My username, on the other hand, is what happens when UNIX systems have 8 character limits. (IRIX, 1994; my father was "percival" so I got cperciva as my university systems account, and I've used it ever since.)
My sister's university (1990's) gave out email addresses to students, using the first initial, first three letters of the last name, and a number to designate how many times that combination had been used. Her name is Amanda Nal___. Her email address was <anal1@___.edu>.
I feel your pain. Mid 90s Unix systems seemed to really start butting heads with this limitation. Before then, there were so few users of the system that it didn't really matter (there was only one Bob), but the explosion of the Internet in the mid 90s meant that lots and lots of people suddenly needed usernames and that limit started to be a real problem.
, either ("Usernames can only contain letters, digits, dashes and underscores, and should be between 2 and 15 characters long. Please choose another.")
I hope I remember this word if I ever have to teach how LZW compression works. It's better than 'banana'.
> Recorde proposed three mathematical terms by which any power (that is, index or exponent) greater than 1 could be expressed: zenzic, i.e. squared; cubic; and sursolid, i.e. raised to a prime number greater than three, the smallest of which is five. Sursolids were as follows: 5 was the first; 7, the second; 11, the third; 13, the fourth; etc.
> Table of powers, symbols and names or descriptions form 0 to 24 by Samuel Jeake, written in 1671
Therefore, a number raised to the power of six would be zenzicubic, a number raised to the power of seven would be the second sursolid, hence bissursolid (not a multiple of two and three), a number raised to the twelfth power would be the "zenzizenzicubic" and a number raised to the power of ten would be the square of the (first) sursolid. The fourteenth power was the square of the second sursolid, and the twenty-second was the square of the third sursolid.
I see you've never had to deal with IUPAC systemic naming? For example paracetamol's systemic name is N-(4-hydroxyphenyl)-Acetamide. Cholesterol's is 2,15-dimethyl-14-(1,5-dimethylhexyl)tetracyclo[8.7.0.02,7.011,15]heptadec-7-en-5-ol.
That at least has a logical meaning, and you really can't simplify it because the thing it represents is that complicated. Using a complicated system of strange words to represent numbers is incomparably worse.
Meh...it's more of a relic of a weird transition point when chemists had the tools to discover complete structures but didn't have the tools to communicate them robustly other than "in words". These days most non-trivial chemicals are described using their "common name" (e.g. Cholesterol), and if you need a robust way of communicating structure there's SMILES (e.g. C[C@H](CCCC(C)C)[C@H]1CC[C@@H]2[C@@]1(CC[C@H]3[C@H]2CC=C4[C@@]3(CC[C@@H](C4)O)C)C for Cholesterol).
Almost no one will attempt to parse SMILES for non-trivial molecules by hand, but almost every computational package can easily parse them.
That's a tree based molecule language with the parens right? This must be for organic molecules mostly then because you can repeat atoms as in, i.e. crystal molecules?
SMILES is usable for any molecule (i.e. any compound with covalent bonds; crystals are generally not considered molecules), though it's most common use is for organic molecules. Parens indicate branching, and cycles are conceptually treated as branches that revisit the same atom twice.
Yes and no. Yes the system is logical and systematic. However, there are tons of named groups and then there are groups of groups and they all have different and arbitrary priority rules. It's actually quite a clusterfuck for what it does and doesn't really have any real purpose on mildly complex molecules.
Nobody in it's right mind will ever call it by it's real name except maybe to sound more intelligent. Most important molecule have short names either based on the standard or given names.
Math was far wordier back then, the only other weird part is breaking the exponent into its prime factors which doesn't feel terrible to me if that is how people computed out or manipulated them.
Eg 2^12 being written 2^(223) has clear advantages
I first saw this word in the roguelike game "Dungeons of Dredmor". I didn't know it was a real English word until now. In the game it's a a magic power buff, stackable up to three castings, which makes sense given the real meaning. Doesn't raise the stat to the 8th power though, which would be very overpowered.
And this is what exponents replaced. I think mathematical notation is like vim, it's easy to use but hard to learn. For example, I cannot imagine having to express perturbative series in this notation or any serious physical model with the "wordy" descriptions from history.
Things like this, or the advantages of Arabic numerals instead of Roman, make me wonder what sort of mathematical insights we're missing out on today due to our current notation.
Sure. Any choice might have some downside. One thing very interesting I always think about is, as I hinted at in my post, is the concept of series. Consider this example
1 + c_1 a^2 + c_2 a^3 + ...
Common syntax makes multiplication between factors implicit, basically grouping factors together in a compact form. The exponents further compactify the visual presentation. This, in a way, makes each term in the series stand alone and appear as a unit. It seems then no coincidence that in physics and math we often consider partial sums as approximations, and concentrate on particular terms seperately. In physics for example, people think of "order \alpha^2 terms" or "higher order corrections" or physical quantities which are often taylor series cut off at some order, of course, assuming the quantity. A very familiar example of this are Feynman diagrams which are fancy taylor series in powers of force coupling constants.
