The manual

1What it is

live-math is a maths whiteboard you type on. You write in the editor, a line at a time, in AsciiMath; whoever is watching sees the sheet typeset, with its plots and charts, as you type. It is made to explain: it draws what helps students notice things, and never solves the exercise for you.

It has three parts:

• The editor (/editor): where you write. Only you see it, with your private notes.
• The viewer: the sheet alone, typeset, with no buttons. It opens in a tab next to the editor (Alt+V), to share in a video call.
• The live: the same viewer on the students' own devices, through a link (Alt+L). For authorised teachers.

You need a computer with a recent Chrome, Edge, Firefox or Safari. Students only need a browser, a phone's too, and no account.

The editor, the viewer and the exports are free and need no account. The live, the sheets in the cloud, the recorded lessons and the published sheets need a login, which is free too: ask for it.

2First steps

2.1Where the sheets go

The first time, the editor asks where to keep your sheets. Each sheet is a text file, name.math; what you can choose depends on the browser:

• A folder on the computer (Chrome, Edge): “Choose the sheets' folder”. The browser asks for permission; next time the editor reopens that folder and the last sheet by itself, or asks for one click to reopen it. “Choose another folder” picks another.
• This browser's storage (Firefox, Safari): “Start”, or “Import sheets…” to start from sheets you already have. Clearing the site's data, or private browsing, loses them: keep a backup (8.5). In a private window of Firefox there is no storage at all.
• The cloud, for authorised teachers: “Log in and use the cloud”, with a Google account or a code by email. The same sheets on any computer and browser.

The local place and the cloud can be used side by side: each sheet lives in one of them, and one list shows both (8.4). On the first start the editor writes a demo sheet, demo, with a bit of everything.

2.2The screen

The editor is a single sheet of text, with nothing around it but a bar at the bottom. Everything else is in the palette (Ctrl+K) and in the help (F1).

The bar at the bottom shows, from the left:

• the sheet's name, followed by “(cloud)” when it is in the cloud;
• whether it is saved: “saved”, “modified”, “saving…”, or why it is not (“not saved: cloud unreachable”, “conflict”…);
• “viewer” with a green dot when a viewer tab is open;
• during a live, “live” with a dot (green when connected) and how many are watching; “rec” while the lesson is being recorded;
• with the curtain on, how many lines it hides;
• on the right, short messages in the accent colour: what just happened, or what to do.

Sometimes a bar appears above the sheet, with its buttons:

• “The file changed on disk.” (or the cloud, or another tab): “Reload” takes the other version, “Keep mine” saves yours over it.
• “Unsaved version found.”: text that was not saved last time (the browser closed first). “Restore” or “Discard”.
• The folder must be reopened to see the computer's sheets, while a cloud sheet is open: “Reopen”, or “Forget it”.

Ctrl+\ opens the side preview: the viewer, at the right of the editor. Drag the line between them to give more room to one side; a double click brings it back to half.

2.3The palette and the help

Ctrl+K opens the palette: every command, with its shortcut on the right. Type a few letters to filter (they need not be adjacent: vbl finds “Viewer: blackboard theme”), move with the arrows (or Ctrl+N, Ctrl+P), Enter runs, Esc or a click outside closes. The same box asks for a name, or for a confirmation (“Yes” or “No”). The full list is in §11.

F1 shows the syntax, the shortcuts and the snippets on one page, with links to the guide and to this manual; F1 or Esc closes it.

3Writing a sheet

3.1Lines and blocks

Every line is a formula, written in AsciiMath (§4). A few characters at the very start of the line make it something else:

An empty line leaves some space; several empty lines, the same space. Spaces at the start of a line do not count for a formula, but they do for a prefix:  # Title is a formula.

3.2Titles, text, results, notes

# Limits
## A notable one
" For **every** `epsilon > 0` there is a `delta > 0`
% remind them about x -> 0^-
lim_(x->0) sin x / x = 1
! e^(i pi) + 1 = 0
---

In a text line, a formula starts and ends with a backtick (`); one still open shows as a formula anyway, while you type it. **bold** works only in text lines. The note is gone from the viewer and from every export.

