Generalization
is a very important tool for obtaining new results in mathematics. We first
experiments with numbers and geometrical shapes and observe some patterns
emerging and then we make some conjectures on the basis of these observations.
These conjectures do not get the status of theorems unless these can be
deducted either from certain axioms or can be deduced from other results which
have been earlier proved in mathematics.
Let us consider some
examples of valid as well as invalid generalizations.
CONJECTURES BASED ON
PRIME NUMBERS:
SOME MORE CONJECTURES:
Conjectures are obtained by generalizations but only those generalizations are valid which can be proved rigorously. We must continue to generalize, but we must generalize with care.
So, what did you conjecture today!
Note: I could not type mathematical symbols on the blog. So, I have attached images in this post. Click here to download pdf file for printing.
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