One wonders which came first. It may be that the notation followed this focus on individual terms, but it is interesting that no one considers "expanding" transition amplitudes in infinite products or continued fractions. Also, there certainly exists much more knowledge (theorems, technology) around series than products AFAIK. Again, I don't know which came first because I don't know too much about math history, but it seems reasonable that the causation may be reversed.
> It seems then no coincidence that in physics and math we often consider partial sums as approximations, and concentrate on particular terms seperately. In physics for example, people think of "order \alpha^2 terms" or "higher order corrections" or physical quantities which are often taylor series cut off at some order, of course, assuming the quantity.
There are very good reasons for this. The whole observation is basically the same phenomenon as someone observing, "you know, 34,825,119,276 and 35,174,884,395 are basically the same number for my purposes; I'll just call it 3e10 or, if I'm being really fancy, 3.5e10".
In these applications, the series variable is a very small number. The higher exponents given to it in later terms of a taylor series cause those terms to be very small compared to the early terms. That's why we take the early terms as an approximation to the whole thing -- we have chosen our representation so that that will be true.
(This is the entire reason for Taylor series in the first place -- a Taylor series is a Maclaurin series adjusted so that the variable can be small for purposes of the series, no matter what its absolute value might be.)
Yes, I think the parent realises that. The point I think is that this stuff would be much harder to realise if we used verbose and prosaic descriptions instead of the visually suggestive modern notation (implicit multiplication etc)
As I read the parent comment, it suggests that the reason we think of the first few terms of a Taylor series as approximating the whole thing is that the notation suggests that the series is composed of a sequence of discrete terms. We know that this is wrong; Taylor series were developed so that the first few terms would approximate the whole.
The notation came from trying to write down short polynomials, I think (quadratics, cubics, etc). Which before algebraic notation, was a huge pain in the butt.
Series notation is just "this is a like a polynomial, but it goes on forever, so here are the first few terms."
By the way, this system of naming exponents looks a lot like Roman numerals to me. Pre-pending a name = raise to higher powers.
It's a bit more ad hoc, because pre-pending a name multiplies the exponent (rather than adding, as Roman numerals do) -- making primes impossible to express. So they need the notion of a "sursolid" to express exponents (like 5 or 7) that do not factor into twos and threes.
So, amazingly, he was able to create a system more ad hoc than Roman numerals.
For any notation, there are trade offs between ease of learning and power after learning. This is the same for mathematical notations as it is for programming languages. If you can effectively subsidize a complex notation/language such that everyone knows it (for example, through primary education), it pays dividends in many locations. There's a reason we use '+' for addition in many programming languages, and it's because it's familiar to everyone. Why is that? Because we all learned it long ago and have been using it ever since.
This is the difference between easy and simple. Easy to use is often hard to learn. This is simple. Many of the best tools are this way. Violins are another great example. I like clojure for a similar reason:
Rich Hickey explaining the difference at rails conf: http://m.youtube.com/watch?v=rI8tNMsozo0
I look at it kind of like Perl. You can express ideas in very compact and powerful ways, but the more clever you get the harder it is for other people to read what you've written. Eventually people start to complain that you're working in some sort of write only language that looks like gibberish to all but a small handful of experts.
Which book, the Principia Mathematica Philosophiae Naturalis?
The thing is, he worked really hard to avoid calculus because it was too new and not widely accepted. Whenever he could give an argument without calculus, he would. It is hard to read but because he's trying to write calculus in the style of Euclid. So most proofs that would involve limits or derivatives would be written in a really roundabout way in terms more familiar to people at the time used to Euclid's geometry. This style of argument survives today in relics such as the geometric proof that lim sin(x)/x = 1 as x -> 0:
Not only supermen were able to read the Principia, as obviously it was read and its ideas spread far and wide, but perhaps modern supermen would be required in order to see the actual calculus behind the veil of Euclid that Newton had to cast it at the time.
I actually do read an English translation of the Principia from time to time for bedtime reading, and it's not that impenetrable.
And yet James Clark Maxwell expressed EM Theory without vector calculus! I guess they had more time back then, without TV or iPhones to soak up their day.
Scientists today are inventing things 1000x more difficult, and any undergrad understands Maxwell's equations better than he did. That ridiculous quip is moot.
Right. Now what was "deep math" in 1860 is learned by every physics undergrad the world over. I would count vector calculus notation as helpful in that regard in the same way that gamma matrix technology helps graduate students grasp spin-1/2 transition amplitudes. Imagine doing these writing out the explicit matrices!
In fairness, versine is "archaic" in the "my parents used this when they were in school, but I didn't need to" sense. It only went out of use when people stopped using trig tables, replacing them with digital calculators.
