10^{24}
yotta Y 10^{–1}
deci d
10^{21}
zetta Z 10^{–2}
centi c
10^{18}
exa E 10^{–3}
milli m
10^{15}
peta P 10^{–6}
micro µ (upright)
10^{12}
tera T 10^{–9}
nano n
10^{9}
giga G 10^{–12}
pico p
10^{6}
mega M 10^{–15}
femto f
10^{3}
kilo k 10^{–18}
atto a
10^{2}
hecto h 10^{–21}
zepto z
10^{1}
deca (or deka) da 10^{–24} yocto
y
## 13.3 Formatting
Mathematical expressions are frequently complex with many components, for
example in a given expression some symbols may represent operators, some
represent functions and some represent variables. The most common question when
typesetting mathematical notation is, "Should symbol "x" be set to upright or
set in italic?" ### 13.3.1 Upright Characters
Do not italicize numbers, punctuation, parentheses, common function names
(see Sect. 13.3.3), units, or mathematical signs (e.g.,
"+") wherever they occur (regardless of whether or not they appear in an
equation or as a superscript or subscript).
### 13.3.2 Variables
We have decided upon the following rule of thumb:
manually converted up to here
– If a symbol in an equation takes a value which needs to be
substituted into it before the result of the equation can be calculated, then it
should be set in* italic*. This symbol is a container for a value. The
contents of the container can either be a constant value or it can vary, but the
important point is that the symbol, at the final reckoning, will be replaced by
something else.
– If the symbol is a label, either for the name of a particle for
example, or attached to a variable giving information about that variable, then
it should be set in* upright*. This label does not act as a container for a
value.
Here are some examples. When talking about the mass of the electron we
use* m* _{e}. The "*m*" will eventually be replaced by the
number for the electron mass, the "e" simply labels this mass as being that of
the electron. When we talk about the electronic charge of the electron we use
"*e*". Here, when calculating, "*e*" will be replaced by the number
for the electronic charge! If the author is talking about some point*
x*, some distance* d,* or some time* t*, all of these are set to
italic in the expectation that later they will be replaced be the value of the
coordinate, distance or time respectively. If the author wishes to label these
as being in some positive region of some space then he might mark them as
*x* _{pos}, * d* _{pos}, or * t* _{pos} .
Here the "pos" is set to be upright as it denotes "positive." If a
variable has a running index such as *x* _{ i }, where* i*
runs between 0 and 3 for example, then this index* i*, though a label, is
set to italic. In this instance the label* i* is a container for a set of
values and in each instance *x* _{0}, * x* _{1}, *
x* _{2}, and * x* _{3} each take an independent
value. The numbers themselves do not act as containers for other values
and so are set to upright. The complex number "i" we consider to be a creature
of the same genus as the numbers and so we set it upright. Often it is
represented as "j." Upright presentation is also a great aid in differentiating
the complex number "i" from "*i*" used as a running index. The mixed use of
"i" is very common. The Roman and lowercase Greek alphabets provide the
most common examples of variables. The use of upright and italic display for
Greek letters follows the same convention as for Latin letters. Sometimes the
Hebrew alphabet is used and sometimes the Frankfurter alphabet is used
(mathematicians and physicists are fond of using many symbols). An
exception to these rules is made for a multiletter abbreviaton. It is usually
written upright rather than in italics even when it represents a value, for
example, IP (for ionization potential) and BER (for bit error rate). (An
exception is the dimensionless Reynolds number "*Re*" (note that "Re"
commonly denotes the "Real" part of a complex number). Use pH, p*I*, and
p*K*a.) Here are some examples of commonly used combinations:
*?* _{B}("Bohr"), *T* _{C}("Curie"), *m*
_{p}("proton"), *m* _{e}("electron"), *e* (value of
the elementary charge), *N* _{A}("Avogadro"), *k*
_{B}("Boltzmann"), *c* or *c* _{0}(speed of light in
vacuum), *h* and
(Planck's constant), *?* _{0}and *?* _{0}(magnetic and
electric constants), *n*-ary, *p*-adic, etc. Note "e" is upright when
it stands for the word "electron."
### 13.3.3 Functions
In an equation with an arbitrary function the function has to be replaced by
the operation that it represents before a final value for the equation can be
obtained. General functions are therefore set in italic. In the equation
before calculating the value of* y* we need to know what the functions
*f*(*x*) and* g*(*x*) are. These functions act as containers
for some proper given operator on* x* and therefore are set in italic. When
the function is named explicitly, it does not act as a container for some
arbitrary operation but tells us exactly what the operations to be performed on
the argument is. In this case these function names should be set upright.
