words

Proof Without Words #3

Proof of the identity

Screen Shot 2017-07-30 at 5.34.22 PM.png

HCfGOYp.gif


dkoCZ.png

The figure for general n is similar, with n right pyramids, one with an (n-1)-cube of side length xk as its base and height xk for each k=1,….,n.


The derivative of sin is cosine.

g4n8SZX.jpg



From ‘Proof without words‘ by Roger Nelsen

Screen Shot 2017-07-30 at 5.47.56 PM.png

By Sidney H. Kung

Screen Shot 2017-07-30 at 5.48.12 PM.png

By Shirley Wakin


Previous ‘Proof Without Words‘: Part 1 | Part 2

M x

Advertisements

Proof Without Words #2

My last post seemed to go down well so I thought I’d compile a few more images of proofs without words!

C93PNFXXUAAfP7z.png

Determinant is the area of a parallelogram, by Solomon Golomb, Mathematics Magazine, March 1985.

C6fZNSAWUAAdyzq.jpg

Jensen_graph.png

A visual proof of Jensen’s inequality, found on Wikipedia. Jensen’s inequality states that

Screen Shot 2017-05-02 at 7.51.38 AM.png

In the diagram, the dashed curve along the X axis is the hypothetical distribution of X, while the dashed curve along the Y axis is the corresponding distribution of Y values. Note that the convex mapping Y(X) increasingly “stretches” the distribution for increasing values of X.

EXwbl.png

 

Hope you enjoyed! M x

Proof Without Words #1

I decided to start a new series called ‘Proof Without Words’: a collection of pictures which prove a mathematical fact with an image. Remember, these are in no way rigorous and are just meant to give an idea of why the fact is true!

1/2+1/4+1/8+1/16+1/32+ … = 1

Sum of the first n positive odd numbers = n^2

arctan(1) + arctan(2) + arctan(3) = π

Screen Shot 2017-02-21 at 9.21.36 PM.png

Screen Shot 2017-02-21 at 9.22.18 PM.png

Viviani’s Theorem

The sum of the distances from any interior point to the sides of an equilateral triangle equals the length of the triangle’s altitude.

Screen Shot 2017-02-21 at 9.24.11 PM.png

M x