And of course the actual calculation of this function had to be done (up to a factor of 2) all the time during navigation until things like GPS appeared. See https://en.wikipedia.org/wiki/Haversine_formula
wow, thanks! that is an incredibly interesting article, and way to learn trig, I think I would have gained more insights had that been part of my introduction.
I'm sort of wondering if that terminolgy sounded more natural then because in "older" English, closer to Germanic roots, it was natural to create new words by pasting together morphemes.
It's so cute! This reminds me a lot of the naming schemes for large numbers, like Enneadekillion and Quinquagintaquadringentilliard. Some of these are still "modern", even if nobody really uses them.
German here; Never ever heard zenzic.
Zen could mean "Zehn" which means ten, where Zic might be an old variant of the suffix -zig which is used to build numbers between 20-99.
Vierzig for example means Forty (Vier = four, plus the suffix). Interestingly it's not as simple for other numbers where the base of the word gets butchered a bit like in english. So as it's "Forty", not "Fourty" it's also "Zwanzig" (20) not "Zweizig".
All that makes me wonder if it really ever meant 'squared' or was rather a old form to build number >100. If Neunzig is 90, Zenzig sounds like it could be 100.
Probably because so much of math was derived from couching things in terms of geometry and geometric proofs. For quite a while squares and cubes were fundamental operations (hence the fact that we use names for them to this day), but general reasoning about exponents independently of geometry was more of a gradual development.
For example, x^4 would have been thought of as a square of a square, with all the geometric meaning that that entails, moreso than as a convenient way to express x * x * x * x. People wouldn't have cared about x * x * x * x, but they might have been interested in the properties of squared squares, hence the oddball naming schemes.
People reasoning about problems involving x^4 would have likely been using verbal proofs involving geometric concepts than using the notation everyone uses today, and in that context x^4 is the square of a square rather than simply being an instance of x^y in which y = 4.
Grain of salt though- this isn't my field, somebody please correct me if I'm wrong!
It's more how the information is presented, and the style of deduction used by mathematicians of the time. For over 2000 years, Euclid's Elements (13 incredibly dense volumes of geometric deduction) were the foundation of mathematical education in academia:
They aren't really books of equations, just lots and lots of verbose axioms and theorems and conclusions about properties of points/lines/triangles/circles/etc.
A square (the squaring operation that is, not the shape) in this context still means x * x, but it would be described (and reasoned about) in terms of being the surface area of a shape or the length of a line, and logically used in comparison to the surface areas of other shapes or the lengths of other lines. A squared square would still mean x^4, but it would be presented in terms of some quantity (x^2) which was shown to be the square of some other quantity (x), which also happens to be the square root of yet another quantity (x^4) according to the logic of whatever property the proof was trying to establish.
An example of a geometric property of the squaring operation could be a simple statement along the lines of, if the side of one (geometric) square is shorter than the side of another square, then the area of the first square is guaranteed to be less than the area of the second square. Which all sounds painfully trivial, to the point of not even being worth talking about, but in the context of Euclid it would have been a meaningful relationship that could be exploited as part of some much more elaborate proof.
At no point would any of this be expressed in terms of modern mathematical notation, it would all be semiformal verbal descriptions of the relationships between 2D geometric primitives. In the world of philosophizing about 2D geometry, squares and square roots have a conceptual primacy that other exponents do not have (likewise for cubes and cube roots in 3D geometry).
What little I've read of this stuff comes across like the machine language of mathematical deduction, where everything is built up from a very long series of very simple statements that (given enough statements) can eventually produce very sophisticated results. The way math is taught now (at least to non-math majors) comes across more like a very high-level language where we're simply taught rules like x^y * x^z = x^(y + z), and we aren't required to understand it in terms of low-level geometric proofs.
That's sort of what I figured, that it was necessarily hyperdimensional. But then I don't see why geometers of this era would have been willing to ponder hyperdimensional solids, but aren't willing to admit arbitrary exponents. Path-dependent historical attitude, I guess.
I think it was probably the first hints of polynomials that could stand on their own without a sane geometric interpretation. As in, "this makes no geometric sense, but I can use the same geometry-based techniques I used on squares and cubes, and it all works out, so let's just go with it."
Another way to interpret it could be, "okay, take this cube with a volume, take that volume and pretend it's actually the width of another special mega-cube and never mix mega-cubes and regular cubes".
Not really. A sudoku puzzle is a set of squares arranged inside another, bigger square. If it were a square of squares, it would mean that you square the initial square's volume including the unit. But in this case, you're only multiplying with a count (or rather, a squared count, since we're talking about two dimensions).
? We're not talking about any units, we're talking about numbers. There are 3^4 squares in a Sudoku puzzle. You don't need a hypercube or any unit of measurement to see that's true.