The most common functions that are set upright are:
C.C., c.c. for complex conjugate
cos for cosine
cot for cotangent
csc, cosec for cosecant
curl for curl
d for differential
delta (greek "D")
det for determinant
div for divergence
erf for error function
exp or e for exponential
grad for gradient
H.C., h.c. for hermetian conjugate
Im for imaginary part
Log or log for logorithim
Ln, ln for natural logorithim
max for maximum
min for minimum
mod for modulo
∏ for product (the mathematical form of the greek letter "P")
Re for real part
sgn for sign
sec for secant
∑ for summation (greek "S")
sin for sine
sqrt for square root
tan for tangent
tr, Tr for trace
Here are several equations containing such functions:
## 13.4 Vectors, Tensors, and Matrices
These are multidimensional objects. They have more than one component and so
some way is needed to distinguish them from simple objects. **Vectors
** Vectors are represented differently in different fields. For products in
the physics program, use bold and italics for characters representing vectors
(e.g., *v*). In the mathematics program, use bold and upright (e.g.,
**v**). For other disciplines, check with your Springer contact. Vectors are
often represented using other styles, such as an overhead arrow,
; this is acceptable but deprecated. Any style mixtures should be corrected.
Vector components should be in italics, not in bold, e.g.,
. For the norm of a vector, follow the author's usage of
or
, or simply *v*. **Matrices ** The characters representing
matrices may be bold, bold and italic, or italic. Matrix elements should be in
italics and nonbold. The superscripts "T" or "t" (transpose) and "H" (Hermitian)
should be upright. Full matrices should be written as displayed equations. For
matrix dimensions use "×," e.g., "a 3×3 matrix" or "an
*n*×*m* matrix." Matrix determinants can be represented using
straight lines, e.g.,
, or as "det *B* ." Follow the author's style for representing omitted
elements (e.g., use ellipses). Examples:
**Tensors **
The characters representing tensors are generally displayed using a sans serif
font (e.g., Arial).
## 13.5 Displayed Equations
### 13.5.1 General
Copy edtiors are, as a rule, not responsible for making improvements with
regard to issues such as page breaks and equation positioning. They are
responsible, however, for some matters of presentation and for the text. For
example, phrases such as "the following equation" or "the equation below" in the
text may frequently be removed as redundant or superfluous. For citation rules
see Sect. 9.5.1 The *alignment* of parts
of an equation may be important. The analogy of "verbs" and "conjunctions," such
as described in the *Chicago Manual of Style*, may be helpful in this
context. "Verb" characters include while "conjunctions" are "Verbs" in successive lines should be aligned beneath each other. A
"conjunction" at the start of a line should be aligned under the first element
to the right of the "verb" in the line above it. Example:
A "verb" or "conjunction" should not be repeated across a
linebreak; it should appear once only as the first element of the following
line. Example:
### 13.5.2 Punctuation
It is useful to assume that a displayed equation is part of a sentence and
that punctuation follows normal language rules, which is demonstrated in the
following examples showing combinations of text and equations (see Chap. 5). Unless otherwise instructed, follow this usage unless an
author has been consistent in not punctuating displayed equations.
Equation ends the sentence:
... where z is fixed and
From this relation ...
Sentence continues after equation:
... where z is fixed,
and *w* takes the value 1.
Combined example:
... because
where x refers to ..., and
In this example, the addition of punctuation would be incorrect:
... where
and *b* are known.
Punctuation within an equation environment:
Another example of punctuation within an equation environment:
Sometimes it is useful to consider equations to be the items in a
list. Here is such an example:
... our list of results:
where *x*, *y*, and *a* refer to ..., respectively,
where *w* and *z* are defined elsewhere, and
As in any list, be careful not to misplace commas or other punctuation.
## 13.6 Inline Equations
Grammatically, an inline equation is a normal part of a paragraph and
sentence. It follows normal punctuation rules (see Sect. 13.5.2 and Chap. 5). It is also
never numbered; if the author has numbered an inline equation, either it must be
changed to a displayed equation (to retain the numbering) or the numbering must
be dropped. Because of the consequent renumbering of following equations, such
an equation should as a rule be changed from inline to displayed. You may
suggest to an author that certain mathematical content be changed from inline to
displayed for better presentation. Examples are fractions and content with
"tall" symbols. Inline fractions should employ a solidus (virgule or slash)
instead of a built-up construction wherever sensible, for example using
instead of the built-up construction If tall symobls are used inline, suggest to the author that the
more convenient inline forms be used; for example, use
for
, use
for
, or use
for
. RPN, or reverse Polish (postfix) notation, cannot be changed to infix
form; do not change "a b +" to "a + b." RPN is sometimes used in texts dealing
with microprocessors, stack-based programs, etc.
## 13.7 Theorems, Definitions, and Proofs
**Theorem Style ** In styling a theorem, the word "Theorem" and the
number are in bold, and the following text in italics. Example:
**Theorem 12** (Fermat's Little Theorem 28) *Let p be a prime and
assume that*
, *then*
Numbers, punctuation, parentheses, etc. should not be in italics. Edit the
text if the author has used a "dangling" theorem:
... according to the following
**Theorem 12 ** *Let x be a prime* ...
**Proof Style ** In styling a proof, the word "Proof" is in italics
and the text follows in upright. An end-of-proof box may be used, the preferred
position being flush right. Example:
*Proof* From (12.9) we know that a polynomial in *x*
with
... and the rest follows by induction.
?
**Definition Style ** In styling a definition, the word "Definition"
is in bold, the text is upright, and the terms being defined are italic.