3.3Chains of steps

A line that starts with a relation continues the formula above it, aligned under its first relation: the steps of a calculation, one per line.

f(x) = x^2 e^x
f'(x) = 2x e^x + x^2 e^x
= x e^x (2 + x)
>= 0 " for " x >= 0

The relations that start a step: = != < > <= >= ~~ ~= ~ -= => <=> -> :=. The first line aligns on its first relation outside brackets, absolute values and quoted text. A note between the lines does not break the chain.

3.4The decimal comma

With the decimal comma on (the default in Italian; palette, “Decimal comma: on” and “Decimal comma: off”), a comma between two digits is a decimal comma: 3,14 is $3,14$. With a space after it, it is a separator: (1, 2) is a pair. A list of numbers without spaces, {1,2,3}, stays a list, and so do the cells of a matrix. With it off, 3.14 is the decimal point.

pi ~~ 3,14
(1, 2) " is a pair, " {1,2,3} " a set"

3.5When a line is wrong

A line that cannot be typeset gets a wavy red underline in the editor; the mouse over it shows why (such as “tangent(f, a): the function and the point”). The viewers never see an error: they keep showing that line as it was the last time it worked, so half-typed formulas never flicker on the shared screen.

3.6Snippets and typing help

Type one of these words, then Tab: it becomes the template on the right, and Tab / Shift+Tab move between its fields. Elsewhere Tab indents.

Brackets ( [ { close themselves and the matching one lights up; quotes do not, since " starts a text line. The editor colours the syntax: functions, Greek letters, numbers, prefixes. Ctrl+Z undoes, Alt+↑/Alt+↓ move a line.

4AsciiMath, symbol by symbol

AsciiMath is a short way of writing formulas, close to how you would type them in a message. Below is everything it knows, in the version used here: what you type on the left (aliases side by side do the same), what it becomes on the right.

The same symbols on one page, each with its LaTeX, and a box to try them: the AsciiMath cheat sheet.

4.1Fractions, scripts and roots

/ makes a fraction of what is right before and after it: a symbol, a number, or a group in brackets. ^ and _ do the same for exponents and indices. The brackets around a numerator, a denominator, an exponent or an index are not shown: (a+1)/(b+1) is $\frac{a+1}{b+1}$, x^(n+1) is $x^{n+1}$. To keep them there, double them: ((a+b))/2 is $\frac{\left(a+b\right)}{2}$. Elsewhere brackets stay: (a+b)^2 is ${\left(a+b\right)}^2$.

Spaces only separate: sinx and sin x are the same, $\sin x$. Letters side by side are a product: ab is $ab$. Mind the names made of letters: xx is $\times$, in is $\in$, oo is $\infty$; a space keeps letters apart, x x.

color colours only its argument; it takes the names of colours (red, blue, green, orange, purple…).

4.2Greek letters

4.3Operations

4.4Big operators, limits, integrals

The limits go after _ and ^; with brackets when they are more than one symbol. On their own line they are set below and above the operator.

4.5Relations

4.6Logic

4.7Brackets

Brackets grow with what they hold. {: x :} is an invisible pair, useful to group; (: :) are angle brackets. |x| is the absolute value: the bars pair up inside brackets, so |x - a| < delta works as it reads.

4.8Arrows

4.9Functions

A function's name is set upright and its argument follows, with or without brackets: sin x, sin(x), sin^2 x. f and g are functions only before a bracket: f(x), but f g stays a product.

Capitalised names

Italian names and linear algebra

The names of Italian textbooks, and the operators of linear algebra. Im is the image only before a bracket: Im(f), while Im z stays a product.

4.10Accents

Fonts

4.11Text inside a formula

Between double quotes, words stay words: x = 1 " if " y > 0. A thin space is added where the text touches a symbol, so "for all"x does not stick. and, or, if as words are already spaced.

