(2005-06-11) Remnants of an era when Latin was the language of knowledge.
Quid quid latine dictum sit, altum videtur. Anything stated in Latin seems profound.
No commas should follow any of the following Latin abbreviations, with onlytwo exceptions ("i.e." and "e.g.") that must be followed by a comma.
Abbr.
Read as (Latin)
English translation
Usage notes
c.
circa
around
Indicatesan approximate date.
et al.
et alii
and others
Endsan incomplete list of people.
etc.
et cetera
and so on
Endsan incomplete list.
viz.
videlicet (videre licet)
that's to say
Introducesa more explicit statement.
sc.
scilicet (scire licet)
also known as
Gives a synonym to lift ambiguities. More rarely used than viz.
i.e.,
id est
that is
Introducesan equivalent paraphrase.
e.g.,
exempli gratia
for example
Introducesa specific example.
vs.
versus
opposed to
Oppositionor contradistinction.
cf.
conferre
compare to
Suggestsa comparative reference.
q.v. qq.v.
quod vide quae vide
which see
Aself-reference, for one item. A self-reference, for several.
N.B.
nota bene
note well
Introducesa delicate precision.
Q.E.D.
quod erat demonstrandum
which was to beproved
Marksthe end of a proof. (Halmos symbol is recommended.)
sic
sic erat scriptum
Exacly as was written
Standsby itself to mark disapproval of a statement quoted verbatim.
ita est
it is so...
... like it or not.
Abbr.
Language
Meaning
Usage notes
wlg WLG
English
without loss ofgenerality
Introduces an arbitrary break of a symmetry amongparameters.
Margaret Marks (2006-11-18) Is the abbreviation "resp." acceptable in a mathematical context?
Yes, but the word "respectively" and the symbol "resp." have different syntaxes. The latter should probably be used exclusively in a mathematical context. It's not a general-purpose abbreviation of the former...
In herblogtoday (2006-11-18) Margaret Marks was kind enough to quote mytwo-line definition of signed infinities asan actual example of the mathematical use of "resp.". (I just noticed this when someone triggered her link to <Numericana>.)
A (contrived) example would be: "The square of 2 (resp. 3) is 4 (resp. 9)."
That could also be stated: "The squares of 2 and 3 are 4 and 9 respectively."
The two syntaxes are different, as advertised. Occasionally, the former syntax can be more convenient andmathematicians find it easier to parse.
It's not a bad thing to have a few keywords which allow the readerto recognize an English sentence as a mathematical statement,because that may deeply affect the meaning... The most critical feature is undoubtedly that mathematical discourse(expressed in any "natural language") isinclusive"by default" whereas everyday speech is not. To a mathematician, the statement "a circle is an oval" is clearly true. It would seem nonsensical to people accustomed to a common "exclusive" definition stating, at the very outset, that an oval is anon-circular shape...
By comparison, the use of "resp." (which puzzled Margaret) is a minor issue ! However, this and other mildly jargonistic terms may actually be helpful in warningthe reader that a sentence is meant to be a mathematical one (which isto be interpretedas inclusively as possible,even if only "common" words are used). Such clues help build what linguists call a "context". Mathematical texts may acquirestrange meaningswhen an attempt is made to read them "out of context".
Separators are common in a financial context, but they are rarely used withinscientific equations, where they would decrease readability rather than improve it.
Also, there are cultural differences for separators (dots or commas). In English, the comma is used to separate 3-digit groups, whereas aperiod is used for the "decimal point". In French it's the exact opposite. A Frenchman who's otherwise fluent in English could easilymisinterpret 999,998 to mean 999.998.
The 22nd CGPM (October 2003)reaffirmedResolution 7 of the 9thCGPM (1948) by ruling that both the comma and the dotwere allowed as decimal markers. Therefore, neither is acceptable inan international context as a typographical separator between groups of digits. The international convention is to use some spacing between 3-digit slices. If at all possible, such spacing should be thinner than the regular spacing between words.
When separators are used, the need for an unsightly single-digit "group" is normallyavoided by allowing the leading group to include up to 4 digits. In particular, separators are not used for integers up to 9999.
Number theorists who deal routinely withlarge integersnever use any separation between groups of digits (separators or spacing) becausevery large integers are unreadable anyway. In this digital age, a marginal improvement in readability isless important than the ability to cut and paste such long numbers uniformly.
Non-blank separators are never used to the right of the decimal marker. Spacing between decimals is best restricted to the (rare) tabulardisplay of many decimalsfor a specific mathematical constant (in which case groups of 5 or 10 decimalsmay be more typical than groups of 3). The ISO 31-0 standardsays:
(2006-10-24) Use only square brackets;outward brackets forexcluded extremities.
For example, [0,1] is the set of all real numbers between 0 and 1, both extremitiesincluded, whereas ]0,1] is the set of all nonzero such numbers.
We do not recommend a notation with ordinary parentheses for excluded extremities; namely (0,1] in the latter example above. It's unfortunately dominant in "domestic" English texts (it puzzles international audiences).
(2006-11-01) The above date is November 1st, 2006.
