Difference between revisions of "Proofs without words"
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The following demonstrate [[proof]]s of various identities and theorems using pictures, inspired from [http://usamts.org/Gallery/G_Gallery.php this gallery]. | The following demonstrate [[proof]]s of various identities and theorems using pictures, inspired from [http://usamts.org/Gallery/G_Gallery.php this gallery]. | ||
+ | == Summations == | ||
<center><asy>unitsize(15); defaultpen(linewidth(0.7)); | <center><asy>unitsize(15); defaultpen(linewidth(0.7)); | ||
int n = 6; pair shiftR = ((n+2),0); real r = 0.3; | int n = 6; pair shiftR = ((n+2),0); real r = 0.3; | ||
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The sum of the first <math>n</math> positive integers is <math>n(n+1)/2</math>.<br><br> | The sum of the first <math>n</math> positive integers is <math>n(n+1)/2</math>.<br><br> | ||
</center> | </center> | ||
+ | |||
+ | == Geometric series == | ||
+ | <!--<center><asy></asy> | ||
+ | The infinite [[geometric series]] <math>\frac 12 + \frac {1}{2^2} + \frac {1}{2^3} + \cdots = 1</math>.<br><br> | ||
+ | </center>--> | ||
<center><asy> defaultpen(linewidth(0.7)); unitsize(15); | <center><asy> defaultpen(linewidth(0.7)); unitsize(15); | ||
− | int n = 10; real h = 6; pen colors[] = { | + | int n = 4; real h = 2; pen colors[] = {rgb(0.8,0,0),rgb(0,0.8,0)}; |
+ | void drawTriGrid(real s){ | ||
+ | for(int i = 0; i < 4; ++i){ | ||
+ | draw( (-s*3/2,s*(3/2 - i)) -- (s*3/2,s*(3/2 - i)), linetype("2 2")); | ||
+ | draw( (s*(3/2 - i),-s*3/2) -- (s*(3/2 - i),s*3/2), linetype("2 2")); | ||
+ | } | ||
+ | } | ||
+ | void fillrect(pair A, pair B, pen p){ filldraw(A--(A.x,B.y)--B--(B.x,A.y)--cycle, p, linewidth(1)); } | ||
+ | |||
+ | for(int i = 0; i < n; ++i) { | ||
+ | fillrect( ((-1)^i*-h/3^i*(3/2),-h/3^i*(3/2)) , ((-1)^i*-h/3^i*(1/2),h/3^i*(3/2)) , colors[0]); | ||
+ | fillrect(-((-1)^i*-h/3^i*(3/2),-h/3^i*(3/2)) ,-((-1)^i*-h/3^i*(1/2),h/3^i*(3/2)) , colors[1]); | ||
+ | fillrect( (-h/3^i*(1/2),(-1)^i*h/3^i*(1/2)) , (h/3^i*(1/2),(-1)^i*h/3^i*(3/2)), colors[0]); | ||
+ | fillrect(-(-h/3^i*(1/2),(-1)^i*h/3^i*(1/2)) ,-(h/3^i*(1/2),(-1)^i*h/3^i*(3/2)), colors[1]); | ||
+ | drawTriGrid(h/3^i); | ||
+ | } | ||
+ | </asy><br> | ||
+ | |||
+ | The infinite [[geometric series]] <math>\frac 13 + \frac {1}{3^2} + \frac {1}{3^3} + \cdots = \frac 12</math>.<br><br> | ||
+ | </center> | ||
+ | |||
+ | <center><asy> defaultpen(linewidth(0.7)); unitsize(15); | ||
+ | int n = 10; real h = 6; pen colors[] = {rgb(0.9,0,0),rgb(0,0.9,0),rgb(0,0,0.9)}; | ||
void drawEquilaterals(pair A, real s){ | void drawEquilaterals(pair A, real s){ | ||
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</center> | </center> | ||
+ | <center><asy> defaultpen(linewidth(1)); unitsize(15); | ||
+ | int n = 8; /* number of layers */ | ||
+ | real h = 3; /* square height */ | ||
+ | pen colors[] = {rgb(0.8,0,0),rgb(0,0.8,0),rgb(0,0,0.8)}; | ||
+ | pair shiftL = (-3*h,0); /* amount to shift second square left by */ | ||
+ | |||
+ | void drawSquares(real s, pair A = (0,0)){ | ||
+ | filldraw(shift(A)*shift(-2*s, -s)*xscale(s)*yscale(s)*unitsquare,colors[0]); | ||
+ | filldraw(shift(A)*shift(-2*s,-2*s)*xscale(s)*yscale(s)*unitsquare,colors[1]); | ||
+ | filldraw(shift(A)*shift(-s ,-2*s)*xscale(s)*yscale(s)*unitsquare,colors[2]); | ||
+ | } | ||
+ | for(int i = 0; i < n; ++i) | ||
+ | drawSquares(h/2^i); | ||
+ | drawSquares(h,shiftL); draw(shift(shiftL+(-2*h,-2*h))*xscale(2*h)*yscale(2*h)*unitsquare); | ||
+ | label("$\cdots$",shiftL+(-h/2,-h/2)); | ||
+ | </asy><br> | ||
+ | |||
+ | Another proof of the identity <math>\frac 14 + \frac {1}{4^2} + \frac {1}{4^3} + \cdots = \frac 13</math>. <br><br> | ||
+ | </center> | ||
+ | |||
+ | == Geometry == | ||
+ | |||
+ | == Miscellaneous == | ||
<center><asy> unitsize(15); defaultpen(linewidth(0.7)); | <center><asy> unitsize(15); defaultpen(linewidth(0.7)); | ||
real r = 0.3, row1 = 3.5, row2 = 0, row3 = -3.5; | real r = 0.3, row1 = 3.5, row2 = 0, row3 = -3.5; |
Revision as of 19:14, 18 March 2010
The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery.
Contents
[hide]Summations
The sum of the first odd natural numbers is .
The sum of the first positive integers is .
Geometric series
The infinite geometric series .
The infinite geometric series .
Another proof of the identity .
Geometry
Miscellaneous
Fermat's Little Theorem: for (above ).