{"id":6043,"date":"2025-02-24T15:12:02","date_gmt":"2025-02-24T23:12:02","guid":{"rendered":"https:\/\/wonghoi.humgar.com\/blog\/?p=6043"},"modified":"2025-02-26T22:05:34","modified_gmt":"2025-02-27T06:05:34","slug":"tricks-you-eventually-pick-up-with-math","status":"publish","type":"post","link":"https:\/\/wonghoi.humgar.com\/blog\/2025\/02\/24\/tricks-you-eventually-pick-up-with-math\/","title":{"rendered":"Tricks you eventually pick up with math"},"content":{"rendered":"\n

This is a note-to-self page which I’ll update as I naturally revisit these ideas opportunistically.<\/p>\n\n\n\n

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Special numbers<\/h2>\n\n\n\n

\"-1\": alternating signs through odd\/even powers \"(-1)^k\"
\"0\": null (trivial additive solution), invariant (sums to zero)
\"1\": identity (trivial multiplicative solution), invariant (multiplies to 1)
\"[0,1)\" shrinks with growing powers
\"2\": \"2*2=2+2\"
Odd: \"2k+1\", Even: \"2k\"<\/p>\n\n\n\n

Problem solving approaches<\/h2>\n\n\n\n

Properties of linearity, aka superposition<\/p>\n\n\n\n

Find ways to see a raw definition of a concept hidden in the problem you’re solving.<\/p>\n\n\n\n

Plugging in easy\/obvious examples to verify a hypothesis (often used in differential equations) during exploration<\/p>\n\n\n\n

Make up a convenient term or multiplier that you wish you could and hopefully the counteracting term can be pushed out or used somewhere, like \"-1+1\" or multiply the numerator and denominator both by \"\sqrt{2}\"<\/p>\n\n\n\n

Special functions<\/h2>\n\n\n\n

Things only a constant function can do<\/p>\n\n\n\n

Small \"\mathrm{sinc}\" goes to \"1\" is the same as the small angle approximation for \"\sin(x)\approx x\"<\/p>\n\n\n\n

Quadratics: Exploit \"x^2 - (\alpha+\beta)x + (\alpha\beta)\" (e.g. used in trace and det to infer eigenvalues)<\/p>\n\n\n\n

Probe and extract with indicator function \"I_{x\in C}\", elementary vector \"\mathbf{e}_i\" and elementary matrices \"\mathbf{E}\", Dirac or Kronecker delta.<\/p>\n\n\n\n

Calculus<\/h2>\n\n\n\n

Symmetric integrals cancels for odd function and doubles of one side for even functions<\/p>\n\n\n\n

<\/p>\n\n\n\n

Series<\/h2>\n\n\n\n

Spotting hidden famous series (such as geometric sums)<\/p>\n\n\n\n

Series expansion dropping terms<\/p>\n\n\n\n

\"\cos(x)\" is even terms of \"e^x\" with alternating signs starting with 1,
\"\sin(x)\" is odd terms of \"e^x\" with alternating signs starting with \"x\"<\/p>\n\n\n\n

Taylor series always have factorial at the bottom (denominator) of the coefficient matching the n-th derivative at the top (numerator) for the n-th power term. <\/p>\n\n\n\n

Telescoping series (adjacent terms cancels)<\/p>\n\n\n\n

Use derivative to bring down polynomial power by \"1\" and create a shifted series (which can be used to recurse or cancel)<\/p>\n\n\n\n

Topology<\/h2>\n\n\n\n

In real line topology, outside the intuitive examples (singletons included), consider universal and empty set first, rationals and irrationals, then blame Cantor.<\/p>\n\n\n\n

Discrete Math (or Primes)<\/h2>\n\n\n\n

Modulos: generate all possible remainders of a certain modulo by multiplying<\/a>.<\/p>\n\n\n\n

<\/p>\n\n\n\n

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This is a note-to-self page which I’ll update as I naturally revisit these ideas opportunistically. Special numbers : alternating signs through odd\/even powers : null (trivial additive solution), invariant (sums to zero): identity (trivial multiplicative solution), invariant (multiplies to 1) … Continue reading →<\/span><\/a><\/p>\n

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