%% This is part of the OpTeX project, see http://petr.olsak.net/optex % Run optex optex-math (two times) to generate this document % or look at PDF here: http://petr.olsak.net/ftp/olsak/optex/optex-math.pdf \fontfam[lmfonts] \font\ttlib=[LiberationMono-Regular] \def\ttspec{\let\_ttfont=\ttlib} \report \def\new #1 {\mnote{\Red$\blacktriangleleft$\,\sans\setfontsize{at9pt}\rm#1}} \fixmnotes\right \enquotes \verbchar` \catcode`\<=13 \def<#1>{$\langle\hbox{\it#1\/}\rangle$} \everyintt={\catcode`\<=13 } \def\ss#1{$\vrule height3pt#1\vrule height3pt$} \addto\_secfont\Blue \addto\_seccfont\Blue \_def\_printsec#1{\_par \_abovetitle{\_penalty-400}\_bigskip {{\_secfont \_noindent \_raggedright \llap{\_printrefnum[@\_quad]}#1}\_nbpar}\_insertmark{#1}% \_nobreak \_belowtitle{\_medskip}% \_firstnoindent } \_def\_printsecc#1{\_par \_abovetitle{\_penalty-200}\_medskip {{\_seccfont \_noindent \_raggedright \llap{\_printrefnum[@\_quad]}#1}\_nbpar}% \_nobreak \_belowtitle{\_medskip}% \_firstnoindent } \def\i #1 {\ii .#1 \iis .#1 {{\code{\\#1}}}} \def\x`{\bgroup\_setverb\xx} \bgroup \lccode\string`\.=\string`\` \lowercase{\egroup \def\xx #1#2.{\i #2 \egroup `#1#2.}} \bgroup \globaldefs=-1 \def\refs{} \tmpnum=0 \def\_Xtoc#1#2#3{\incr\tmpnum \ea\addto \ea\refs \expanded{{\noexpand\sdef{r:#3}{\the\tmpnum}}} \scantoeol\ignoreit } \isfile{optex-doc.ref}\iftrue \input{optex-doc.ref}\fi \ea \egroup \refs \def\urloptexdoc{https://petr.olsak.net/ftp/olsak/optex/optex-doc.pdf} \def\optexdoc[#1]{\ea\ulink\expanded{[\urloptexdoc\#toc:\trycs{r:#1}{}]}} \hyperlinks\Green\Green \insertoutline{MATH} \outlines{0} \tit Typesetting Math with \OpTeX/ \hfill Version 03, January 2021 \author Petr Olšák This document is a brief summary of typesetting math. It describes \TeX/, Plain \TeX/ and \OpTeX/ features concerned to math. The first two types of features are documented in \TeX/book in chapters 16, 17, and 18, but it is summarized here in short again in order to give a complete guide about math typesetting for \OpTeX/ users. \new {} The \OpTeX/ features which differs from standard \TeX/ or Plain \TeX/ are documented with the red triangle at the margin (like in this paragraph). Reader can simply distinguish between \"standard" features (given by \TeX/ or Plain \TeX/) and new \OpTeX/ features. There are more types of extensions: e\TeX, lua\TeX/, Unicode math and \OpTeX/ macros. The appropriate label (e\TeX, Lua\TeX/, Unicode, \OpTeX/) is appended to the red triangle to inform you about the extension type. Nevertheless, \OpTeX/ user doesn't have to worry about it, all extensions are available if Unicode Math font is loaded (e.g., by the command `\fonfam[lmfonts]`). See \optexdoc[1.3.3]{section 1.3.3} in \OpTeX/ documentation about loading Unicode math fonts. {\iindent=2em \bigskip \maketoc } \vfil\break \sec Basics structure of math formulas \secc General rules and terminology The \ii in-line/math in-line math (in the paragraph) is created by `$$`. The \ii display/math display math (a standalone line between paragraphs) is created by `$$$$`. More than one line can be here if an appropriate macro is used. In-line math is processed in a \TeX/ group in \ii in-line/math/mode {\em in-line math mode}. The display math is processed in a \TeX/ group in \ii display/math/mode {\em display math mode}. Spaces are ignored in math modes, so `$x+y$` and `$x + y$` gives the same result: $x+y$. The \ii math/list is a sequence of \ii math/atom,atom {\em math atoms} and \ii other/material {\em other material}. The math atoms are \ii single/math/object {\em single math objects} or \ii composed/math/atom composed math atoms. \begitems \hfuzz=.6pt * The single math object is a single character to be printed in math mode like `x`, `+`, `\int`. * The math atom is constructed in general by `{}^{}_{}`. It consists from \ii nucleus {\em nucleus} , \ii exponent exponent and \ii subscript subscript . Each part of the atom should be empty. If or is empty, we need not to write brackets and the prefix `^` or `_`. If the or or consist only from a single math object then we need not use brackets. For example `x^2` is a math atom with `x` in the nucleus, `2` in the exponent, and with empty subscript. Or `a_{i,j}` is a math atom with `a` in the nucleus, empty exponent, and `i,j` in the subscript.% \fnote{In \OpTeX/, the character `_` can be interpreted as a part of the control sequence name, not as the subscript constructor. But in common cases, constructions of math atoms are interpreted exactly as in plain \TeX. See sections \optexdoc[2.2.2]{2.2.2} and \optexdoc[2.15]{2.15} of \OpTeX/ documentation for more details. If you want to be sure that `_` is just a subscript constructor in \OpTeX/ then you can set \code{\\catcode`\\_=8} but after this, you cannot use control sequences with `_` character.} The constructors for exponent `^` and subscript `_` can be used in arbitrary order after the nucleus, for example, `z_1^{x+y}` is the same math atom as `z^{x+y}_1`. The single math objects not followed by `^` nor `_` are considered as math atoms with this object in the nucleus and with empty exponent and subscript (this is a very common case). \TeX/ assigns the \ii class {\em class} for each math atom, see section~\ref[class]. * Other material can be \TeX/ box or glue (space) or `\kern` or `\vrule` etc. \enditems Example: The `Z = \int``_\Omega x^{2y} + z\, dx` generates $Z = \int_\Omega x^{2y} + z\, dx$ and it is which consists from: \begitems * `Z` is math atom with empty exponent and subscript, class: Ord, * `=` is math atom with empty exponent and subscript, class: Rel, * `\int``_\Omega` is math atom with empty exponent and with subscript `\Omega`, class: Op, * `x^{2y}` is math atom with exponent `2y` and empty subscript, class: Ord, \begitems \let\_bullet=\circ * `2` is math atom with empty exponent and subscript, class: Ord, * `y` is math atom with empty exponent and subscript, class: Ord, \enditems * `+` is math atom with empty exponent and subscript, class: Bin, * `z` is math atom with empty exponent and subscript, class: Ord, * `\,` is another material, the glue (space) in this case, * `d` is math atom with empty exponent and subscript, class: Ord, * `x` is math atom with empty exponent and subscript, class: Ord. \enditems \secc[class] Classes of math atoms \TeX/ assigns \ii class {\em a class} for each math atom.\fnote {Using terminology of \TeX/book, each single math object has its {\em class} but the math atom has its {\em kind} derived from this class. I use only one word for both meanings in this document.