Margaret Marks (2006-11-18)
Resp.
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 her blog
today (2006-11-18) Margaret Marks was kind enough to quote my
two-line definition of signed infinities as
an 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 and
mathematicians find it easier to parse.
It's not a bad thing to have a few keywords which allow the reader
to 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") is inclusive
"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 a non-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 warning
the reader that a sentence is meant to be a mathematical one (which is
to be interpreted as inclusively as possible,
even if only "common" words are used).
Such clues help build what linguists call a "context".
Mathematical texts may acquire strange meanings
when an attempt is made to read them "out of context".
Miguel Angel Urrutia
(Panama, 2003-08-07) Typography of Numbers Why use
999998 when 999,998 seems more readable?
Separators are common in a financial context, but they are rarely used within
scientific 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 a
period is used for the "decimal point".
In French it's the exact opposite.
A Frenchman who's otherwise fluent in English could easily
misinterpret 999,998 to mean 999.998.
The 22^{nd} CGPM
(October 2003)
reaffirmed Resolution 7 of the 9^{th}
CGPM (1948) by ruling that both the comma and the dot
were allowed as decimal markers. Therefore, neither is acceptable in
an 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.
Unless their rough magnitude is clear from the context,
it's a good idea to avoid decimal numbers featuring exactly 3 digits after
the decimal marker and 3 digits or less before it
(except in the unambiguous case of a single leading zero digit).
Many people could otherwise misinterpret such numbers by a factor of one thousand.
When separators are used, the need for an unsightly single-digit "group" is normally
avoided 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 with large integers
never use any separation between groups of digits (separators or spacing) because
very large integers are unreadable anyway.
In this digital age, a marginal improvement in readability is
less 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) tabular
display of many decimals
for a specific mathematical constant (in which case groups of 5 or 10 decimals
may be more typical than groups of 3).
The ISO 31-0 standard
says:
" To facilitate the reading of numbers with many digits,
these may be separated into suitable groups, preferably of three,
counting from the decimal sign towards the left and the right.
The groups should be separated by a small space,
and never by a comma or a point, nor by any other means. "
(2006-10-24) Denoting Intervals
Use only square brackets; outward brackets for excluded extremities.
For example, [0,1] is the set of all real numbers between 0 and 1, both extremities
included, 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).
The chief reason why the "domestic" notation is unacceptable is that (0,1) is universally
accepted as denoting an ordered pair (an element of a cartesian product)
in any modern context.
Using that same notation to denote ]0,1[ is confusing at best,
for any audience.
(2006-11-01) International dates in numerical form
The above date is November 1^{st}, 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 the
international ISO 8601 standard
in its simplest form...
That standard specifies that the most significant numbers must be listed first,
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 times
which 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 week
number 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 the
above 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 reminiscent
of 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 hours
before 10am.
For added precision, another ":" may be added, followed by a two-digit
number of seconds, from 00 to 59.
For the ultimate in precision, this number of seconds may be given with as many
decimals 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 the
correct chronological order
(whether the time stamps are given at the same level
of 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
"Who
Wants 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.
Note the consistent use of the suffixes, which are of Latin origin:
Dividend= "That which is to be divided"
(the orator Cato ended all of its speeches with
the famous quote "Carthago delenda est": Catharge is to be destroyed).
In a nonmathematical context, dividends are profits that are to be divided among
all shareholders.
Divisor = "That which is to do the dividing".
A director does the directing, an advisor
does the advising, etc. In ancient Rome, the Emperor was the
"Imperator", the one supposed to issue the orders (Latin: "Imperare")
(2007-03-06) Pronouncing Numbers
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) Pronouncing Formulas
How do native speakers pronounce formulas in English.
There's also the issue of indicating parentheses
and groupings when pronouncing expressions.
If the expression is simple 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) PEMDAS? Not.
The Meaning of it All.
From a semantical viewpoint, the meaning of complex arithmetical expressions
involving elementary operators is the value obtained by applying the operators
in their conventional order of precedence.
Unfortunately, English-speaking schoolchildren are often taught to associate this
with the mnemonic sentence "Please Excuse My Dear Aunt Sally":
Parentheses first,
then Exponentiations, Multiplications, Divisions, Additions and Subtractions...
This should not be relied upon, for the following reasons:
It tells only part of the whole story.
The other essential part is that you must group things left to right
when encountering operators of the same precedence.
For example:
9-1-2-3 actually means
(((9-1)-2)-3) =_{ }3.
Addition and subtraction actually have the same precedence so that
9-3+2 is universally understood to mean
(9-3)+2 = 8
(see previous point) and is not equated to the expression
9-(3+2) = 4,
as it would be if addition had higher precedence than subtraction._{ }
With many computer languages (and/or scientific calculators),
multiplication and division have the
same precedence as well.
This means that 9/3´2
is actually worked out left to right to denote
(9/3)´2 = 6
on many calculators, whereas the written expression
may be intended to mean
9/(3´2) = 1½,
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!
(2012-11-02) Physical Units
Products and ratios of physical units.
The above issue 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 unit
is 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
(the solidus) entirely when expressing physical units.
According to this relatively new fashion, the abbreviation for the previous unit would be:
J . K^{-1} . kg^{-1} =
m^{2} . s^{-2} . K^{-1}
I still favor parsimony and prefer to express
gravitational fields in N/kg (newtons per kilogram)
rather than in
m/s/s , m/s^{2} or m.s^{-2}