of this idea is the most familiar story for the `construction' of real numbers: to imagine that the limit of a Cauchy sequence of rationals can be defined to be essentially the sequence itself! The `essentially' refers to the fact that different Cauchy sequences can head in the same direction: so it means that we must introduce a complicated notion of equivalence into the story (more infinite numbers of checks, repeated an uncountably infinite number of times!).

There is a good reason why pure mathematicians cling so tenaciously to the idea of real numbers. They provide us with the ostensible `values' that lengths, areas, values of functions and solutions to equations seem to require. But is this all really just a dream??

In this video we have a new look at these notions, with a view of examining whether they really do support `real values', or whether perhaps they are intrinsically approximate notions. To motivate the discussion, we go back to a crucial calculation of Archimedes.

Essentially, the "length of the circumference" or "area of a circle" is not defined. It has only approximate meaning. We "need" real number to define what do we mean by these terms.

See also:
Measurement, approximation and interval arithmetic

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