Let’s look at how it works for pi. John Heidemann at the Information Sciences Institute at USC has a list of all the best rational approximations (of the first kind) of pi with denominators up through about 50 million. The numbers 3/1, 13/4, 16/5, 19/6, and 22/7 are the first few fractions on this list. 13/4 is a little closer to pi than 3 is. It’s about 0.1084 away instead of 0.1416. But if we multiply the differences by the denominators, 13/4 doesn’t do so well. We get 0.1416×1=0.1416 and 0.1084×4=0.4336, so 13/4 loses pretty badly. The same is true for 16/5 and 19/6. Both of them are a little closer to pi, but they aren’t enough closer to make up for their larger denominators. Thus, 13/4, 16/5, and 19/6 are best approximations of the first kind but not of the second kind. But when we get to 22/7, things change. It’s quite close to pi. Its difference, 0.00126, is very small. If we multiply it by its denominator, we get 0.00126×7=0.00882. This beats 0.1416 pretty handily, so it’s a best approximation of the second kind. It’s also the next convergent in the continued fraction for pi.
My favorite convergent in the continued fraction for pi 355/113. It’s a really good approximation. It’s a continued fraction mic drop. The next best approximation of the first kind has the denominator 16,604. But it isn’t even that much better than 355/113. We don’t get another decimal digit of accuracy, but we have to increase our denominator by two decimal places. No thanks. We have to increase to the denominator 33,215 before we get a new approximation that’s worth bothering with.
More retrocomputing .. My high school physics class had a storage room filled with demo apparatus from earlier years. Our teacher had little knowledge of what his predecessors managed to acquire and squirrel away. Two of us managed to get an old Heath EC-1 analog computer working. A great way to learn a bit about electrical engineering - how op amps work, the need for precision components, oscilloscope use, etc. This type of computer is also known as a differential analyzer as it is very well suited for solving differential equations. Trying to program it taught me a fair amount of basic physics and math. You had to understand how to formulate an equation to get the wiring right.
Here is a sample worked problem to solve the motion of a mass on a spring. You would connect up four op amps and other elements like precision potentiometers, a power supply and an output oscilloscope with what could be a rat's nest of patchcords.
Analog computers were heavily used in the aerospace industry post WWII up through much of the 1960s.
Heath had a larger model intended for engineers and universities - the ES-400. I haven't been able to find a full manual, although it is made of basic elements similar to the EC-1 .. just more of them. It was probably users knew what they were doing. You bought module kits and put them in a big chassis. The "full computer" cost $945 in the day - about $8,200 in 2015 money. It lacked a means to display the output and important input elements like function generators. No big deal as anyone who would possibly use an analog computer already had such kit.
For regions like the US with a mm/dd/yy calendar order today gives 3/14/15 -- the first five digits of Pi 3.1415. going a bit further you get the first ten digits - 3.141592653 - at 9:26:53.
Only one day every century has this relation. And that brings up a curious relation with Einstein whose birthday is today. In 1915 he published his paper on General Relativity. To call it remarkable is understatement of the grandest order.
There is a foundational paper published a bit earlier with an astonishing opening paragraph. A non-technical phrasing might be: “Here is my theory of the dynamics of space and time, with an introduction to its mathematical underpinnings, as well as derivations of all the previous laws of physics within this new framework.” He mentions Grossman, but that was help with the math rather than the underlying physics. I doubt we’ll ever see such a dramatic leap penned by an individual for a theory that holds up to experiment. And it has held solidly for a century!
What you need to do is celebrate. He had a sweet tooth and liked vanilla ice cream cones with chocolate sprinkles. He also had a fondness for fruit, particularly cherry, pie - so perhaps a slice of cherry pie with a scoop of ice cream is appropriate. You might throw care to the wind and add some chocolate shavings or sprinkles...
And it is a day to celebrate William Jones - the person who believed (but didn't prove) the ratio of the circumference to the diameter of a circle was irrational and should have it's own symbol.
the symbol for this ratio known today as π (pi) dates from the early 18th century. Before this the ratio had been awkwardly referred to in medieval Latin as: quantitas in quam cum multiflicetur diameter, proveniet circumferencia (the quantity which, when the diameter is multiplied by it, yields the circumference).
It is widely believed that the great Swiss-born mathematician Leonhard Euler (1707-83) introduced the symbol π into common use. In fact it was first used in print in its modern sense in 1706 a year before Euler's birth by a self-taught mathematics teacher William Jones (1675-1749) in his second book Synopsis Palmariorum Matheseos, or A New Introduction to the Mathematics based on his teaching notes.
Before the appearance of the symbol π, approximations such as 22/7 and 355/113 had also been used to express the ratio, which may have given the impression that it was a rational number. Though he did not prove it, Jones believed that π was an irrational number: an infinite, non-repeating sequence of digits that could never totally be expressed in numerical form. In Synopsis he wrote: '... the exact proportion between the diameter and the circumference can never be expressed in numbers...'. Consequently, a symbol was required to represent an ideal that can be approached but never reached. For this Jones recognised that only a pure platonic symbol would suffice.
I have not come across any indication of Jones' dessert preferences. Then again he was Welsh. Pie in the day was likely meat filled.
From a paper by Francis Glaton that appeared in the April 5, 1977 issue of Nature. He published a series on heredity which is seen as a big step in the development of correlation and regression analysis.
For the handful of RPN calculator fans - a community project that goes well beyond the capabilities of the old HP 42S. The WP 34S does Riemann's Zeta functions and Legendre polynomials. I have more function with Wolfram alpha on my iPhone, but this is strangely attractive. There is also an iPhone version, but somehow the physical unit is desirable.