The year is listed first as a 4-digit number. Dashes are used, not slashes.
Taken together, those two clues properly indicate that the date identification uses theinternationalISO 8601 standardin its simplest form...
That standard specifies that themost significant numbers must be listedfirst,as is the case with digits in ordinary decimal numeration (thus, the month follows the year and the day of the month is listed in third position). Months and days are always given as two-digit numbers, with a leading zero if necessary. This design makes lexical and chronological orders coincide, as is most desirable.
This is simple and logical enough when it comes to identify a specific day.
Unfortunately, the standard also allows other formats for dates and timeswhich cannot be assumed to be self-explanatory. We think these are best ignored outside of specialized contexts For example, separating dashes may be dropped, time can be included and a weeknumber could be specified instead of a month number... Such add-ons to the above basics only hinder general acceptance.
Compatible Customary Time Stamps
In digital contexts, ISO 8601 dates of theabove type are sometimes followed by separate (nonstandard) time stamps.
Such customary time stamps feature a column character ":"between hours and minutes, from 00:00 to 23:59. This is reminiscentof the ubiquitous displays for digital 24-hour clocks,which need no introduction. The only "unusual" part of the convention is a leading zero for the hoursbefore 10am.
For added precision, another ":" may be added, followed by a two-digitnumber of seconds, from 00 to 59. For the ultimate in precision, this number of seconds may be given with as manydecimals as needed (using a decimal point).
Basic ISO 8601 dates followed by such time stamps (with a single blank space inbetween) identify a precise time. By design, the lexicographical ordering of such identifications is thecorrect chronological order (whether the time stamps are given at the same levelof precision or not). Examples:
With thanks to Steve Healey ("BonusSpin") of Edison, NJ, a math & physics junior at Rutgers (New Brunswick, NJ) who appeared on the ABC TV show "WhoWants to Be a Millionaire?" on Sept. 14 & 17, 2000.
Thanks also to Keith McClary for suggesting (2004-05-24) multiplier and multiplicand, especially for noncommutative multiplications.
(2007-03-06) High-school parlance and straight talk.
In high-school parlance, the negative integer "-7" is pronounced"negative seven". Outside of the classroom, almost everybody says:"minus 7".
The number 0.7 (namely seven tenth or 70% of a whole) is also written .7 and pronounced either"decimal seven" or "zero point seven" (the latter being preferred in modern"straight talk").
(2000-12-07) How do native speakers pronounce formulas in English.
If A and B arestrings, I call their concatenation "A before B".
There's also the issue of indicating parenthesesand groupings when pronouncing expressions. If the expression issimple enough,a parenthesis is adequately pronounced by marking a short pause. For example, x (y+z) / t could be spoken out"x times ... y plus z ... over t" (pronouncing "y plus z" very quickly).
The locution "outside of" can also be used to introduce an opening parenthesis,matching a closing parenthesis corresponding to a short pause.
When dictating more complex expressions involving parentheses, it's best to say open parent' and close parent' as needed.
(2000-12-07) The Meaning of it All.
From a semantical viewpoint, the meaning of complex arithmetical expressionsinvolving elementary operators is thevalue obtained by applying the operatorsin their conventional order of precedence. Unfortunately, English-speaking schoolchildren are often taught to associate thiswith the mnemonic sentence "Please Excuse My Dear Aunt Sally": Parentheses first,then Exponentiations, Multiplications, Divisions, Additions and Subtractions...This shouldnot be relied upon, for the following reasons:
It tells only part of the whole story. The otheressential part is that you must group thingsleft to rightwhen encountering operators of the same precedence. For example: actually means =3.
Addition and subtraction actually have thesame precedence so that is universally understood to mean = 8(see previous point) and isnot equated to the expression = 4,as it would beif addition had higher precedence than subtraction.
With many computer languages (and/or scientific calculators),multiplication and division have thesame precedence as well. This means that is actually worked outleft to right to denoteon many calculators, whereas thewritten expression may be intended to mean ½,according to the aforementioned "PEMDAS" rule (wrongly) taught to schoolkids. When the latter is meant, it's probably better to use 9/3/2
Use parentheses to make sure you're understood!
To improve readibility, other styles of so-called grouping symbols are available besides ordinary parentheses. In a valid expression, these are always used inmatching pairs enclosing a valid expression:
The aboveissue arises for physical units, especially when an electronic calculator is involved.
For example, the SI unit for entropy or thermal capacity is the joule per kelvin (J/K). A specific thermal capacity is a thermal capacity per unit of mass, so the SI unitis the joule per kelvin per kilogram or J/K/kg. It's a bad idea to write that down as J/K.kg (joule per kelvin-kilogram) because of the above ambiguity, although most professionals won't even blink. A modern trend (which I dislike) is to forgo the use of the division sign(thesolidus) entirely when expressing physical units. According to this relatively new fashion, the abbreviation for the previous unit would be:
J . K-1 . kg-1 = m2 . s-2 . K-1
I still favor parsimony and prefer to expressgravitational fields in N/kg (newtons per kilogram) rather than in m/s/s , m/s2 or m.s-2
