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Using the History of Calculus to Enrich our Teaching
Euclid answers this question (ca. 300 BC), building on
earlier work of Theaetetus of Athens (417–369 BC) and
Eudoxus of Cnidus (408–355 BC).
Even more basic: What do we mean by an irrational
The quotients will
be the same
regardless of which
Every ratio of integers corresponds to a
unique finite sequence of integers.
α = β if given any rational number m/n:
m < nα implies m < nβ, m/n < α m/n < β
m = nα implies m = nβ, m/n = α m/n = β
m > nα implies m > nβ, m/n > α m/n > β
Eudoxus is also credited with being the first to use the
method of exhaustion for finding areas and volumes,
which would be used to great effect by Archimedes
This means: Pick any rational number m/n less than
From some point on, all of the finite sums are > m/n.
Pick any rational number m/n larger than ln 2.
From some point on, all of the finite sums are < m/n.
Pick any number b larger than f ´(x); for all h
close to (but not equal to) 0, [f(x+h) – f(x)]/h < b.
Pick any number a smaller than f ´(x); for all h
close to (but not equal to) 0, [f(x+h) – f(x)]/h > a.
ε is the distance above or below f ´(x). δ is a limit on
distance between h and 0 that will guarantee that [f(x+h) –
f(x)]/h is within this distance of f ´(x).