Pi Day Paradox

Here’s the infamous “proof” that shows that  π is equal to 4.


The zig-zagged polygon, whose perimeter always remains 4, does appear to approach the circle as we repeat these steps to infinity. Ergo, π is equal to 4!

The problem, as Vi Hart explains in this amusing math doodle video, is that while the area of the polygon does approach the area of the circle, the actual perimeter of the polygon is much larger than the circumference of the circle. Confused? Think of it this way: if you put a jumbled up 10 feet long rope into a 1 foot long container, you wouldn’t say that the rope is now 1 foot long, would you? You would take out the rope, and extend it fully to measure its actual size.

See some more discussion here. Also, here’s another video that “proves” — using a similar approach — that π is equal to 2, and the square root of 2 (another one of my favorite irrational numbers) is also equal to 2!


By the way, if π were actually equal to 4, all circles would be squares. What a terrible world that would be! Similarly, if π were equal to 3, all circles would be hexagons. (For a hexagon, the ratio of its circumference to its diameter is equal to 3.)

Oh, and happy pi day! Don’t forget to have some delicious pie, and as you eat it, marvel at the glory of this magnificent, transcendental, and most importantly, irrational number.

i ate some pie***

Previously on this blog: Proof Without Words IVHappy Pi Day 20133.14A Sanskrit Mnemonic for πHappy Pi Day!A Mathematical Conundrum.


One Trillion

One hundred dollar bill ($100)


Ten thousand dollars ($10,000)

ten thousand

One million dollars ($1,000,000)

one million

One hundred million dollars ($100,000,000)

hundred million

One billion dollars ($1,000,000,000)

one billion

One trillion dollars ($1,000,000,000,000)

one trillion

That escalated quickly, huh?!



Previous “Sense of Proportion” posts: IIIIII, IVVVI, VII.

Belphegor’s Prime

1000000000000066600000000000001  is a palindromic prime number, which contains the number of the beast (666) surrounded by thirteen zeros on each side. The number is called Belphegor’s prime, after the demon (depicted below) with the same name.





“Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. Immortality may be a silly word, but probably a mathematician has the best chance of whatever it may mean.” — G H Hardy

[See Why is the number 1,729 hidden in Futurama episodes?]

Proof Without Words VII

pww vii

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

Also check out Proof Without Words IIIIIIIVV, and VI.

Proof Without Words VI

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

04strogatz5-custom1Click here for details.


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:

One nano-century is approximately π seconds long.


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 IV

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

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


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.”

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