4.12Matrices, vectors, systems

A matrix is a list of rows, each in brackets, separated by commas: [[1,2],[3,4]] with square brackets, ((1,2),(3,4)) with round ones, |(a,b),(c,d)| for a determinant. A column vector is a matrix of one column. A system is a matrix with { on the left and nothing on the right, :}:

A = [[1,2],[3,4]]
det A = |(1,2),(3,4)| = -2
vec v = ((x),(y),(z))
{(x + y = 1),(x - y = 2):}
|x| = {(x, " if " x >= 0),(-x, " if " x < 0):}

4.13Other symbols

4.14Characters with a meaning

&, #, $ and % are shown as they are; ~ alone is $\sim$. ', '', ''' are primes. A single \ followed by a space is a thin space; \\ is a backslash. dx, dy, dz, dt are differentials, with a little space before them: int x^2 dx.

5Figures

A line that starts with plot:, gon:, sign: or system: is drawn. Its parts are separated by semicolons; their order does not matter. The figures are drawn in the viewer's colours, sharp at any size, and their labels are typeset like the rest of the sheet.

5.1Expressions in figures

Inside a figure the expressions are computed, so they use a smaller language than AsciiMath:

• the variable is x (in the complex plane, z, with i the imaginary unit); constants: pi, e;
• operations: + - * / ^, with the usual precedence (-x^2 is $-\left(x^2\right)$); a product can be implicit: 2x, x sin x, 2pi; |x| is the absolute value;
• functions: sin cos tan cot (and sen tg cotg), arcsin arccos arctan (and arcsen arctg), sinh cosh tanh (and sh ch th), ln, log (natural, as ln), exp, sqrt, root(n)(x), abs, floor, ceil, sgn;
• a function's argument goes in brackets, or is what follows up to the next function: sin 2x is $\sin \left(2x\right)$; sin^2 x is ${\left(\sin x\right)}^2$; the odd roots of negative numbers are real (root(3)(-8) is −2);
• numbers: 0.5, or 0,5 with the decimal comma; a decimal comma has no space after it.

5.2Plots: plot:

Up to four functions of x, with their colours and a legend. y = f is the same as f. Points are pairs in brackets; x = a is a vertical line.

plot: sin x; x - x^3/6; x in [-2pi, 2pi]; y in [-2, 2]

plot: x^2; (1, 1); (-2, 4); x = 2

The window. x in [a, b] and y in [c, d] choose what is shown. Without them, x goes from −5 to 5 (wider if a point or a line is outside) and y follows the functions, leaving out what shoots off near an asymptote. When the interval of x is written with pi, the ticks are multiples of π. 5.10 moves the window from the keyboard.

At most four among functions and vertical lines; a function is drawn once even when it appears in several parts.

5.3Zeros, equations, inequalities

roots(f) draws f and its zeros, with their values. An equation draws both sides and their intersections; an inequality marks on the x axis where it holds, with full and empty dots at the ends. A double inequality, -1 < f(x) <= 2, works too. The values are exact when they are notable (5.9).

plot: roots(sin x); x in [-2pi, 2pi]

plot: x^2 = 2x + 1

plot: x^2 - 1 > 0; x in [-3, 3]

A condition with no solution says so in the legend. A function with infinitely many zeros in the window (sin(1/x) near 0) asks for a narrower window.

5.4Tangents, derivatives, areas

tangent(f, a) draws f, the tangent at x = a and its equation (a vertical line where the derivative is infinite). derivative(f) draws the curve of f'. area(f, a, b) fills the region between f and the axis from a to b, with the integral, and the area too when f changes sign.

plot: x^2; tangent(x^2, 1)

plot: sin x; derivative(sin x); x in [-2pi, 2pi]

plot: area(x^2 - 1, 0, 2)

Their arguments are separated by a comma and a space, so that a decimal comma inside them still works: area(x^2, 0,5, 2). Without any space, every comma separates. These words are recognised only as whole words: tan(1) is still the tangent function.

5.5The complex plane

With i or z the plot becomes the Gauss plane, with the same scale on both axes. A number is a point, with its ray from the origin. A root of a whole expression, w^(1/n), root(n)(w) or sqrt(w), shows all its values, on their circle, named $z_0,z_1,\dots$. An equation in z must be a polynomial: its roots are drawn.

plot: (3+4i)^(1/8)

plot: z^2 + 2z + 5 = 0; 2e^(i pi/3)

A plot is either real or complex: x and z do not mix. Up to four elements; no inequalities.

