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I would like to collect a "big-list" of ideas and methods from different areas (although I'm particularly interested in elementary number theory, algebra, calculus, linear algebra, geometry, physics, probability, combinatorics) that you find enlightening because they have changed (for the better) your approach to a broader set of problems, theorems, or to mathematics in general by giving you some brilliant insight and useful intuition.


To make an example, I think that Apostol's geometric methods collected in New horizons in geometry deserves a place in the list.

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The probabilistic method. $~~~$

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Many things in calculus can be derived easily just by using the approximation \begin{equation} f(x + \Delta x) \approx f(x) + f'(x) \Delta x. \end{equation}

For example, the chain rule. Let $h(x) = f(g(x))$. Then \begin{align} h(x + \Delta x) &= f(g(x + \Delta x)) \\ &\approx f(g(x) + g'(x) \Delta x) \\ &\approx f(g(x)) + \underbrace{f'(g(x))g'(x)}_{h'(x)} \Delta x. \end{align}

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