Adding Time On A Calculator: Fill & Download for Free

Let’s do an apples-to-apples comparison.“Hello, World!” is 13 characters long. Assuming an 8x8 bitmap font, let’s say we can fit it 37 times on a line of a 4K screen, which is 3840 pixels wide (in order to do that we’ll shave off one pixel from the blank space). The same 4K screen has a height of 2160 pixels and can therefore hold 270 lines. So we can fit “Hello, World!” 37*270 = 9990 times on a 4K screen. Let’s round that up to 10k, for convenience.Assuming a frame rate of 120 Hz, we can display “Hello, World!” about 1.2 million times per second. That means it will take about 834 seconds, i.e. just under 14 minutes to display “Hello, World!” a billion times. Let’s say it takes 15 minutes max. Scale that up by the appropriate small integer factors for a smaller display and/or lower frame rate. On a 1920x1080 display that means 1 hour at 120 Hz as before, 2 hours at 60 Hz, or 4 hours at 30 Hz. Definitely not forever.To do this we only have to render “Hello, World!” as a 103-by-8 bitmap once and then repeatedly copy that into a frame buffer. For fairness, we’ll clear every frame and build it up from scratch. A game engine would do a whole lot more to fill up the frame buffer. It’s a completely safe bet that rendering “Hello, World!” this way is not any slower than rendering a smoothly running game.In this setup, the dominating bottleneck is the time it takes to display a frame. This is controlled by the screen size and refresh rate. Those parameters will dominate any calculation of effective running time. You can’t render the information any faster than screen size and refresh rate will allow.As the other answers have said, if you print to a terminal, then you’re employing a completely different output mechanism. It’s no longer a fair comparison. You might reasonably ask why terminals are so slow, and the simple answer is that they don’t have to be fast. You don’t need high refresh rates for displaying text at a reasonable size (presumably larger than 8x8 pixels) that people want to read.Edit: Added image; fixed a thinko.

The modern way of doing that is with series, but that wasn't the way it was done in antiquity.Creating trig tables was a major task. Ptolemy (100–178) produced one of the earliest tables for trigonometry in his work, the Almagest, and he included the mathematics needed to develop that table. It was a table of for every angle from 1/2° through 180° in intervals of 1/2°. Also he explained how to interpolate between the given angles.First, based on the Pythagorean theorem and similar triangles, the sines and cosines of certain angles can be computed directly. In particular, you can directly find the sines and cosines for the angles 30°, 45°, and 60°. Ptolemy knew two other angles that could be constructed, namely 36° and 72°. These angles were constructed by Euclid in Proposition IV.10 of his Elements. He then used complementary angle formulas, half angle formulas, sum formulas and difference formulas to get trig functions for all angles which were multiples of 1 1/2°. Interpolation gave him the rest.Using essentially these methods, Ulugh Beg (15th century) constructed sine and tangent tables for every minute of arc to about nine digits of accuracy!Ulugh Beg Observatory, Samarkand, UzbekistanI should mention also that Aryabhata (476–550) in India developed a method to create table of sines using second differences, a discrete analog of the second order differential equation that the sine function satisfies.Slide rules are good for finding trig functions to three decimal places. They're essentially analog trig tables.Power seriesIn the late 17th century, Newton and other mathematicians developed power series. A power series is like a polynomial of unbounded degree. Here are the power series for sine and cosine (where x is an angle measured in radians):The three dots ... mean that the expression is to go on forever, adding another term, then subtracting a term, etc. The exclamation point ! is to be read “factorial”, and it means you multiply together the whole numbers from 1 up through the given number. For example, 5!, “five factorial”, equals 1 times 2 times 3 times 4 times 5, which is 120, and so, 6! = 720.These power series have infinitely many terms, but they get small so very fast that only the first few terms contribute much.Suppose you want to compute the sine of 45°, correct to some number of places, using this power series. First convert 45° to radians to get π/4, which is 0.78539816 to eight places. Then compute the value of[math]0.78539816 - \frac{0.78539816^3}{3!}+\frac{0.78539816^5}{5!} [/math][math]{}- \frac{0.78539816^7}{7!} + \frac{0.78539816^9}{9!}-\cdots[/math]You’ll find the following partial computations[math]0.78539816 = 0.78539816[/math][math]0.70465265 = 0.78539816 - \frac{0.78539816^3}{3!}[/math][math]0.70714304 =\mbox{ that } + \frac{0.78539816^5}{5!}[/math][math]0.70710647 =\mbox{ that } - \frac{0.78539816^7}{7!}[/math][math]0.70710678 =\mbox{ that } + \frac{0.78539816^9}{9!}[/math]The correct answer is the square root of 1/2, which is 0.70710678. Only four terms of the power series were needed to get the first five places, and the next term gave the next two places.Computers and calculators use power series to compute sines and cosines.Reference: Computing Trigonometric Functions

Ask and ye shall receive.Now, I feel like I should build up to this because it took me about 3 hours to count every line in that show.God I wish I were joking.Ok, here was the process:First, I went through Hamilton: The Revolution page by page and counted the number of lines per character per song and noted it on a sticky note. To count a “line”, I followed a combination of capitalization and the line breaks denoted in the script. I only counted direct references to a character (“company”, “ensemble” and “men/women” were disregarded). The results are, of course, a rough estimate.A lot of sticky notesThen I took those numbers, and wrote them on paper per character.Gross.Then I added up each character’s total line count from every song. I did use a calculator, because I don't hate myself that much.Based on this, the character with the most lines is…Hamilton!This actually surprised me, to be honest. I expected Burr to win, only because he is the narrator.But Jordan, you say. I would also love a full breakdown of every character in the show and their exact line counts.If you insist.Alexander Hamilton: 916Aaron Burr: 655Eliza Hamilton: 324George Washington: 300Angelica Schuyler: 246Thomas Jefferson: 234John Laurens: 163Marquis de Lafayette: 129Hercules Mulligan: 115James Madison: 112King George: 100Philip Hamilton: 85Peggy Schuyler: 42Maria Reynolds: 31Samuel Seabury: 28Charles Lee: 15George Eacker: 11James Reynolds: 10Doctor: 6I hope at least one person finds this information useful. This was an obnoxious amount of time spent on a fairly useless task.But I did find it extremely interesting.Edit: I feel compelled also to break this down by actor (assuming original cast). Please excuse me.Lin-Manuel Miranda: 916Leslie Odom, Jr. : 655Daveed Diggs: 363Phillippa Soo: 324Christopher Jackson: 300Anthony Ramos: 248Renee Elise Goldsberry: 246Okieriete Onaodowan: 227Jonathan Groff: 100Jasmine Cephas Jones: 73Thayne Jasperson: 28Sydney James Harcourt: 16Jon Rua: 15Ephraim Sykes: 11This moves Tony-award winner Daveed Diggs up significantly, which I find interesting. It's worth noting that the show’s Tony nominees (with the exceptions of Anthony Ramos, who was not nominated, and Jonathan Groff, who was) had the most lines overall. However, the show’s winner for Best Lead Actor did not have the most lines (Leslie Odom, Jr.) as compared to the show’s other nominee in this category (Lin-Manuel Miranda).Interesting.