} This data type is used when \TeX/ decides about \ii horizontal/spacing horizontal spaces between atoms in the output. (Note that spaces in the input are ignored.) For example, `$xy$` prints two atoms without space between them but `$x+y$` is printed with small spaces around the `+` binary operator. Compare: $xy$ and $x+y$. The class is assigned depending on the nucleus of the atom. If the nucleus is not a single math object, i.e. it is constructed by `{}` with braces then the atom has its class Ord. If the nucleus is a single math object constructed without braces then the class of the atom depends on this single math object. Each single math object must be declared in \TeX/ with its default class. The following table lists the classes with typical examples. The full set of all math objects used in math typesetting is listed in section~\ref[objects] with their default classes. \bigskip \noindent\hfil\table{llll}{ & \ii class Class & Meaning & Example \crl 0 & \iid Ord & ordinary object & variables, digits, $x, {\bbchar R}, \Gamma, 0, 1$ \cr 1 & \iid Op & big opertator & $\sum, \int, \bigcup$ \cr 2 & \iid Bin & binary operator & $+, \times, -, \pm, \cup$ \cr 3 & \iid Rel & reations & $=, \ne, \leq, \supseteq, \succsim$ \cr 4 & \iid Open & opening bracket & $\{, (, [, \langle$ \cr 5 & \iid Close & closing bracket & $\}, ), ], \rangle$ \cr 6 & \iid Punct & punctuation & comma \cr & \iid Inner & left-right & \code{\\left...\\right} outputs, see section~\ref[delims] } \bigskip There are \ii horizontal/spacing,spacing three space types used by the algorithm for horizontal spacing in the math formulas. \begitems * \ii thin/space Thin space: \x`\thinmuskip` primitive register, `\,` macro. Used around Op atoms. * \ii medium/space Medium space: \x`\medmuskip` primitive register, `\>` macro. Used around Bin atoms. * \ii thick/space Thick space: \x`\thickmusip` primitive register, `\;` macro. Used around Rel atoms. \enditems \puttext 7.8cm -3.7cm {\rotbox{90}{Left atom}} \puttext 11.5cm -.5cm {Right atom} \puttext 8.2cm -4.4cm {\typosize[9/11] \thistable{\def\_enspace{ }}% \table {l|8c|} { \omit & Ord & Op & Bin & Rel & Open & Close & Punct & \omit \hfil Inner \crlp{2-9} Ord & 0 & 1 & 2 & 3 & 0 & 0 & 0 & 1 \cr Op & 1 & 1 & & 3 & 0 & 0 & 0 & 1 \cr Bin & 2 & 2 & & & 2 & & & 2 \cr Rel & 3 & 3 & & 0 & 3 & 0 & 0 & 3 \cr Open & 0 & 0 & & 0 & 0 & 0 & 0 & 0 \cr Close & 0 & 1 & 2 & 3 & 0 & 0 & 0 & 1 \cr Punct & 1 & 1 & & 1 & 1 & 1 & 1 & 1 \cr Inner & 1 & 1 & 2 & 3 & 1 & 0 & 1 & 1 \crlp{2-9} }} \hangindent=-8.7cm \hangafter=0 Ord atoms are printed without spaces between them. The spaces are not cumulated, so the rule about spaces mentioned above is only a rough idea. The exact rule for horizontal spaces is given for each pairs of atoms in the table here. The symbol 0 means no space, 1 thin space, 2 medium space, and 3 means thick space.\fnote {The table presented here is built into \TeX. On the other hand, \LuaTeX/ allows to set arbitrary spaces between arbitrary pairs, see section 7.5.2 in the \LuaTeX/ manual.} \hangindent=-8.7cm \hangafter-2 The Bin atom is automatically transformed to the Ord atom if no atom precedes or if Op, Bin, Rel, Open, or Punct atom precedes. And it is transformed to the Ord atom if Rel, Close or Punct atom follows. This corresponds to the empty cells in the table. Why such behavior? Compare \"\hbox{$0-3$}" and \"$-3$". The Bin atom in the second case behaves like Ord atom because it is \ii unary/minus {\em unary minus}. There is no space between the unary minus and the following object. All medium spaces and thick spaces and some thin spaces from this table are omitted if the is processed in \ii script/style,scriptscript/style script or scriptscript styles (smaller size). See section~\ref[styles] about math styles. You can overwrite the default class derived from the nucleus of the atom by \TeX/ primitives \x`\mathord`, \x`\mathop`, \x`\mathbin`, \x`\mathrel`, \x`\mathopen`, \x`\mathclose`, \x`\mathpunct` and \x`\mathinner`. They can precede a nucleus of the atom and they set the class of the atom. For example, `x \mathrel+ y` behaves like `x = y` from a spacing point of view but + is printed. Another example: `\mathop{\rm lim} z` creates the atom `lim` in roman font of class Op. So, the thin space is inserted between lim and $z$. There are more special kinds of math atoms: fractions, math accents, radicals. They are constructed in a special way (see next sections) but they behave like Ord atom in the horizontal spacing algorithm. \secc[styles] Math styles When a formula (or a sub-formula) is processed by \TeX/ then one from four \ii math/style,display/style,text/style,script/style,scriptscript/style styles is active: display style ($D$), text style ($T$), script style ($S$) or scriptscript style ($SS$). \ii T/style,D/style,S/style,SS/style The $T$ style is started in in-line math mode `$...$` and the $D$ style is started in display math mode `$$...$$`. The first level of exponents or subscripts is processed in $S$ style and the second and more levels of exponents or indexes are processed in $SS$ style. There are special rules for math styles when fractions are constructed, see section~\ref[frac]. The $D$ and $T$ styles use basic \ii font/size font size, $S$ uses smaller font size (typically 70~\%) and $SS$ style uses more smaller font size (typically 50~\%). Next levels of \"more smaller fonts" are not used due to classical typographic rules. The \ii nucleus nucleus of \iid Op atoms (big operators, $\sum$, $\int$, etc.) have typically bigger versions of the character shape for $D$ style than for $T$ style. So, there are four sizes for such math objects: one size for each math style. All other math objects (with non Op class) are printed only in three sizes: The sizes for $T$ and $D$ styles are equal. The \iid Op atom puts its \iid exponent and \iid subscript above and below the nucleus in $D$ style but right to the nucleus in other styles: $$ `\sum``_{i=1}^\infty` \quad \hbox{gives}\quad \sum_{i=1}^\infty \hbox{ in $D$ style and} \quad \textstyle \sum_{i=1}^\infty \hbox{ in $T$ style}. $$ This default behavior of the Op atom can be modified by placing \x`\limits` or \x`\nolimits` or \x`\displaylimits` \TeX/ primitive just after its nucleus before the constructors of exponent and/or index. The `\nolimits` puts exponent and subscript right to the nucleus (regardless of the current style) and `\limits` puts these