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A beautiful proof without words

While surfing the Web for Math-related stuff I happened upon this wonderful "proof" without words:

Can you figure out what the image illustrates? Can you figure out what two facts you need to know to do the "proof?" Yes, I realize that visual demonstrations are not proofs.

If you need a hint, check out the original document by Professor Osler.

Comments (9) Trackbacks (1)
  1. The above illustration shows sin^2x+cos^2x=1.The area under the shaded curve is also= the area under the unshaded curve

  2. Adding onto Ankita’s answer, the image illustrates the Pythagorean Theorem.

  3. Pat, Im sure you mean the “Pythagorean Identity”. And that IS what Ankita said, using math instead of words… so you hardly “added” to anything.

    Frankly, I disagree with Ankita on that whole bit on the areas being equal. Yes, it may be true, but it is not obvious from the illustration. In fact, I dont see how anyone can deduce that from the drawing without knowing it to be true already beforehand.

    What “two facts”, Sol? I dont know what youre getting at. There are probably hundreds or more “facts” one must know, as prerequisite, before they can even get this far in the first place.

    Sol, why dont you consider a drawing proof? What is the difference between a diagram of a problem or a relationship and the pretty pictures we call “math symbols”? Each of these letters you are reading on the screen is a picture unto itself. They are all symbolic, pictures… both of which translate into abstractions within the mind. If it can be conceived properly then it can be reasoned about properly, regardless of how we illustrate our thoughts. If one constitutes a valid proof then why cant another? Do you not remember doing geometric proofs using pictures? Who said proof was necessary anyway? Axioms are axioms for a reason. They are accepted as intuitively obvious, first principle knowledge. If an illustration can accomplish that task where symbols cannot, then I say is permissible. As valid as any other notion of “reason”.

  4. Hi Sol. I found a neat proof on YouTube that shows (n)^2 – (n-1)^2 = n + (n+1). Do you know of a resource of very simple algebraic proofs like this one that I can do with my high schoolers? I’m not looking for geometric ones, just algebraic ones.

    I suppose I could look for “fun math facts” and build a proof from them.

  5. Shana,

    I don’t know of a resource off-hand but someone reading your comment might. Also, I’d do a Google search for algebra proof without words and look at image results as well as web results.

  6. Ok, thank you. The search continues….!

  7. Here is my favourite proof without words:
    This first appeared in a Chinese textbook about 3000 years ago!
    And of course, the picture proves ….

  8. Shana, just a remark that there is a typo. The correct identity is

    n^2 – (n-1)^2 = 2n – 1

    with -1 on the right.

    You can get plenty of such identities by picking a polynomial and getting its Taylor series in a point other than 0. For example, if P(n) = n^2, then also

    P(n) = P(1) + P'(1)(n-1) + P”(1)(n-1)^2/2

    which gives you

    n^2 = 1 + 2(n-1) + (n-1)^2.

    Which you can rewrite as

    n^2 – (n-1)^2 = 2n – 1.

    Taking the expansion at n = 2 you get

    n^2 = 4 + 4(n-2) + (n-2)^2

    so that

    n^2 – (n-2)^2 = 4(n – 1).

    If you start with P(n) = (n+1)^2 you get

    (n+1)^2 – (n-1)^2 = 4n.

  9. Thank you, Alexander. I had changed all the n+1’s to plain old n’s but went the wrong way on the right side of the equation. Thank you for your correction :^)

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