 
Summary: 8 CASIM ABBAS
Lecture 2
2. Mathematical proofs
Today we will explore the question 'What are mathematical proofs and why do
we need them ?' Apart from learning Linear Algebra the other purpose of this
lecture is to teach you how to write mathematically correct proofs.
In the following example we divide a circle into regions by placing points on the
boundary and connecting them with line segments as in figure 8 It seems that the
1 42
8
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Figure 8
number of regions doubles each time we add another point on the boundary. In
Mathematics a statement like this does not establish the truth of a fact. Math
ematicians demand a proof beyond all possible doubt, not beyond all reasonable
doubt. We will look as an example at the proof that
2 (the positive number whose
square equals two) is irrational. The proof is what we call an indirect proof, i.e. we
assume that