objects above and below the nucleus (regardless of the current style). There can be more such primitives in a queue (due to a macro expansion, for instance). Then the last primitive in the queue wins. If the last primitive is \x`\displaylimits` then the default behavior is processed regardless there are \x`\limits` or \x`\nolimits` before it. $$ `\sum\nolimits``_{i=1}^\infty` \quad \hbox{gives}\quad \sum\nolimits_{i=1}^\infty \hbox{ in $D$ style and}\quad \textstyle\sum\nolimits_{i=1}^\infty \hbox{ in $T$ style}. $$ Atoms of all other classes have their exponents and/or subscripts only right to their nucleus without any exception. The primitives \x`\displaystyle`, \x`\textstyle`, \x`\scriptstyle` and \x`\scriptscriptstyle` set the given style regardless the default rules. For example, you can create a formula in in-line math mode and in $D$ style by `$\displaystyle $` or a formula in display mode and $T$ style can be printed by `$$\textstyle $$`. If a subformula is placed below something (below a line from root symbol, below a fraction line), then the processed style $D, T, S$ or $SS$ is \ii cramped/style {\em cramped}. The exponents are positioned slightly lower than in \ii non-cramped/style non-cramped style. The selectors `\displaystyle`\,\dots `\scriptscriptstyle` mentioned above select non-cramped style. The non-cramped style is selected when math mode starts too. \new \OpTeX/ You can select a cramped style by the macro \x`\cramped` at the start of the math formula or after the math-style selectors: `\scriptstyle\cramped` for example. Several macros need to know what math style is currently processed (for example they need to draw something in an appropriate size). But it not possible simply due to the syntax of fractions (section~\ref[frac]). This syntax requires to process all math lists in two steps: the first step expands all macros and creates structured data of processed math list. The second step reads the output of the first step, switches between math styles and creates definitive output. So, macros (working in the first step) cannot know the current math style because it is set only in the second step. \TeX/ supports the primitive \x`\matchchioce``{}{}{}{}` which prepares four math lists in the first step and only one of these four lists are used in the second step. We can put different macros into each of the four parameters of `\mathchoice`. Plain \TeX/ supports the macro \x`\mathpalette` which gives a more comfortable interface of \x`\mathchoice` to the macro programmer. The cramped/non-cramped variants of the current style are kept when `\mathchioce` is used. \new \OpTeX/ We describe another interface for creating macros depending on the current style. You can use \x`\mathstyles``{}`. It behaves like `{}`, moreover, you can use the following commands inside such : \begitems * The macro \x`\currstyle`. It expands to `\displaystyle`, `\textstyle`, `\scriptstyle` or `\scriptscriptstyle` depending on the current math style when the `\mathstyles` was opened. * The \x`\dobystyle``{}{}{}{}` is expandable macro. It expands its parameter ``, ``, `` or `` depending on the current math style when `\mathstyles` was opened. * The value of the \x`\stylenum` register is 0, 1, 2 or 3 depending on the current math style when `\mathstyles` was opened. \enditems % Example of usage of \x`\mathstyles`: \def\mysymbol{\mathbin\mathstyles {\kern1pt\vrule height\mysymbolA width\mysymbolA\kern1pt}} \def\mysymbolA{\dobystyle{5pt}{5pt}{3.5pt}{2.5pt}} \begtt \def\mysymbol{\mathbin\mathstyles {\kern1pt\vrule height\mysymbolA width\mysymbolA\kern1pt}} \def\mysymbolA{\dobystyle{5pt}{5pt}{3.5pt}{2.5pt}} Test: $a\mysymbol b_{c \mysymbol d}$ or $a\mysymbol b\over c$. \endtt This example gives Test: $a\mysymbol b_{c \mysymbol d}$ or $a\mysymbol b\over c$. The \x`\mathstyles` macro mentioned above uses \TeX/ primitive \x`\mathchoice`, so it creates four math lists and only one is used. It may take more computer time in special cases. \new Lua\TeX/ Lua\TeX/ supports the \x`\mathstyle` primitive (no \"`s`" at the end of this control sequence) which expands to values 0 to 7 depending on the current style: $D, D', T, T', S, S', SS, SS'$ (where $X'$ means cramped variant of the style). This primitive does not use `\mathchoice` but it simply ignores the fraction syntax, so `$a\mysymbol b\over c$` cannot work if `\mysymbol` is defined using the `\mathstyle` primitive. See section 7.3.1 of Lua\TeX/ documentation for more information. \secc[frac] Fractions The \iid fraction can be constructed by `{`\x`\over``}`. If the fraction is only a single object in the whole math mode (between dollars), you need not use the outer braces, so you can write `$1\over2$` to get $1\over2$. The \ii numerator,denominator and are printed in \"smaller" math style than current math style. More exactly the following schema is used. $D$: $T\over T$., $T$: $S\over S$, $S$: $SS\over SS$, $SS$: $SS\over SS$. For example $$ `{a+b \over c}` \quad \hbox{is printed as } {a+b\over c} \hbox{ in $D$ style and as } \textstyle {a+b\over c} \hbox{ in $T$ style}. $$ The \LaTeX/ macro \x`\frac``{}{}` is not supported in Plain \TeX/ nor in \OpTeX/ but you can define such macro if you want. The syntax with \x`\over` is more preferred because it is more human-readable notation. You can write the fraction in the same manner as you can read it. You can compare: `$1\over2$` (one over two) with `$\frac12$` (frac twelve). Besides the `\over` primitive, there are analogical \TeX/ primitives which create \"generalized" fractions. The result is similar to `{\over }` but there is something extra: \begitems * `{`\x`\atop``}` does `{\over}` but without the fractional rule. * `{`\x`\above``}` creates fractional rule with thickness. * `{`\x`\overwithdelims``}` adds the left to the fraction and the right to the fraction. * `{`\x`\atopwithdelims``}` is analogical to `\overwithdelims` but without fractional rule. * `{`\x`\abovewithdelims``}` behaves as `\overwithdelims` but the fractional rule has thickness. \enditems % The `\...withdelims` variants read and , they must be declared as {\em math delimiter} in \TeX. They are vertically scalable math objects, typically brackets. See section~\ref[delims] for more information about math delimiters. Example: $$ `{n \atopwithdelims() k}`\quad \hbox{ creates } {n \choose k} \hbox{ in $D$ style and } \textstyle {n \choose k} \hbox{ in $T$ style}. $$ The \x`\choose` macro is defined by `\def\choose{\atopwithdelims()}`, so the user can write `{n\choose k}` in order to get binomial coefficients. \secc[delims] Vertically scalable objects: math delimiters The vertically scalable objects are called \ii delimiters {\em delimiters}. For example, all types of brackets are declared as delimiters. This means that you can use a bracket in arbitrary vertical size.