How the number zero was ‘discovered’
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(Counterpoint; the ones we like are the ones we can grasp intuitively, and they receive more complimentary names: positive, natural, rational, prime. "integer" is Latin for "complete, sound, healthy")
It won't end, we'll always be discovering new number systems, so perhaps we should reserve terms now. Paradoxical numbers, anyone? Or repugnant?
> early 16th cent. (as an adjective meaning ‘entire, whole’): from Latin, ‘intact, whole,’ from in- (expressing negation) + the root of tangere ‘to touch.’ Compare with entire, also with integral,integrate, and integrity.
Indeed, whole numbers is another word for integers, dated in English, but that's what they are called in some other languages like German and Russian.
I also suspect natural is like in natural philosophy: occurring in nature, not like the other meaning of innate, although the two meanings are closely related.
Imaginary is due to Descartes (https://en.wikipedia.org/wiki/Complex_number#History):
> [...] sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.
It says Descartes didn't like them, but imaginary literally means not occurring in nature, and this was presumably before people accepted mathematical concepts as being as real as anything else.
Transcendental also literally means beyond the algebraic numbers, since that's the definition, that's what they transcend.
Surreal numbers are surreal because somebody (Conway?) needed to attach a prefix to real, and my dictionary says sur- is the same as super-. My dictionary says it is surreal that is the odd word here: a backformation for surrealism, which is some kind of art movement (I wouldn't trust some French painters to get English etymology right, eh? ;)).
Also, what about dyadic, and p-adic, numbers? Ordinal numbers? Hyperreal?
They look similar because (per [1]) they both come from Latin ratio ("reckoning, accounting"), which derives from reri (think, judge).
[1] http://www.dictionary.com/browse/ratio?s=t
[2] They're the numbers expressible as one integer divided by another.
Anyway, all the people who said stuff like "When are we going to use X math topic in real life" were even more flummoxed by random math historical facts that weren't going to be on the test.
Anyway, at least for me, I think I passed a point of maturity where I don't feel "bombarded" when I encounter additional information in regards to a math topic. I did poorly in math in high school, and, at the time, I felt that the less information, the better-- I could only integrate so much. But now at my age (almost 40) learning additional historical facts seems to help me understand the topic better.
Maybe as "unreal as anything else" would be a more accurate way to put it. Mathematical realism and even any kind of Platonism have a lot of detractors. I think the attitude change was more due to what is now known as Logicism (https://en.wikipedia.org/wiki/Logicism) and Formalism ( https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe... ) when people realized that you did not need to appeal to the physical world to come up with useful things.
I would be tempted to say "pictured", "in a picture".
https://en.wiktionary.org/wiki/imago#Latin
But imago also has a lot of other meanings, and so "ghostly" and "conceptual" are also possibilities. :-)
The quantity zero is empty, so the name is clearly appropriate.
negare is "deny" in classical Latin, but the term "negative number" is much more recent than any form of Latin, and refers to the "taking away" sense of "negative", just as in "negative reinforcement". Negative numbers are related to subtraction.
Supernatural numbers are an extension of the natural numbers. The terminology is pretty straightforward. Also, the word "supernatural" is generally complimentary, not derogatory. Similarly, surreal numbers extend, and are named in reference to, the real numbers.
Irrational numbers can't be expressed as ratios (of integers). It's not the term's fault if you confuse it with another word.
Like life, once you get out of a black and white understanding of your world, it's suddenly more complex...
It's so natural, too. Numbers, you count, 1, 2, 3, 4... it makes so much sense. And of course, rational numbers aren't too hard to grasp within that context. They're expressible with the nice numbers we know.
Even today, there are a lot of people who consider pi a strange, unusual number, an oddity. If they're aware of the square root of two, or euler's constant, they may consider them to be unusual as well.
At least in the US, it isn't until you've studied math for a while that you really do get into the notion of limits. A professor in Calc might talk about this in more depth (maybe when Taylor series come up) or high school geometry might mention that pi isn't actually that unusual. But I honestly don't blame people that this is hard to grasp. It's a long way from the "numbers" you understand from an early age, when you learn to count.
But yeah, it says something that numbers that occur "naturally" are almost never representable as the ratio of two integers. The ratio of a circumference to a radius. The "long" side of a right triangle with "legs" of a single unit. The number whose natural log is 1. Almost every number that took us by surprise, that we didn't invent but were instead required to measure and explain, remarkably, isn't expressible the way we thought numbers would be.
Can I go on a bit here and say that math gets hard the moment almost everything else gets hard, when you get out of your own head and try to see things as they actually are? When I was a kid, I loved "mechanical drawing", because it all looked so good on the page. I drew human creations, cars, houses, everything with nice lines, it all looked so sharp and snappy. And then you try to draw nature... where are the lines? It turns out you can't represent nature very well with lines. You have to draw what isn't there, not what is. The first time this comes up is when you try to draw someone's face.
