32

According to the paper of Davies cited by MJD, Archimedes actually gives a double inequality $$\frac{265}{153}<\sqrt{3}<\frac{1351}{780}.$$ As both of these fractions are not just random approximations, but are the 9-th and 12-th convergents for the continued fraction expansion of $\sqrt{3}$, there is no doubt that Archimedes used the continued ...


30

The idea was to simplify multiplicaton of numbers. If you ever tried to multiply $10$-digit numbers by hand you will see what I am talking about. The idea is this. We have $a^{m+n}=a^m a^n.$ On the right hand side we have a product of $a^m$ and $a^n$, while on the left hand side a sum $m+n$. So if you write two progressions, one arithmeric and one geometric ...


25

The literature is ambiguous because the question is ambiguous. The characteristics of these early machines are well known, but one can set the criteria for being "electronic computer" according to their preference. Does it have to be completely non-mechanical? Turing universal? digital? binary? History can tell us what happened, not what choice of words we ...


18

There could not have been a $\pi$ day in 1592 regardless of calendar conventions for the simple reason that there was no such thing as $\pi$ back then. The symbol was introduced by William Jones in 1706 and did not come into common usage until after 1737, when Euler popularized it in his texts. This was similar to zero, which got a placeholder symbol long ...


17

From the MacTutor biography of Kepler: Calculating tables, the normal business for an astronomer, always involved heavy arithmetic. Kepler was accordingly delighted when in 1616 he came across Napier's work on logarithms (published in 1614). However, Mästlin promptly told him first that it was unseemly for a serious mathematician to rejoice over a mere ...


13

Perhaps one of the most famous computational model discoveries of the 20th century is Lorenz's observation in 1961 of chaotic behavior in a weather model. It overturned the existing consensus in meteorology, and led him to the discovery of a strange attractor in a simplified model of atmospheric convection. The implications are described in the now famous ...


13

Yes, it has been: , or more stylized , the depression made by the tip of a Babylonian wedge shaped stylus on a clay tablet. When a circular stylus was used (rarely) the symbol was just $\bigcirc$. The earliest positional system was sexagesimal, with base 60, so it had cuneiform symbols for all digits from 1 to 59. Babylonians used it since before 2000 BC for ...


9

If you are interested in descriptions of “everyday life” of human computers, here is an excerpt from Stan Ulam’s autobiography, Adventures of a Mathematician (University of California Press, 1991) concerning the years 1949–1952, when he was a part of the team working in Los Alamos on thermonuclear explosion. The account of the mathematical problems involved ...


7

Why does being decimal disqualify ENIAC? Decimal electronic stored-program computers used to be a thing, for example the commercially successful IBM 1401. It was a matter of "binary for scientific use, decimal for business use". My own bias as a programmer disqualifies ENIAC for not being a stored-program digital computer, or more accurately, I'm more ...


6

Posting History It was already mentioned that the Mesopotamian (hexagezimal system) used a special symbol for our "10". Also, In the modern hexadecimal system we have A for "10". The Roman and the Greek specialties were also mentioned. New stuff To add something to the already existing posts I copy here a paragraph from my own lecture notes about numbers ...


6

Lamb discusses the issue in How Much Pi Do You Need?: "I asked a NASA scientist how many digits of pi the agency uses for its calculations. Susan Gomez, manager of the International Space Station Guidance Navigation and Control (GNC) subsystem for NASA, said that calculations involving pi use 15 digits for GNC code and 16 for the Space Integrated ...


6

To keep secret the level of expertise in cryptanalysis so future opponents wouldn't put effort into improving their own codes. Probably pointless because it was inevitable that some details of Bletchley would leak to the USSR, in the same way as secrets of the Manhattan project. And probably equally pointless in that, although every side in WWII broke almost ...


6

There's a very simple geometrical response that provides a rounded solution which might reflect on this problem at http://www.gjbath.com/SQR3.htm. It's so much easier seeing it! This involves circumscribing an equilateral triangle and taking a diameter from the apex. Then, applying $26/15$ as an approximation to $\sqrt 3$ ($\tan 30^\circ\approx 15/26$) and ...


6

Multiplication is a lot of work; in numerical computing it is considered evil and many tricks are used to avoid it. So Napier created the table of logarithms. (Briggs worked with Napier to make the table more useful.) IIRC, Napier's logarithms were to base 0.9999999. However, keep in mind that not all multiplication is evil. Particularly multiplication ...


6

Archimedes might, of course, have used a better method. (He also produces the approximation $\frac{1351}{780}$, for which the foregoing is not obviously practical...) I disagree on the foregoing not being practical: it's a trivial computation compared to some which were carried out by hand by later mathematicians. But I think that really is the nub: any ...