\fnote{ This is not exactly true, because traditional typography says that they cannot be scaled continuously but by visible steps. This means that there is a sequence of increasing brackets in the font, the reader must see a difference between every two sizes of brackets.} The following objects are declared as delimiters (i.e.\ vertically scalable): \bigskip {\tt \adef!{\bslash} \table{l 14c}{ \rm source: & ( & ) & [ & ] & \code{\\\{} & \code{\\\}} & / & !backslash & !langle & !rangle & | & \code{\\|} & \cr \rm output: &$($&$)$&$[$&$]$& $\{$ & $\}$ & $/$ & $\backslash$ & $\langle$ & $\rangle$ & $|$ & $\|$ } \medskip \table{l 14c}{ \rm source: & !lfloor & !rfloor & !lceil & !rceil \cr \rm output: & $\lfloor$ & $\rfloor$ & $\lceil$ & $\rceil$ } \medskip \table{l 14c}{ \rm source: & !uparrow & !Uparrow & !dowarrow & !Downarrow & !updownarrow & !Updownarrow \cr \rm output: & $\uparrow$ & $\Uparrow$ & $\downarrow$ & $\Downarrow$ & $\updownarrow$ & $\Updownarrow$ }} \bigskip \noindent \new Unicode If you can produce the characters $\langle$, $\rangle$,\fnote {Do not confuse $\string<, >$ and $\langle, \rangle$. The first pair are Rel atoms with meaning \"less than" or \"greater than", but the second pair are special types of brackets. They are not directly available at computer keyboards without using a keyboard macro.} $\lfloor$, $\rfloor$, ... $\updownarrow$, $\Updownarrow$ directly in your text editor then you can use these Unicode characters in your source instead of control sequences `\langle`, `\rangle`, `\lfloor`, `\rfloor` ... `\updownarrow`, `\Updownarrow`. For many users (including me), there is more simple to type `\lfloor` than to find how to create the $\lfloor$ character in my text editor. Note that there exist text editors (Emacs, for example) enabling you to type `\lfloor` and this control sequence is immediately converted to the $\lfloor$ Unicode character. Your source text looks pretty and you can use classical \TeX/ sequences. \new Unicode There are more \ii delimiters delimiters, but it heavily depends on loaded Unicode Math font. For example, this document is printed in `latinmodern-math` font and there are six more delimiters `\lBrack`~$\lBrack$, `\rBrack`~$\rBrack$, `\lAngle` $\lAngle$, `\rAngle` $\rAngle$, `\lgroup` $\lgroup$, `\rgroup` $\rgroup$. See section~\ref[objects] for table of all Unicode symbols for math typesetting. Arbitrary tall formula can be surrounded by a pair of delimiters using \x`\left` and \x`\right` \TeX/ primitives: `\left \right`. The delimiters are scaled to the height and depth of the and vertically centered to the {\em math axis}.\fnote {Math axis is a horizontal line passing through the center of symbols $+$ and $-$. All vertically scalable objects are vertically centered with respect to this axis.} Example: $$ `+ \left\{ \sum_{i=1}^\infty x_i \right)` \quad \hbox{ gives } + \left\{ \sum_{i=1}^\infty x_i \right). $$ The pair `\left \right` creates the formula in a \TeX/ group. Such group can be nested with another groups. Each `\left` must have its `\right` counterpart at the same group level. If you don't want to create visible delimiter, use dot instead . Example: $$ `\left. \int``_0^t e^{x^2}\,dx\, \right|_{t=42}` \quad \hbox{ gives } \left. \int_0^t e^{x^2} \,dx\, \right|_{t=42} $$ \new e\TeX/ You can use \x`\middle``` inside the which is surrounded by `\left...\right`. Then the given is scaled to the same size like their `\left` and `\right` counterparts. When a delimiter is used without `\left` nor `\right` prefix, then it is the Open, Close, Ord or Bin atom by its natural meaning: $(, [, \{, \ldots, \lfloor, \lceil$ are Open atoms, $], ], \}, \ldots, \rfloor, \rceil$ are Close atoms, $/, \backslash, |, \|$ are Ord atoms and $\uparrow, \Uparrow, \ldots, \Updownarrow$ are Bin atoms. You can overwrite this default setting, for example `\mathclose(`. If delimiters are used with `\left` and `\right` prefixes then `\left` behaves like Open atom, `\right` behaves like Close atom and the math list `\left\right` is encapsulated as a single Inner atom. The `\middle` behaves like Open atom at its left side and like Close atom at its right side. The sequence of increasing delimiters can be printed by the following macros: $$ `(` \to (,\quad \x`\big``(` \to \big(,\quad \x`\bbig``(` \to \bbig(\fnote{Provided only in \OpTeX/.}, \quad \x`\Big``(` \to \Big(,\quad \x`\bigg``(` \to \bigg(, \quad \x`\Bigg``(` \to \Bigg(. $$ The `\Bigg` is not the maximal size of the bracket. Try `\left(\vbox to5cm{}\right.`, for example. You can see that the font \"cheats" from certain sizes, because there are not all infinity number of sizes of brackets drawn in the font, of course. The `\big` creates Ord atom. We need to create Open atom for opening bracket and Close atom for closing bracket more often. Then we can use macros \x`\bigl```, \x`\bbigl```, \x`\Bigl```, \x`\biggl```, \x`\Biggl``` for creating Open atoms and \x`\bigr```, \x`\bbigr```, \x`\Bigr```, \x`\biggr```, \x`\Biggr``` for creating Close atoms. Unfortunately, the source is not too attractive when more sizes of brackets are used, but typographic traditions say that we have to distinguish brackets by the size in math mode if they are in equal types: $$ `\bbigl( f\bigl( 2(x+y) + z\bigr) \bbigr)'` \quad \hbox{gives } \bbigl(f\bigl(2(x+y)+z\bigr)\bbigr)'. $$ The `math.opm` package from \OpTeX/ provides doing this more comfortable. \secc Horizontally scalable objects: math accents Arbitrary wide formula can be covered by \ii math/accent,accent {\em scalable math accent}. Example: $$ \x`\overrightarrow`` {a+b+c+d+e+f}` \quad \hbox{gives } \overrightarrow {a+b+c+d+e+f}. $$ The usage is: control sequence of selected math accent followed by `{}`. Standard scalable math accents are: \x`\overline` $\overline{abc}$, \x`\overbrace` $\overbrace{abc}$, \x`\overrightarrow` $\overrightarrow{abc}$, \x`\overleftarrow`~$\overleftarrow{abc}$, \x`\underline` $\underline{abc}$, \x`\underbrace` $\underbrace{abc}$. An Op atom is created. The exponents and subscripts are centered above and below the nucleus of this atom (regardless of the current style). Example: $$ `\overbrace {b\cdot b\cdot b \cdots b}^{k\times}` \quad \hbox{gives } \overbrace {b\cdot b\cdot b \cdots b}^{k\times} $$ There are scalable accents with a limited maximum width: \x`\widehat` $\widehat{abc}$ and \x`\widetilde` $\widetilde{abc}$. If the formula is wider than the font can cover then the widest variant from the font is used and it is horizontally centered. \new Unicode There are more scalable accents in Unicode math fonts: \x`\overparen` $\overparen{abc}$, \x`\underparen` $\underparen{abc}$, \x`\overbracket` $\overbracket{abc}$, \x`\underbracket` $\underbracket{abc}$, \x`\overleftrightarrow` $\overleftrightarrow{abc}$, \x`\overleftharpoon` $\overleftharpoon{abc}$, \x`\overrightharpoon` $\overrightharpoon{abc}$, \x`\wideoverbar` $\wideoverbar{abc}$, \x`\widebreve` $\widebreve{abc}$, \x`\widecheck` $\widecheck{abc}$,\nl \x`\wideutilde` $\wideutilde{abc}$, \x`\mathunderbar` $\mathunderbar{abc}$, \x`\underleftrightarrow` $\underleftrightarrow{abc}$, \x`\widebridgeabove` $\widebridgeabove{abc}$,\nl \x`\underrightharpoondown` $\underrightharpoondown{abc}$, \x`\underleftharpoondown` $\underleftharpoondown{abc}$, \x`\underleftarrow` $\underleftarrow{abc}$,\nl \x`\underrightarrow` $\underrightarrow{abc}$. \secc Fixed math accents Fixed \ii math/accent,accent math accents can be applied to single math object or to the `{}`. The accent is centered (with respect of slanting axis) and the result is a nucleus of Ord atom. For example `\dot x` gives $\dot x$. The list of fixed math accents follows: \x`\acute`` x` $\acute x$, \x`\bar`` x` $\bar x$, \x`\breve`` x` $\breve x$, \x`\check`` x` $\check x$, \x`\dot`` x` $\dot x$, \x`\ddot`` x` $\ddot x$, \x`\grave`` x` $\grave x$, \x`\hat`` x` $\hat x$, \x`\vec`` x` $\vec x$, \x`\tilde`` x` $\tilde x$. \new Unicode The additional fixed accents depends on used Unicode math font. The `latinmodern-math` supports: \x`\ovhook`` x` $\ovhook x$, \x`\ocirc`` x` $\ocirc x$, \x`\leftharpoonaccent`` x` $\leftharpoonaccent x$, \x`\rightharpoonaccent`` x` $\rightharpoonaccent x$, \x`\dddot`` x` $\dddot x$, \x`\ddddot`` x` $\ddddot x$, \x`\widebridgeabove`` x` $\widebridgeabove x$, \x`\asteraccent`` x` $\asteraccent x$. There exist one special math accent `'` (single quote, ASCII 39) which can be appended after a symbol like this: `f'` and it creates $f'$ (typical meaning is the derivation of the given function). You can put more such accents, for example `g'''` gives $g'''$. \secc Roots There is a macro \x`\sqrt``{}` to create \ii square/root square root. For example: $$ `\sqrt{\sqrt{\sqrt{x+1}+1}+1}` \quad \hbox{gives } \sqrt{\sqrt{\sqrt{x+1}+1}+1} $$ The \ii root $n$-th root is created by the macro \x`\root`` \of{}`. For example `\root k+1\of x` gives $\root k+1\of x$. \secc Math alphabets Letters \ii math/alphabet $a\dots z$, $A\dots Z$ and $\alpha$\dots$\omega$ are printed in italic in math mode. This follows the traditional typographic rule. All other math symbols, digits, and uppercase Greek letters must be upright.\fnote {French typographic convention says that uppercase Greek letters have to be in italic too. Use `\_itGreek` declaration in this case.} These rules are independent of the current variant of surrounding text font. If we want to use the letters or digits in another than this default shape, then we can use \ii math/alphabet/selector {\em math alphabet selectors}: \x`\mit`, \x`\rm`, \x`\it`, \x`\bf`, \x`\cal`. \new \OpTeX/ \OpTeX/ supports more such selectors \x`\script`, \x`\frak`, \x`\bbchar`, \x`\bi`, see \optexdoc[1.3.3]{section 1.3.3} in the \OpTeX/ documentation. The math selectors have local validity in the group. The control sequences \x`\rm`, \x`\it`, \x`\bf`, and \x`\bi` act as variant selectors of fonts in non-math mode (text mode) and they act as math alphabet selectors in math mode. This \"overlaying" concept is given by Plain \TeX/. Example: math operators lim, sin, cos, log, etc.\ must be printed unslanted. We are using `\lim`, `\sin`, `\cos`, `\log` etc.\ in math mode in order to comply with this typographic convention. For example `\sin` is defined as: \begtt \def\sin {\mathop{\rm sin}\nolimits} \endtt The `\rm` is used here as math alphabet selector, no variant selector of text fonts. The list of all predefined `\rm`-like math operators follows: \x`\arccos`, \x`\arcsin`, \x`\arctan`, \x`\arg`, \x`\cos`, \x`\cosh`, \x`\cot`, \x`\coth`, \x`\deg`. \x`\det`, \x`\dim`, \x`\exp`, \x`\gcd`, \x`\hom`, \x`\inf`, \x`\ker`, \x`\lg`, \x`\lim`, \x`\liminf`, \x`\limsup`, \x`\ln`, \x`\log`, \x`\max`, \x`\min`, \x`\Pr`, \x`\scs`, \x`\sin`, \x`\sinh`, \x`\sup`, \x`\tan`, \x`\tanh`, You can define another such operator analogically. \new Unicode Unicode font can include the following math alphabets: \begtt \typosize[9/11] \_rmvariables % upright letters A-Z, a-z \_bfvariables % bold letters A-Z, a-z \_itvariables % italic letters A-Z, a-z \_bivariables % bold italic letters A-Z, a-z \_calvariables % calligraphic letters A-Z, a-z \_bcalvariables % calligraphic letters A-Z, a-z \_frakvariables % fraktur A-Z, a-z \_bfrakvariables % bold fraktur A-Z, a-z \_sansvaraibales % sans serif letters A-Z, a-z \_bsansvaraibales % bold sans serif letters A-Z, a-z \_isansvaraibales % slanted sans serif letters A-Z, a-z \_bisansvaraibales % bold slanted sans serif letters A-Z, a-z \_ttvariables % monospace, typewriter letters A-Z, a-z \_bbvariables % double struck A-Z, a-z \_rmdigits % upright digits 0..9 \_bfdigits % bold digits 0..9 \_sansdigits % sans serif digits 0..9 \_bsansdigits % bold sans serif digits 0..9 \_ttdigits % monospace typewriter digits 0..9 \_bbdigits % double-struck digits 0..9 \_rmgreek % upright Greek letters \alpha-\omega \_itgreek % slanted Greek letters \alpha-\omega \_bfgreek % bold Greek letters \alpha-\omega \_bigreek % bold italic Greek letters \alpha-\omega \_bsansgreek % bold sans serif Greek letters \alpha-\omega \_bisansgreek % bold slanted snas serif Greek letters \alpha-\omega \_itGreek % slanted Greek letters \Alpha-\Omega \_bfGreek % bold Greek letters \Alpha-\Omega \_biGreek % bold italic Greek letters \Alpha-\Omega \_bsansGreek % bold sans serif Greek letters \Alpha-\Omega \_bisansGreek % bold slanted snas serif Greek letters \Alpha-\Omega \endtt % Not all Unicode math fonts include all math alphabets listed here. Typically, the lowercase letters of calligraphic shape and all letters of bold calligraphic shape are missing. \new \OpTeX/ \OpTeX/ defines internal math alphabet selectors as mentioned in the previous listing of math alphabets and