**Other ** Authors may use the theorem, definition, or proof styles
for algorithms, assumptions, axioms, cases, claims, conjectures, corollaries,
demonstrations, examples, exercises, hypotheses, lemmas, notes, problems,
properties, propositions, questions, remarks, rules, solutions, etc.
Furthermore, an author may use offset text, or different or smaller fonts for
such environments. This should be done consistently.
## 13.8 Other Issues
**Functions and Symbols ** For clarity, functions and operands are
usually distinguished using parentheses, e.g., "*f*(*x*)" and "sin
(?+?)." A symbol may act as the verb in a sentence where it can be translated
into an English-language phrase such as "is equal to" or "is an element of."
Examples:
Here the function
The modulus is large and
**Multiplication Symbol ** Center dots (·) or line dots (.)
are often used for multiplication. In many cases they are redundant and can be
removed, i.e.., "a·b" or "a.b" be changed to the "ab" form. For a text
involving vectors, an author may have used the center dot to indicate scalar
(inner) multiplication
as distinct from vector multiplication *u*×*v*,
and in such cases the center dot must be retained. Use the "times" (×)
symbol, rather than an asterisk (*) or the letter "x," in constructions such as
"10 mm×0.7 mm×0.3 mm," "10,000×*g*
centrifuge," and "100× lens." **Similar Symbols ** There are
many symbols that are very similar in meaning. Be extremely careful in changing
an author's usage of such symbols, but query any apparent misuse. For example,
some scientists distinguish between "≈" meaning "is approximately equal
to" and "~" meaning "is similar to," "is of the order of," or "is
asymptotically equal to," while others consider "≈," "~," "≅," and
"?" to be interchangeable, each meaning "is approximately equal to." The
symbols "≅" and "≡" are often used to mean "is congruent to," a
specific mathematical relation. Many references avoid defining these symbols
because they vary in usage, even within disciplines. It's the author's
responsibility to use the correct symbols consistently. Take into account the
scale and measurement precision implied in the text and query the author if you
suspect a mistake or inconsistent usage. **Equation References **
On general usage, see Sect. 9.5.1. It may sometimes be
useful to retain specific qualifiers such as "congruence" or "inequality" but
remove normally redundant qualifiers such as "equation," "relation," or
"expression." Some people, however, distinguish between a "relationship" and an
"equation" while others distinguish the term "relationship" (e.g., "in this
chapter we examine the relationship between *x* and *y*") and a
specific instance of a relationship at some point in time, a "relation" (e.g.,
"here the relation
applies"). Accept an author's usage; do not change it. **Derivatives
** An author may use schemes such as dots and double dots above characters to
represent derivatives, e.g., double derivative
. **Ellipses ** Centered dots are preferred for "=" and for
operators such as "+," "–," and "×," and line dots are preferred for
commas. Alternatively, line dots may be used throughout. Symbols and commas
should precede and follow ellipses. Examples:
**Parentheses ** In general, apply the hierarchy
, e.g.,
. If necessary, this nesting hierarchy may be repeated in bold, working
outwards. Alternatively the parentheses may be nested using "taller" symbols,
i.e.,
. Parentheses of different style are not required in unambiguous cases, e.g.,
. Unmatched ("lonely") parentheses in equations often reveal a typographical
error; sometimes the required correction is obvious, but more often it's safer
to ask the author to correct the problem. Parentheses should be "taller" than
the enclosed content and most software packages automatically handle this; such
issues are usually more efficiently corrected by the author or typesetter, so
where possible use "global" instructions with specific examples. The "less than"
and "greater than" signs ("<" and ">") should not be mixed up with angle
brackets ("⟨" and "⟩"). Note any special cases. In many scientific
treatments parentheses have strict meanings, and the style should not be
changed, e.g., {111} vs ⟨111⟩ vs (111) in solid-state physics. Some
styles involve deliberately asymmetrical notation. Examples: the mathematical
intervals (*a*,*b*] and " vs "O III." Query the author
if you suspect an error. **Signs ** In* descriptive*
treatments, terms such as "greater than" are preferred. For example, use "the
number of patients was greater than 30," not "... was > 30." Where the text
is more "mathematical" than descriptive, or there are numerous relations, signs
are more efficient for the reader and should be used. **Prime ** A
prime should appear adjacent to the relevant character. In terms of meaning, the
character and its prime form a unit, which itself may, for example, have a
superscript attached to it. The usages
and
are not equivalent. **Special Fonts ** Some fonts have specific
mathematical meaning and usage. Examples include number sets (
,
,
,
,
, etc.) and symbols with defined or "local" meaning. **Abbreviations
** The term "iff" (if and only if) is in Merriam–Webster's and does not
require explanation. For abbreviations that do not appear in this dictionary,
such as "wlog" (without loss of generality) and "wrt" (with respect to), add a
first-use definition. Confirm proper use using a suitable dictionary. The
abbreviations for Hermitian conjugate ("H.c."), Hermitian adjoint ("H.a."), and
complex conjugate ("c.c.") should be upright, as should terms such as "const."
for "constant". |