Proof Without Words VII

From here (PDF link). Mouse-over the image for a verbal description of the proof.

Also check out Proof Without Words I, II, III, IV, V, and VI.

Proof Without Words VI

Proof that the area of a circle is equal to π × r².

Click here for details.

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Previously on this blog: Proof Without Words I, II, III, IV, and V.

Happy Pi Day 2013

From this year onwards, in addition to celebrating Pi Day on March 14th, I am going to celebrate Pi Approximation Day on July 22nd (22/7) as well. I can really use an additional day of celebration for my favorite mathematical constant!

By the way, here’s an interesting approximation of π: A nano-century is approximately π seconds long. In other words, if you divide the number of seconds in a century by one billion (nano = 1 billionth), you’ll get a result that’s close to π:

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The comic is from Dinosaur Comic.

Previous posts involving π: 3.14, A Sanskrit Mnemonic for π, Happy Pi Day!, A Mathematical Conundrum.

Proof Without Words V

Another geometric series followed by a visual proof:

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Previously on this blog: Proof Without Words I, II, III, and IV.

Proof Without Words IV

Here’s an infinite geometric series followed by a visual proof:

[Source: Mathematics Magazine, Vol. 62, No. 5]

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Previously on this blog, Proof Without Words I, II, and III.

The Square Root of 2

The square root of 2 is another one of my favorite irrational numbers. It is equal to the hypotenuse of a right triangle whose sides are equal to 1. Unlike π, e and φ, the irrationality of this number is not very apparent, hence it’s not as popular as some of the other irrational numbers. But I feel that it deserves more popularity as there’s something that we use pretty much every day that involves this fascinating number: the standard size paper.

The ratio of the two sides of an A4 size paper – the standard letter format in many countries including US – equals the square root of 2 (~ 1.41421). Because of this aspect ratio, when you cut the paper in half along its longer side, the resulting halves retain the same ratio.

To understand why the square root of 2 is irrational, we can start by asking the following question: ‘Can it be expressed as a ratio of two whole numbers?’ If the answer to this question is ‘yes’ than it is a rational number which can be expressed as n/m, where both n and m are whole numbers. By squaring both sides and multiplying by m², we get 2m²=n². This implies that it is possible to have two identical squares with their sides equal to m, whose total area is identical to a bigger square whose sides are equal to n. A visual proof emerges based on this proposition – check out this link (PDF) if you’re interested in understanding how the irrationality of the square root of 2 can be proven based on this hypothesis.

Let me end this post with a nerdy pick-up line: “You must be the square root of two, because I feel irrational around you.”

3.14

Happy Pi Day!

Remember that the “Real Pi Day” will be celebrated on March 14th, 2015 at 9:26:53 AM. These numbers (3/14/15, 9:26:53) correspond with the first 10 digits of pi (3.141592653).

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Previously on this blog: A Sanskrit Mnemonic for π, Happy Pi Day!, A Mathematical Conundrum.

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.

Proof Without Words II

Conjecture: For every natural number n, the sum of the first n odd numbers is n².

In other words:

1 + 3 + 5 + . . . + (2n – 1) = n²

For example, starting with n = 1:

1 = 1²

1 + 3 = 2²

1 + 3 + 5 = 3²

1 + 3 + 5 + 7 = 4²

1 + 3 + 5 + 7 + 9 = 5²

1 + 3 + 5 + 7 + 9 + 11 = 6²

And so on.

This conjecture can be proven by induction, but I think this little picture below does it marvelously and most succinctly:

If you start from the lower left corner, you can see that the first two odd numbers (1 and 3) make a 2 x 2 square. Hence, 1 + 3 = 2². Similarly, the first three odd numbers (1,3, and 5) make a 3 x 3 square. Hence, 1 + 3 + 5 = 3². How cool is that?!

And there’s more: this elegant scheme is also a geometric depiction of the identity (k + 1)² = k² + 2k + 1. Can you spot this proof?

[Source: Excursions in Calculus]

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Also check out an earlier post: Proof without Words.

Proof Without Words

A couple of months ago, Google honored one of the most famous problems in mathematics — known as Fermat’s Last Theorem — on the 410th birth anniversary of the man who came up with the theorem: Pierre De Fermat. Here’s the Goodle doodle from August 17th, 2011:

The Fermat’s Last Theorem is really a higher-dimensional extension of the Pythagoras theorem that operates in 2 dimensions. Do you remember the proof (or, a proof) of the Pythagorean theorem? Here’s one:

Take a look at the right angle triangle shown below, the sides are: a=3, b=4 and c=5. Every side of the triangle is expanded to create squares A, B and C. If we slice square A into smaller squares of 1×1, we get 9 small squares. Similarly, by cutting square B we get 16 small squares. In total, we have 9+16=25 small squares — which can fit into square C which needs 25 1×1 squares.

In other words, the area of square C is equal to the combined areas of A and B. Hence, a² + b² = c²! This is a visual proof of the Pythagoras theorem — a proof without words.

And Fermat’s Last Theorem is a projection of this into a higher dimensional space. He hypothesized that it is not possible to do this kind of perfect-fit re-arrangement for objects that have more than two dimensions. For example, instead of making squares by projecting the sides of the triangle, if you make cubes (i.e. 3 dimensions instead of 2) then you can’t cut those smaller cubes (A and B) in any way that can give you enough to fill the larger cube. Dissecting A and B into cubes of 1x1x1 would give us 27+64=91 small cubies, but the larger cube needs 125 to be completely filled.

Hence, aⁿ + bⁿ is not equal to cⁿ when n>2. And that’s known as Fermat’s Last Theorem.

Fermat himself left no proof of his conjecture, and it remained one of the most famous unsolved problems in mathematics. It took around 350 years for mathematicians to prove Fermat’s Last Theorem! The proof involves some heavy mathematics, and beyond the scope of this post (as well as my limited mathematical knowledge.)

What intrigues me though is the idea of ‘proof without words’ (like a layman’s proof I explained above.) For Pythagorean theorem there exists numerous such proofs that require not even a single word to explain. Here are few of my favorite examples of such visual proofs:

I think the following two are really splendid and quite elegant:

Click to embiggen – it may take some time to spot the proofs in these patterns. The second proof by dissection above is actually ascribed to Bhaskara, the great Indian mathematician from the 12th century.

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Image courtesy: The first demonstration (proof by rearrangement) is from the Wikipedia link on Pythagorean Theorem, and the other two (proofs by dissection) are from this paper. There are many more such proofs on this wonderful page.

A 48-minute documentary about the mathematician who solved Fermat’s Last Theorem is available here.