sets as default: \begtt \typosize[10/12] \_itvariables \_rmdigits \_itgreek \_rmGreek \endtt % Moreover, it defines the alphabet selectors at user level (see section 1.3.3 of the \OpTeX/ manual). For example \begtt \typosize[10/12] \_def\_marm {\_inmath{\_rmavariables \_rmdigits}} % \_marm is used in \rm \endtt % `\_inmath` runs its parameter only in math mode and sets the math alphabets. You can see the file `unimath-codes.opm` where all these selectors are defined. You can redefine them. For example, \OpTeX/ defines `\bf` as a math alphabet selector that selects sans serif bold in math. This is the common notation for vectors, tensors, and matrices. If you dislike this (maybe because plain \TeX/ has different defaults), then you can define: \begtt \typosize[10/12] \_def\_mabf {\_inmath{\_bfvariables\_bfdigits\_bfgreek\_bfGreek}} % used in \bf \endtt \secc[objects] List of single math objects \new Unicode All \ii single/math/object single math objects are listed in the `unimath-table.opm` or `unicode-math-table.tex` file. You can look into this file. The codes, \TeX/ sequences, classes, and comments for all possible math codes are here. Maybe, your Unicode math font which is loaded does not support all these codes. \new \OpTeX/ You can try all codes of the currently loaded font by \begtt \input print-unimath.opm \endtt The `unimath-table` is printed with characters available in the loaded font. \new \OpTeX/ If the character is unsupported by the font then the slot is empty and only \TeX/ sequence and the class of the code is printed in the table. For example, this document loads `latimodern-math.otf` font. And the result from `\input print-unimath.opm` looks like the following ten pages. Unsupported characters can be replaced by characters from other Unicode math font, see section~\ref[newfam]. \label[listA]\wlabel{} \bigskip \input print-unimath.opm \bigskip \label[listB]\wlabel{} It isn't very comfortable to find something in the previous table if you know the shape. You can try the online web tool \ii Detexify \ulink[https://detexify.kirelabs.org/classify.html]{Detexify}. You can draw the symbol here and the suggestion of \TeX/ sequence is printed. \def\g#1{{\tt\string#1}~$#1$} The generic \TeX/ sequences for the Greek letters \ii Greek/letters can be used: \g\alpha, \g\beta, \g\gamma, \g\delta, \g\varepsilon, \g\zeta, \g\eta, \g\theta, \g\iota, \g\kappa, \g\lambda, \g\mu, \g\nu, \g\xi, \g\omicron, \g\pi, \g\rho, \g\varsigma, \g\sigma, \g\tau, \g\upsilon, \g\varphi, \g\chi, \g\psi, \g\omega, \g\vardelta, \g\epsilon, \g\vartheta, \g\varkappa, \g\phi, \g\varrho, \g\varpi\ and \g\Alpha, \g\Beta, \g\Gamma, \g\Delta, \g\Epsilon, \g\Zeta, \g\Eta, \g\Theta, \g\Iota, \g\Kappa, \g\Lambda, \g\Mu, \g\Nu, \g\Xi, \g\Omicron, \g\Pi, \g\Rho, \g\Sigma, \g\Tau, \g\Upsilon, \g\Phi, \g\Chi, \g\Psi, \g\Omega. \new \OpTeX/ The variant of Greek letters in the output (upright, italic, bold, bold sans serif, etc.) written by the sequences `\alpha`, `\beta` etc. depends on the math alphabet selected by `\_itgreek`, `\_rmgreek`, etc. selectors. The user-level selectors `\bf` and `\bi` set `\_bsansgreek` and `\_bisansgreek`, so `{\bi\delta}` produces $\bi\delta$. \new Unicode All characters available in the math font can be accessed by \TeX/ control sequence or by directly using the Unicode character in the document source. Example: \begtt \adef/{} $$ \sum/_{k=0}^\infty e^{(\alpha+i\beta/_k)} = e^\alpha \sum/_{k=0}^\infty e^{i\beta/_k} = e^\alpha \sum/_{k=0}^\infty (\cos\beta/_k + i\sin\beta/_k). $$ \endtt or \begtt \ttspec $$ ∑_{k=0}^∞ e^{(α + iβ_k)} = e^α ∑_{k=0}^∞ e^{iβ_k} = e^α ∑_{k=0}^∞ (\cos β_k + i\sin β_k). $$ \endtt both gives the same result: $$ ∑_{k=0}^∞ e^{(α + iβ_k)} = e^α ∑_{k=0}^∞ e^{iβ_k} = e^α ∑_{k=0}^∞ (\cos β_k + i\sin β_k). $$ \medskip \sec Other specialities \secc The `\not` prefix You can apply \x`\not` before a following math object. The slash $/$ is overprinted such math object, for example `$a \not= b$` gives $a \not= b$. \new \OpTeX/ If there exists a direct Unicode character for the negation of a relation symbol (for example `\ne` creates $\ne$ directly as a character U+2260) then `\not` expands to appropriate Unicode character. For example `\not=` expands to `\ne` or `\not\in` expands to `\notin`. If such character does not exist then the centered $/$ is overprinted over the next character. \secc The `\buildrel` macro: text over the relation The macro \x`\buildrel`` \over ` creates a new atom Rel with the and with the smaller above this . Example: `$M \buildrel\rm def\over= X\cup Y$` gives $M\buildrel\rm def\over= X\cup Y$. \secc Spaces Spaces between atoms are created automatically as were mentioned in section~\ref[class]. But sometimes you have to help \TeX/ to create appropriate space. You can use following macros: \begitems * `\,` is \ii thin/space {\em thin space} used around Op atoms, after comma, etc.: \ss\,, * `\!` is \ii negative/thin/space negative thin space, * `\>` is \ii medium/space {\em medium space} used around Bin atoms: \ss\>, * `\;` is \ii thick/space {\em thick space} used around Rel atoms: \ss\;, * \x`\quad` is \ii em/space {\em em space}: \ss\quad, * \x`\qquad` is \ii double/em/space {\em double em space}: \ss\qquad. \enditems % Of course, you can use \ii direct/space {\em direct space} {\visiblesp`\ `} which is \TeX/ primitive and gives interword space: \ss\ \ or you can use \x`\hskip`` ` to put arbitrary space. The space size of `\,`, `\!` resp. `\>`, resp. `\;` is given by `\thinmuskip`, resp. `\medmuskip`, resp. `\thickmuskip` values. You can see in the `plain.tex` file that these default values differ very little in their basic size but there is no stretchability/shrinkability in the `\,` space, there is small stretchability in the `\>` space, and more stretchability in the `\;` space. The registers \x`\thinmuskip`, \x`\medmuskip`, and \x`\thickmuskip` store so-called \ii mu/values {\em mu values} given by math unit `mu`. It is 1/18 em and this unit depends on the current font size used in the math formula ($S$ or $SS$ styles use smaller font size, the `mu` unit is smaller here). You can use \x`\mskip` instead `\hskip` or \x`\mkern` instead `\kern` if you want to use this special `mu` unit. It is allowed only in math mode. If \x`\nonscript` precedes `\hskip`, `\mskip`, `\kern`, or `\mkern` then this space is applied only in $D$ or $T$ style. The \x`\quad` and \x`\qquad` spaces have fixed width and they can be used in text mode too. (\OpTeX/ allows to use `\,` in text mode too). Use `\quad` or `\qquad` if you want to separate more formulas created in single math mode. Examples of typical usage of spaces: \begtt \adef/{} \typosize[10/12] $$ \alpha\,(x+y), \qquad \int/_a^b \!