5

Some information can be found here 1 Using the secant method on the parabola $y = x^2$, a new estimate $x_{n+m}$ for $\sqrt{3}$ can be obtained from $x_n$ and $x_m$ as $$x_{n+m} = \frac{3 + x_n x_m}{x_n + x_m}$$ where if we start with $x_1 = \frac{5}{3}$ then $x_{2n} > \sqrt{3}$ and $x_{2n+1} < \sqrt{3}$. We therefore obtain $$x_1 < x_3 < \sqrt{...


5

John Napier (1550-1617) published his table of logarithms Mirifici Logarithmorum Canonis Descriptio in 1614 after some twenty years of work and described his method of construction in Mirifici Logarithmorum Canonis Constructio, published posthumously in 1619 (Edinburgh) by his son Robert, with appendices by Napier and Henry Briggs (1561-1630). Briggs worked ...


5

Decimal "10" = Hexadecimal "A" Maybe you don't consider that a "digit", but we'll necessarily have to broaden the definition of "digit" to include anything that's not 0-9.


5

There is a heap of answers to this question, and they cover a wide range of aspects of the early computer industry. As has been pointed out, the very first machines were hand built lab experiments. It was not at all obvious how to build a digital computer that would work. Remember Babbage's Analytical Engine had foundered on the challenge of actually ...


4

Mauro has already mentioned the Roman numerals. The Greeks of the classical period used the letters of the alphabet to represent numbers. The first nine letters (A to Θ) stand for the units from 1 to 9, the next nine letters (I to Ϙ) stand for the multiples of ten (10, 20, 30…), the next nine (from P to ϡ) are the hundreds (100, 200…). You can write (for ...


4

François Viète computed $\pi$ to nine decimal places in 1573. He was also intimately aware of the Gregorian calendar, though as a denouncer rather than a supporter of the calculations used to obtain it. (That said, you could just as well celebrate Ultimate Pi Day in the Julian calendar. What's more important is that you were using Anno Domini years, which ...


4

Quick Internet search says that feather quills were mainly used in 600-1800 AD. After that people gradually switched to steel. First true mathematicians (Babylonians) wrote on clay tablets. In the Greek/Roman Antiquity they wrote on papyrus, presumably with reed pens or brushes. Apparently they switched from brushes to pens in Ptolemaic Egypt, that is ...


4

First machines were one-off affairs, custom built (often for a very narrow purpose). There just weren't enough of those around to make standarization of anything surrounding them worthwhile. Thomas Watson (IBM president at the time) accurately estimated a demand of half a dozen computers worldwide in that timeframe. Remember the idea behind the Multics ...


3

My personal marker is stored program control, ie, the program is stored in the computer memory. Early machines like the original ENIAC, Colossus, Zuse's Z-machines, were all programmed by changing the wiring in some fashion. The Z-machines used relays, ENIAC was an advance because it used electronics (valves, the transistor hadn't been invented). The first ...


3

The first memo, on orbit determination, that Katherine Johnson did with a co-worker is The Determination of Azimuth Angle at Burnout for Placing a Satellite over a Selected Earth Position 1960. T.H. Skopinski, Katherine G. Johnson, NASA TN D-233 This example from the Hidden Figures book provides some details of the actual computations that she did, checking ...


3

According to Wikipedia, it is the Austrian mathematician Gerhard Wanner.


3

This changed in 1964 when IBM introduced System/360, a scalable computing system. It's obvious now that we have scalable computing systems that we can have scalable computing systems. Before then, it wasn't. There were in my opinion two insights that made this possible: Firstly, the separation of architecture from implementation: that the machine as ...


2

Actually you can use any symbol to represent "Zero". Suppose we take it "X". But in whatever index you are working with (let us say our index is "n"). the $n^{th}$ number in that index has to be (#symbol for "1")(symbol for "0"). For example if your arithmetic is based on index of "7" and we use "A" for "1" "B" for "2" "C" for "3" "D" for "4" "E" for "...


2

In cryptography, designs often have "constants": effectively parameters which can use an arbitrary number (or an arbitrary number which doesn't have any simple patterns) but which must use the same number in all implementations. However, it's considered suspicious if the numbers are truly arbitrary: perhaps you calculated them in such a way as to create a "...


1

Radio navigation was used on a large scale during the "Battle of Britain" in WWII, to direct the fighter planes to the enemy bomber formations. Of course, they experimented much earlier than that, even during WWI, but practical use on a large scale started during WWII. See Wikipedia article "Radio navigation".


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