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.