\! f(x)\,{\rm d}x, \qquad \Gamma/_{\!i}. $$ \endtt $$ \alpha\,(x+y), \qquad \int_a^b \!\! f(x)\,{\rm d}x, \qquad \Gamma_{\!i}. $$ \secc Texts in math mode If you write `$Hello world!$` (i.e.\ Hello world in math mode), then you get $Hello world!$. It is interpreted as the product of variables $H$, and $e$, and $l^2$, and $o$, etc., followed by the symbol ! used for factorial. The non-ASCII letters (with accents) don't work at all because they are never used as symbols for variables. Spaces are ignored. If you want to write a short text in the math mode, then you can use `\hbox{}`. The \x`\hbox` primitive initializes text mode regardless of the \"outer mode". Example: \begtt \adef/{} \typosize[10/12] $$ \sum/_{n=0}^\infty (-1)^n a_n \hbox{ converges, if $a_n\searrow 0$.} $$ \endtt $$ \sum_{n=0}^\infty (-1)^n a_n \hbox{ converges, if $a_n\searrow 0$.} $$ % Note the space before the word \"converges". The space before `\hbox` is irrelevant. Second notice: the example shows the text mode inside math mode and the in-line math mode inside this text mode. The same result can be produced by: \begtt \adef/{} \typosize[10/12] $$ \sum/_{n=0}^\infty (-1)^n a_n \hbox{ converges, if } a_n\searrow 0. $$ \endtt % The difference can be visible if the formula $a_n\searrow 0$ includes a fraction, for example ${1\over2}a_n\searrow 0$. The first example prints the fraction in the text style and the second example prints it in the display style. The disadvantage of `\hbox` is that it starts in the text mode independently of the current style, but we want to use smaller font in $S$ or $SS$ styles. \new \OpTeX/ You can use \x`\mathbox``{}` in such situations. This macro behaves like `\hbox` but the text is appropriately smaller in $S$ and $SS$ styles. Example: $$ `{\mathbox{cena}\over\mathbox{výkon}}` \quad \hbox{gives } {\mathbox{cena}\over\mathbox{výkon}} \hbox{ in $D$ style and } \textstyle {\mathbox{cena}\over\mathbox{výkon}} \hbox{ in $T$ style.} $$ Note that $\mathbox{cena}\over\mathbox{výkon}$ means $\rm price\over performance$ and you can write `$\rm price\over performance$` when you are using only words without spaces and accented letters. But phrases with spaces or accented letters should be printed in text mode using `\hbox` or `\mathbox`. \secc `\vcenter` The \x`\vcenter` primitive behaves like `\vbox`, but it can be used only in math mode and its result is vertically centered to the math axis. For example, matrices, are created by tables in `\vcenter`. All big objects in math formula are centered on the math axis and the baseline is ignored. In the following example, we create a new big math operator by `\vcenter`: \begtt $$ \def\myop#1{\mathop{\vcenter{\frame{\vbox to2em{\vss\hbox{ $#1$ }\vss}}}}} F(x) = \myop{x(i)}_{i=1}^\infty $$ \endtt \vskip-2em $$ \def\myop#1{\mathop{\vcenter{\frame{\vbox to2em{\vss\hbox{ $#1$ }\vss}}}}} F(x) = \myop{x(i)}_{i=1}^\infty $$ \secc Three dots You can write `$1,2,\dots,n$` to get $1,2,\dots,n$. The \x`\dots` macro puts thin space between dots and after the last dot, so the five object: comma, dots, comma are exactly equidistant. Typographic conventions say that you have to use the repeating symbol before and after three dots (comma in the previous example) and the three dots should be at baseline if the repeating symbol is at baseline. Or they should be at the math axis if the repeating symbol is at the math axis. We have to use \x`\cdots` instead `\dots` in the second case. Example: $$ \addto\_ttfont{\thefontsize[9.5]} `a_1, a_2, \dots, a_n, \quad a_1 + a_2 + \cdots + a_n` \qquad a_1, a_2, \dots, a_n, \quad a_1 + a_2 + \cdots + a_n $$ \vskip-.7em There are \x`\vdots` $\vdots$, \x`\ddots` $\ddots$ and \x`\adots` $\adots$ which can be used in matrices. \new Unicode Three dots like the output of the `\dots` macro are present as a single character in fonts too. This character is called ellipsis. Font designers typically suggest this character with smaller spaces between dots than we need in math mode. So the rule about equidistant \"comma, three dots, comma" is not met when this character is used. You can try `$1,2,\unicodeellipsis, n$` and `$1+2+\unicodecdots + n$`. You get $1,2,\unicodeellipsis, n$ and $1+2+\unicodecdots + n$. If you feel that this is better, then you can set: `\let\dots=\unicodeellipsis \let\cdots=\unicodecdots`. The Unicode fonts include compact variants `\unicodevdots` $\unicodevdots$, `\unicodeddots` $\unicodeddots$ and `\unicodeadots` $\unicodeadots$ too. \secc Phantoms and `\smash` The \x`\phantom``{}` macro creates an invisible subformula equal to the formula generated by . It has its size, so it can interfere with surrounding visible subformulas. This macro is very useful for aligning with special requirements. Examples are shown in sections~\ref[matrix] and~\ref[elines]. The \x`\vphantom``{}` has only its vertical size (i.e.\ its height and depth), the width is zero. The \x`\hphantom``{}` has only its width. The \x`\smash``{}` is the opposite of `\vphantom`. It creates visible subformula but it has only its width. Its height and depth are zero. The result of these macros is the nucleus of an Ord atom. You can use `\mathop`, `\mathbin`, etc. primitives to change this class. For example `$a\mathrel{\phantom{=}}b$` creates the same formula as $a=b$ with the same distance from $a$ to $b$ but without the equal sign: $a\mathrel{\phantom{=}}b$. These macros work in the text mode too. Then their argument is the . \sec Structured objects \secc[matrix] Matrices The macro \x`\matrix``{}` creates a vertically centered table of items. The includes separated by `&` and rows are separated by `\cr`. The number of columns and rows are unlimited. Columns are printed centered and separated by the `\quad` space. The vertically scalable brackets around the table are not printed. You can use \x`\pmatrix``{}` instead `\matrix{}`: the vertically scalable parentheses () are inserted around the table. Examples: \begtt $$ {\bf A} = \pmatrix{a_{1,1} & a_{1,2} & \cdots & a_{1,n} \cr a_{2,1} & a_{2,2} & \cdots & a_{2,n} \cr \vdots & \vdots & \ddots & \vdots \cr a_{m,1} & a_{m,2} & \cdots & a_{m,n} }, $$ \endtt $$ {\bf A} = \pmatrix{a_{1,1} & a_{1,2} & \cdots & a_{1,n} \cr a_{2,1} & a_{2,2} & \cdots & a_{2,n} \cr \vdots & \vdots & \ddots & \vdots \cr a_{m,1} & a_{m,2} & \cdots & a_{m,n} }, $$ or: \begtt \def\qmatrix[#1]{\left[\matrix{#1}\right]} $$ \qmatrix[a&b&c\cr d&e&f\cr g&h&i] \cdot \qmatrix[x_1\cr x_2\cr x_3] = \qmatrix[b_1\cr b_2\cr b_3]. $$ \endtt $$ \def\qmatrix[#1]{\left[\matrix{#1}\right]} \qmatrix[a&b&c\cr d&e&f\cr g&h&i] \cdot \qmatrix[x_1\cr x_2\cr x_3] = \qmatrix[b_1\cr b_2\cr b_3]. $$ If you need to align the columns by another way than to center, then you can use the phantom. Compare: \begtt $$ \pmatrix{ 1 & -1 & 0 \cr 0 & 2 & 13 \cr -3 & 0 & 5 } \quad \hbox{or} \quad \def\0{\phantom0} \def\+{\phantom+} \pmatrix { \+1 & -1 & \00 \cr \+0 & \+2 & 13 \cr -3 & \+0 & \05 } $$ \endtt $$ \pmatrix{ 1 & -1 & 0 \cr 0 & 2 & 13 \cr -3 & 0 & 5 } \quad \hbox{or} \quad \def\0{\phantom0} \def\+{\phantom+} \pmatrix { \+1 & -1 & \00 \cr \+0 & \+2 & 13 \cr -3 & \+0 & \05 } $$ \new\OpTeX/ Another option to set the right aligned matrix is setting the \x`\lmfil`: Its value is used on the left side in each `\matrix` item. The right side is set directly to `\hfil`. \begtt $$ \lmfil={\hfill} % left matrix filler = \hfill \pmatrix{ 1 & -1 & 0 \cr 0 & 2 & 13 \cr -3 & 0 & 5 } $$ \endtt \new \OpTeX/ If you want to draw a vertical line inside the matrix, you can use `\adef|` as in the following example: \begtt $$ \adef|{\kern-.2em&\strut\vrule&\kern-.2em} \def\+{\phantom+} \pmatrix{1 & 2 & 3 | 0 \cr 4 & 5 & 6 | 1 \cr 7 & 8 & 9 | 2 } \sim \pmatrix{1 & \+2 & \+3 | 0 \cr 0 & -3 & -6 | 1 \cr 0 & -6 & -12 | 2} \sim \pmatrix{1 & 2 & 3 | 0 \cr 0 & 1 & 2 | -1/3 \cr 0 & 0 & 0 | 1 } $$ \endtt $$ \adef|{\kern-.2em&\strut\vrule&\kern-.2em} \def\+{\phantom+} \pmatrix{1 & 2 & 3 | 0 \cr 4 & 5 & 6 | 1 \cr 7 & 8 & 9 | 2 } \sim \pmatrix{1 & \+2 & \+3 | 0 \cr 0 & -3 & -6 | 1 \cr 0 & -6 & -12 | 2} \sim \pmatrix{1 & 2 & 3 | 0 \cr 0 & 1 & 2 | -1/3 \cr 0 & 0 & 0 | 1 } $$ If you want to put something before the opening bracket in the matrix, you can use another `\matrix`. Example: \begtt $$ \adef|{\kern-.2em&\strut\vrule&\kern-.2em} \def\+{\phantom+} \def\r{{\bf r}} \pmatrix{1 & 2 & 3 | 0 \cr 4 & 5 & 6 | 1 \cr 7 & 8 & 9 | 2 } \ \sim \ \matrix{\cr 2.\r - 4\cdot 1.\r: \cr 3.\r -7\cdot 1.\r: } \pmatrix{1 & \+2 & \+3 | 0 \cr 0 & -3 & -6 | 1 \cr 0 & -6 & -12 | 2} $$ \endtt $$ \adef|{\kern-.2em&\strut\vrule&\kern-.2em} \def\+{\phantom+} \def\r{{\bf r}} \pmatrix{1 & 2 & 3 | 0 \cr 4 & 5 & 6 | 1 \cr 7 & 8 & 9 | 2 } \ \sim \ \matrix{\cr 2.\r - 4\cdot 1.\r: \cr 3.\r -7\cdot 1.\r: } \pmatrix{1 & \+2 & \+3 | 0 \cr 0 & -3 & -6 | 1 \cr 0 & -6 & -12 | 2} $$ Plain \TeX/ defines the \x`\bordermatrix` macro which allows you to create a top row above the brackets and left column before the opening bracket. \TeX/book shows the following example: \begtt $$ M = \bordermatrix{ &C&I&C'\cr C&1&0&0 \cr I&1-b&0 \cr C'&0&a&1-a } $$ \endtt $$ M = \bordermatrix{ &C&I&C'\cr C&1&0&0 \cr I&1-b&0 \cr C'&0&a&1-a } $$ The `\matrix` macro used in $D$ style creates all its items in $T$ style. \new \OpTeX/ If you are using the `\matrix` macro in $T$ style or $S$ style (but not in fractions nor another matrices) then the resulting table is appropriately smaller and all its items are processed in $S$ style. If you are using `\matrix` in $SS$ style then the items are in $SS$ style too. The following example shows one-column matrix in script style: $$ `\sum``_{\matrix{i\in M\cr j\in N\cr k\in P}} x_{i,j,k}` \ \hbox{ creates: } \sum_{\matrix{i\in M\cr j\in N\cr k\in P}} x_{i,j,k} $$ \secc Cases The \x`\cases` macro can be used as in the following example: \begtt $$ f(x) = \cases{ 1 & for $x\le 0$, \cr 100-x\over 100 & when $0 < x \le 100$, \cr 0 & in other cases. } $$ \endtt $$ f(x) = \cases{ 1 & for $x\le 0$, \cr 100-x\over 100 & when $0\string< x\le 100$, \cr 0 & in other cases. } $$ The `\cases` macro behaves like a special `\matrix` with two left-aligned columns and with left vertically scaled brace $\{$. The first column is processed in math mode and $T$ style, the second column is processed in text mode. We have to use `$...$` in the second column if there is math material. \sec Lines in display mode \secc General principles The \ii display/math,display/math/mode `$$$$` finalizes previous paragraph, prints centered on single line with a vertical space above and below and opens next paragraph with no indentation. From \TeX/'s point of view, the text above `$$$$` plus text below is a single paragraph interrupted by display . If there is no text above (i.e. the opening `$$` are in vertical mode), then the internal `\noindent` is processed first and the empty line above is created. Thus, it is definitely a bad idea to open display mode in vertical mode: never put an empty line before `$$$$`. On the other hand, the empty line just after `$$$$` says that the paragraph is finalized by the and the next text (after the empty line) opens the next paragraph with indentation. Summary: \begitems * \"Text above `$$$$` text below": the interrupts the paragraph with \"Text above" and \"text below". The \"text below" is without indentation. * \"Text above `$$$$` empty line Text below": the \"Text below" opens new paragraph with indentation. * \"Empty line `$$$$`" is bad practice. \enditems In contrast with the paragraph breaking, there is no built-in algorithm for breaking display to more lines. If the is too wide then overfull `\hbox` occurs and human must decide about splitting the to more lines. The macros `\displaylines` and `\eqalign` are intended to such task, see sections~\ref[dlines] and~\ref[elines]. On the other hand, the in-line math , i.e.\ the `$$` in a paragraph, can be broken after a Bin atom (with penalty `\binoppenalty`) or after a Rel atom (with penalty `\relpenalty`). If you don't want to break such a formula at a specific place then use `\nobreak`, for example `$a+\nobreak b$`. If you want never to break such formulas then you can set `\binoppenalty=10000`, `\relpenalty=10000`. (Default values are 700 and 500.) \secc References to display lines The `$$`\x`\eqno``$$` \ii mark prints centered and the at right margin. The `$$`\x`\leqno``$$` prints centered and the at left margin. Examples: \begtt $$ a^2 + b^2 = c^2 \eqno (1) $$ \endtt $$ a^2 + b^2 = c^2 \eqno (1) $$ \begtt \aftergroup\nobreak $$ a^2 + b^2 = c^2 \leqno \rm Py $$ \endtt $$\predisplaypenalty=10000 a^2 + b^2 = c^2 \leqno \rm Py $$ The \ii mark is processed in math mode and $T$ style. If you want to put a text here then you have to use `\rm ` or `\hbox{}`. \new \OpTeX/ The auto-generated can be created by `$$`\x`\eqmark``$$`. The