Same for numbers. Draw a cube, measure the volume. Ok, easy. Draw a cylinder, measure the volume. A little harder, you have to make a good circle, but it can be done. Measure the volume - now you need this strange, "irrational" number that. Now go draw a tree trunk...
Hard. There isn't even a nice circle there. And to actually take the volume or surface area accurately, you need some way to measure the space under the curve, which requires calculus, little squares above and below the line, which can be made smaller and smaller, but they never reach zero. And the limit of the upper and lower sequence isn't an approximation of the number. The limit is the number. And it turns out that it's exceptionally unusual for that limit to be expressible as a ratio of two integers.
It's just unreal, isn't it? Er, I mean, real. ;)
I think it's just under an hour's worth of videos, in five minute chunks, covering both mathematics and history.
The previous contender was the zero at Gwalior in India, dated 876 AD http://www.ams.org/samplings/feature-column/fcarc-india-zero
>U 0 a 0, but I 0 thee
>O 0 no 0, but O 0 me.
>O let not my 0 a mere 0 go,
>But 0 my 0 I 0 thee so.
As noted in the comments here, "cipher" used to be another name for zero/0, so the above reads as:
>You sigh for a cipher, but I sigh for thee
>O sigh for no cipher, but O sigh for me.
>O let not my sigh for a mere cipher go
>But sigh for my sigh, for I sigh for thee so.
Which, of course, explains why Neo, The One from the Matrix, had an enemy named Cypher.
> "Śūnyatā" (Sanskrit) is usually translated as "emptiness," "hollow, hollowness," "voidness." It is the noun form of the adjective śūnya or śhūnya, plus -tā:
> śūnya means "zero," "nothing," "empty" or "void". Śūnya comes from the root śvi, meaning "hollow".
> -tā means "-ness"
There are many loanwords meaning shunya or zero in many Indian languages and they are all adopted from Sanskrit.
Interesting etymology, from a philosophical perspective. ie. What came first, the circumscribing matter or the non-matter? In this case, via hollowness, the circumscribing matter.
I'm really curious about this part. It sounds like they were already using Arabic numerals. How did they intend to express numbers like the ones mentioned in the first paragraph, like 101?
Anyway, the two most used methods are to call a vector zero when its norm is zero or to call a vector zero when it's orthogonal to all other vectors in the space. The first approach leads to least-squares approaches, which gives things like GMRES or QMR in linear algebra or first-order system least-squares (FOSLS) finite element methods. The second approach leads to Galerkin and Petrov-Galerkin algorithms, which gives things like CG in linear algebra or more standard Galerkin or Petro-Galerkin finite element methods.
Anyway, that's just an aside, but I wanted to add that how we visualize zero has a definite computational, algorithmic affect.
Straight from Buddhist meditation, meaning "emptiness" -> and now it's the basis of math and science. And it's not just zero, take a look at the pronunciation of numerals in Sanskrit:
0 śūnya - Arabic "ṣifr" -> Latin "zephir" -> "zero"
1 éka - Greek "ena" -> Latin "unus" -> English "one"
2 dvi - like the prefix "di-" or "bi-" meaning double, German "zwei" -> English "two"
3 trí - like "three"
4 catúr - like "quatre" in French
5 pañca - like Greek "pénte"
6 ṣáṣ - like six, or "șase" in Romanian
7 saptá - like seven, or "șapte" in Romanian
8 aṣṭá - like German "acht", English "eight"
9 náva - like nine, or "nouă" in Romanian, which also means new in both languages (new sounds like nine)
10 dagan - like Latin "decem"
It's amazing how much Sanskrit is in our languages.
https://en.wikipedia.org/wiki/Proto-Indo-Europeans
Here's a visualization of IE languages, perhaps not accurate in every single detail, but easy to remember:
0 shunya 1 ek 2 do 3 theen 4 char 5 paanch 6 che 7 saath 8 aatuh 9 nau 10 dus
Yeah, PIE (Proto-Indo-European) is very interesting—we think that we're so different but our languages are so similar.
Slovak has two cognates: šifra (cipher: secret writing, plus some related meanings, such as an individual character of a secret writing, an abbreviated signature, or anidentifying mark) and the word cifra (digit). (Probably used a bit less than its synonym číslica).
The word ciferník denotes an instrument or clock dial (also číselník).
http://www.dailymail.co.uk/sciencetech/article-3698184/Liste...
No doubt, the ban was supported by arguments about how it's making everyone more productive by catching problems early and eliminating a whole class of errors.
And so that would be where we can trace the origins of static typing.
Then that is not 0, that is null!