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A Preview of Math 112 (Introduction to Analysis)
In this course we will make a small number of
assumptions
about numbers, (thirteen assumptions in all).
The first twelve assumptions
will be familiar number facts. The last assumption may not
look familiar, but I hope it will seem as plausible as
things you have assumed about numbers in the past.
You will not be permitted to assume any facts about numbers
other than the thirteen stated assumptions. For example,
we will not explicitly assume that 3x0 = 0, or that 2x2 = 4,
so you will not be allowed to assume this. (These facts
will be proved in theorems 2.66
and 2.84.)
You will not be allowed to assume that (-1)(-1) = 1, or that
0 < 1. (These facts will follow from
exercise 2.77c
and corollary 2.104.)
We will not justify the representation of numbers
by points on a line, so no proofs can depend on pictures
of graphs of functions.
On the basis of our assumptions about real numbers, we will construct
a more general class of complex numbers, in which -1, and in
fact every number, has a square root. Many results about the algebra
and calculus of real functions will be shown to hold for complex
functions.
Occasionally I will draw pictures to motivate proofs, but the
proofs themselves will not depend on the pictures.
Sometimes in examples or remarks I will use arguments
depending on similar triangles or trigonometric identities,
but my theorems and definitions will depend only
on my assumptions. I will also refer to integers and rational
numbers in examples before I give the formal definitions,
but no theorems will involve integers until they have been defined.
Nothing in this course will be trivial or obvious or clear.
If you come across these words, it probably means that I am
engaging in a mild deception. Beware.
In theorem 2.66 we will prove the well known
fact that 2x0 =0. In
(12.57) we will prove the less well known fact that e(2 pi i) = 1,
where i represents a square root of -1.
The fact that we can derive the last not-so-obvious result
from our thirteen assumptions is somewhat remarkable.
The prerequisite for this course is a course in one-variable
calculus. From the remarks made above you know that you
cannot assume any facts from your calculus course, but
if you are not familiar with the rules for calculating the
derivatives of the sine and cosine and exponential functions
from an earlier course, our definitions of
the sine and cosine and exponential will seem rather meaningless.
By the time you finish this course, you should be adept at reading,
and constructing, and writing mathematical proofs.
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