# Complex Analysis

This mathematical fallacy is due to a simple assumption, that $-1=\dfrac{-1}{1}=\dfrac{1}{-1}$ . Proceeding with $\dfrac{-1}{1}=\dfrac{1}{-1}$ and taking square-roots of both sides, we get: $\dfrac{\sqrt{-1}}{\sqrt{1}}=\dfrac{\sqrt{1}}{\sqrt{-1}}$ Now, as the Euler’s constant $i= \sqrt{-1}$ and $\sqrt{1}=1$ , we can have $\dfrac{i}{1}=\dfrac{1}{i} \ldots \{1 \}$ $\Rightarrow i^2=1 ## Triangle Inequality Triangle inequality has its name on a geometrical fact that the length of one side of a triangle can never be greater than the sum of the lengths of other two sides of the triangle. If$ a$,$ b$and$ c$be the three sides of a ## Numbers – The Basic Introduction If mathematics was a language, logic was the grammar, numbers should have been the alphabet. There are many types of numbers we use in mathematics, but at a broader aspect we may categorize them in two categories: 1. Countable Numbers 2. Uncountable Numbers The numbers which can be counted in ## Everywhere Continuous Non-differentiable Function Weierstrass had drawn attention to the fact that there exist functions which are continuous for every value of$ x$but do not possess a derivative for any value. We now consider the celebrated function given by Weierstrass to show this fact. It will be shown that if$ f(x)=

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