Proof Without Words III

The beautiful thing about geometric proofs is that – in addition to proving geometric and arithmetic concepts – we can use it to explain some abstract equations in algebra as well. Once you connect the dots between algebra and geometry, abstract algebraic equations start making sense like never before.

For example, think about how we can illustrate squaring a number geometrically. We first start with a single number. And single numbers can be shown as points on a line:

In that sense, numbers live in a one-dimensional space. When we square a number we move to a two-dimensional space, and instead of a line we now have a square.

Obviously, the sides of the square are equal to the original number we started with. (And we’re now in a two dimensional space because in order to specify the location of a point within the square, we need two numbers {x, y}.) Similarly, when we multiply the same number three times, we move from a 2-D space to a 3-D cube.

Now keeping that in mind, consider the equation (a+b)2 = a2 + 2ab + b2:

Of course, we could have proven this equation by opening up the bracket on the left hand side – like (a + b) times (a + b) – and solved it like we did in school. But the above mentioned proof is pretty elegant and sensible, isn’t it?

Also check out Proof Without Words I, and II.

Hat tip goes to a couple of friends on Facebook who posted this video of a desi dude (math teacher?) explaining this “mathemagic” on a blackboard.

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3 responses to “Proof Without Words III

  1. We had this this visual proof in our 7th standard math books (Maharashtra State Board)!

    • Lucky you! We didn’t have such thinking-outside-the-box curriculum/textbook in Gujarat. We proved that equation by hand (by opening up the parenthesis) and didn’t really understood it.

      • Not that lucky ;-) We had that in our books, but they don’t teach it to us as it would not come in the exam. We too learned it through opening the parenthesis. I noticed that in my 7th std book when I was in 8th and was revising my 7th std book for Teacher’s Day (We used to teach lower classes that day)! I did tell it to my Teacher’s Day students!

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