It’s Leonhard Euler’s birthday today! He was born in Basel on April 15, 1707.

Euler was a relentlessly energetic mathematician, physicist, astronomer, geographer, logician and engineer who founded the subjects of graph theory and topology and made pioneering discoveries in analysis, number theory, mechanics, fluid dynamics, optics, and even music theory. When he lost sight in his right eye, he remarked

“Now I will have fewer distractions.”

and even when he became almost blind in his other eye he averaged one paper a week.

I love how he seized the day and explored all the possibilities available to him. Harnessing the tools of calculus, he came up with dozens of amazing formulas, like this:

But how do you find a formula like this???

It’s not magic: it uses a method called ‘Euler’s continued fraction formula’ which can produce many similar formulas. This method is not easy to guess, but you can check it using high-school algebra, especially if you don’t worry about convergence—which Euler didn’t, not much anyway. Then, to apply this method and get Euler’s formula for 4/π, you need to know some calculus.

Let’s see how it works!

Here is Euler’s continued fraction formula:

It’s a way to rewrite a sum

a_0 + a_0 a_1 + a_0 a_1 a_2 + \cdots + a_0 a_1 a_2 \cdots a_n

as a fraction. To check it, just try some small choices of n like 1 and 2 and 3. Use the rules of algebra to simplify the fraction at right. You’ll get the desired sum! With some thought, you should be able to see the pattern of why it works for any n.

There are probably better things to say about this, but I don’t know them. So, I’ll just show you how to use this trick to get Euler’s formula for 4/π.

There are many ways to apply Euler’s continued fraction formula. For example, we can start by remembering that

\displaystyle{ \int_0^x \frac{1}{1+t^2} \, dt = \arctan x }

Then we apply the geometric series

\displaystyle{  \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots }

to write

\displaystyle{  \frac{1}{1+t^2} = 1 - t^2 + t^4 - t^6 + \cdots }

Integrating this from 0 to x, we get a nice infinite series for arctan(x).

To apply Euler’s continued fraction formula, we need to massage this series until it’s of the form

a_0 + a_0 a_1 + a_0 a_1 a_2 + \cdots

Luckily this is not hard to do:

Next, hit this formula for arctan(x) with Euler’s continued fraction formula!

The result looks pretty complicated, but we can simplify it a bit by multiplying the top and bottom of one fraction by 3, another by 5, and so on:

Next, to get something impressive and cover our tracks, let’s choose some very simple number x whose arctangent is also nice. Like x = 1. The arctangent of 1 is π/4, since a line of slope 1 has an angle of 45° from the horizontal. So taking x = 1, we’ll get a cool formula for π/4:

This was not a rigorous proof by today’s standards, since I didn’t investigate whether any of the series converge. But neither did Euler! He probably thought about it, but he didn’t let it slow him down.

You have to imagine Euler spending all day, every day, trying stuff like this.

For more fun examples of what you can do with Euler’s continued fraction formula, go here:

• Wikipedia, Euler’s continued fraction formula.

I think the moral of the story is that to achieve greatness, it helps to take what you’re good at and run wild with it: try everything you can, and let people know about the good stuff.

Happy Euler’s day!

5 Responses to Euler

  1. OkCarl says:

    Very beautiful. Thanks.

  2. Jesús López says:

    An answer in Math Overflow gives a nice application of this Euler equivalence to the \zeta-function.

  3. I feel someone should mention the MAA column “How Euler Did It”,

    • John Baez says:

      The review says:

      After all, it’s quite impossible, in a single book, to cover the total work of a man whose mathematical and scientific output occupies 640 volumes pertaining to number theory, analysis, geometry, astronomy, graph theory, differential equations, logic etc. But whether or not the 40 themes covered in this book are truly representative (and I feel they are), Ed Sandifer’s exposition is nonetheless both informative and entertaining.

  4. Euler’s Equations would’ve been the foundation of fluid mechanics instead of Navier-Stokes … but he encountered friction.

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