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Landau’s Beautiful Proofs

Originally posted on Math Online Tom Circle:
Landau’s beautiful proofs: 1= cos 0 = cos (x-x) Opening cos (x-x): 1 = cos x.cos (-x) – sin x.sin (-x) => 1= cos² x + sin² x [QED] Let cos x= b/c,…

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Open Cubic Root

Originally posted on Math Online Tom Circle:
Mental Trick It was discovered by the Martial Art writer Liang Yusheng 武侠小说家 梁羽生 (《白发魔女传》作者), who met Hua Luogeng (华罗庚) @1979 in England: 2³= [8] 8³= 51[2] 3³= 2[7] 7³= 34[3] The last…

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Quick verify multiplication

Originally posted on Math Online Tom Circle:
Verify: 2,578,314  x 6,321,737 = 16,299,423,011,418 ? 2,578,314  => (2+5+7+8+3+1+4)) =30 => (3+0) = 3 6,321,737 => (6+3+2+1+7+3+7)=29 => 2+9 = 11 => (1+1) = 2 3 x 2 = 6  16,299,423,011,418 => (1+6+2+9+9+4+2+3+0+1+1+4+1+8) = 51 => (5+1) =…

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Solution 2 (Eigenvalue): Monkeys & Coconuts

Originally posted on Math Online Tom Circle:
Solution 2: Use Linear Algebra Eigenvalue equation: A.X = λ.X A =S(x)=  where x = coconuts S(x)=λx Since each iteration of the transformation caused the coconut status ‘unchanged’, which means λ = 1 (see…

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Solution 1 (Sequence): Monkeys & Coconuts

Originally posted on Math Online Tom Circle:
Monkeys & Coconuts Problem Solution 1 : iteration problem => Use sequence (initial coconuts) Let Since is integer  , k+4 ≡ 0 mod() k≡-4 mod() Minimum {k} = = 3121 [QED] Note: The solution was given by Paul Richard…

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Monkeys & Coconuts Problem

Originally posted on Math Online Tom Circle:
5 monkeys found some coconuts at the beach. 1st monkey came, divided the coconuts into 5 groups, left 1 coconut which it threw to the sea, and took away 1 group of coconuts.…

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French Curve

Originally posted on Math Online Tom Circle:
The French method of drawing curves is very systematic: “Pratique de l’etude d’une fonction” Let f be the function represented by the curve C Steps: 1. Simplify f(x). Determine the Domain of definition…

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Cut a cake 1/5

Originally posted on Math Online Tom Circle:
Visually cut a cake 1/5 portions of equal size: 1) divide into half: 2) divide 1/5 of the right half: 3) divide half, obtain 1/5 = right of (3) 4) By symmetry another…

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Tuition That We May Have To Believe In

Originally posted on guanyinmiao's musings (Archived: July 2009 to July 2019):
This commentary, “Tuition That We May Have To Believe In”, is a reply to a previous article on tuition by Howard Chiu (Mr.), “Tuition We Don’t Have To Believe In”…

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Generalized Analytic Geometry

Originally posted on Math Online Tom Circle:
Generalized Analytic Geometry Find the equation of the circle which cuts the tangent 2x-y=0 at M(1,4), passing thru point A(4,-1). Solution: 1st generalization: Let the point circle be: (x-1)² + (y-4)² =0 2nd…

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Math Chants

Originally posted on Math Online Tom Circle:
Math Chants make learning Math formulas or Math properties fun and easy for memory . Some of them we learned in secondary school stay in the brain for whole life, even after leaving…

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What is “sin A”

Originally posted on Math Online Tom Circle:
What is “sin A” concretely ? 1. Draw a circle (diameter 1) 2. Connect any 3 points on the circle to form a triangle of angles A, B, C. 3. The length of…

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