π

I've posted a couple of past columns concluding that humans, ultimately, are not as rational as we like to think ourselves to be.  What we are, I believe, is pattern seekers, and that's the basis of intuition.   You suddenly realize that one thing is like another--like Newton's realization that the falling apple and the earth's moon were bound by the same force, or Philo Farnsworth's inspiration for television from watching a till in the soil.  It's a sudden insight, an epiphany.  And fundamentally, it's irrational. 

But that brings me to another kind of irrationality:  irrational numbers.  Belatedly, I learned that March 14 is known as π day.  π, the lower case Greek letter pi, is equal to 3.14159265359…, and March 14 or 3/14 which is an estimate of π to two decimal places has been designated Pi Day. 

π is an irrational number.  It just goes on without repeating for ever and ever. 

There are lots of irrational numbers.  As a matter of fact there is a famous, elegant proof by mathematician Georg Cantor showing that the set of irrational numbers is larger than the set of rational numbers. That's a lot of irrational numbers considering there are an infinite number of rational numbers.  Cantor showed there are different sizes of infinity.  But that's another Mindfingers post.

π is perhaps the most famous example of an irrational number, and almost certainly the first one practically used.  π is simply the number of times a diameter goes into the circumference of a circle.  That is, if you took the diameter, D, in the circle below and wrapped it around the Circumference, C, it would go just over three times—π times to be exact.  

Circles, of course, are important, now as in antiquity.  The moon is a circle.  The wheel is a circle.  π was the magic number that would turn straight lines into circles.  It was known back as far as the Babylonians and Ancient Egyptians.

Despite the fact that π has been proved to be an irrational number, this hasn’t stopped people from trying to patterns within it. It has been expanded to around ten trillion decimal places, last I checked.  People are forever counting how many 5's there are, or where certain combinations of numbers appear.  There are oodles of “sacred geometry” websites professing spiritual revelation out of the understanding of π and other important irrational numbers.  

Of course, since π just keeps going, it does have the exact phone numbers, consecutively and in alphabetical order, of everyone who has read this post.

But in fact, π is not without pattern.  If it's written in its decimal format, it looks that way, but there are other ways to write numbers.  As an infinite series for example:

Looking at that, π appears quite elegant actually.  It's an accurate representation of π, but it's not very efficient.  To get π to 10 measly decimal places, you'd have to expand that series to 5 billion terms.  There are other infinite series that converge on π much more quickly, but they lack elegance of the above.   This one converges very quickly, but isn't quite as nice to look at:

Messy.  Looks impressive though, right?  And it's hypogeometric.  That sounds important!

The above mess was the work of Srinivasa Ramanujan, a self-schooled brilliant mathematician from one of the poorest parts of India in the late-19^th / early-20^th century who, with practically no formal training in mathematics, wrote to Cambridge professor of mathematics G.H. Hardy about his work on infinite series and continued fractions. Hardy, after reading the unsolicited work Ramanujan, was completely out of his depth.  He famously said that the theorems in Ramanujan's treatise "must be true, because, if they were not true, no one would have the imagination to invent them."  Ramanujan went on to a storied tenure at  Cambridge University and made phenomenal contributions to mathematics.  He died at 32, probably in no small part due to chronic malnutrition and disease as a youngster.

pic of Ramanaujan.

Srinivasa Ramanujan:  "An equation for me has no meaning, unless it represents a thought of God."

Another way of articulate a number is through continuing fractions, where π can be represented as:

These numbers, like humans, may be irrational, but they certainly aren't unintuitive.  They have patterns that can't be seen in a simple decimal expansion.  The patterns can only be seen when you include infinity--such as infinite series or continued fractions.  Rather pretty patterns, and restoring my faith in the underlying elegance of the universe. 

Without getting too geeky (I know, I know--too late), one of the most beautiful equations in mathematics is known as Euler's Identity. I've even seen a tattoo of it.

Here, e is the natural exponent.  If you've ever done much work on growth of bacteria, half-life of radiation, or compound interest, you may have come across it.  It's the "base unit" if you will, of exponential growth or decay.  And, like π, it's irrational.  2.718281828... 

The term i is the square root of -1.  But, of course, negative numbers can't have square roots which is why i stands for "imaginary number."

So you take one irrational number, e, take it to the power of another irrational number, π, times the imaginary square root of negative one, subtract 1 (known as "the multiplicative identity" in mathematics) and you get 0 (known as "the additive identity" in mathematics).  It ties together a number of disparate mathematical terms and concepts in an equation of breathtaking eloquence and simplicity.

Let's close off with a little brain teaser.  What's the solution to the infinite series below.  Answer next time.