5.6The unit circle: gon:

gon: draws the trigonometric circle. An angle (pi/3) is a radius with its projections on the axes, labelled when notable. An arc is an interval of angles: [a, b], (a, b), [a, b), (a, b], closed or open at each end. An equation or inequality in x shows its solutions: points, or arcs.

gon: sin x >= 1/2

gon: cos x > -sqrt2/2; tan x > 1

gon: pi/3; [pi, 3pi/2)

Against a number, sin x, cos x, tan x and cot x also get their line: the horizontal, the vertical, the tangent at $\left(1,0\right)$ or at $\left(0,1\right)$. An angle that is not notable is named after its arc: $\alpha =\arcsin \left(\frac{1}{3}\right)$, then $\pi -\alpha$. x in [a, b], at most one turn, limits where the solutions are looked for (the rest of the circle is dashed); without it, a whole turn, and an arc across 0 starts from a negative angle. Up to four among conditions and arcs.

5.7Sign charts: sign:

sign: takes a product or a quotient, alone or in an inequality with 0 on one side, and draws one row per factor as you wrote it (solid where positive, dashed where negative), the row of the product with its signs, and, for an inequality, the solutions.

sign: (x - 1)(x + 2)/(x - 3) >= 0

It does not factor for you: those steps are yours to show. A zero of a denominator is an empty dot, and the product shows ∄ there. A negative constant gets a row of its own. The critical points are evenly spaced, as on the blackboard, labelled with notable values or with $x_1,x_2,\dots$ defined in the legend. Up to six rows and twelve points; with infinitely many (sin x), x in [a, b] narrows it.

5.8Systems of inequalities: system:

One inequality per part, any sides, double ones too: a row for each, solid where it holds, and the last row where they all hold.

system: x^2 - 1 > 0; x < 3; 2x + 1 >= 0

5.9Exact labels

Zeros, intersections, angles and the ends of areas are labelled with their exact value when it is a notable one: fractions, $\pi$ and its fractions, $\sqrt{2}$ and other roots, $\frac{1+\sqrt{5}}{2}$, $e$, $\ln 3$. Otherwise an approximation, $\approx 1.73$. Labels that would overlap are left out.

5.10Moving a plot's window: Alt+G

With the cursor on a plot: line, Alt+G highlights it and the keys move what is shown: the arrows move the window by a tick, Shift+arrows widen it on that side, + and - zoom (in steps of 1, 2, 5, 10…, multiples of π when x is in π), 0 goes back to the automatic window. Esc, Enter, Alt+G, a click or any other key ends it.

Each key rewrites x in [...]; y in [...] in the line itself, so students see the plot move, and the line stays the only truth. A single Ctrl+Z undoes the whole adjustment. The Gauss plane stays square; the circle has no window.

6Showing it

6.1The viewer

Alt+V opens the viewer in a new tab: the sheet alone, without your notes, updated at every key. Share that tab in your video call (“viewer” turns green in the editor while it is open). Best with two screens: the editor on one, the viewer shared on the other.

Chrome and Edge can share a single tab. Firefox and Safari share windows or the whole screen: drag the viewer's tab out into a window of its own, and share that window.

The viewer follows the line you are on: that block has a bar on its left (the active block), and the sheet scrolls to keep it in sight. Formulas too wide for the column are scaled down a little.

6.2How it looks

These change every viewer at once, the live's and the recordings' included:

• Alt+= / Alt+-: bigger or smaller (from 70% to 200%);
• Alt+T: paper (light) or blackboard (dark);
• Alt+H: the bar of the active block, on or off;
• the decimal comma (palette), which also changes how numbers are written in the figures.

6.3Pointing and marking

On the viewer tab, or on the side preview, the mouse becomes a pointer that every viewer sees. Drag to draw: circle a term, underline a step, cross out a factor. Shift and drag highlights a box. Esc (in the editor or on the viewer) erases the drawings.

Marks stay on their symbols in every viewer, phones included, whatever the size of the screen, and move with their block when you add lines above it. Typing hides the pointer. At most 32 marks: the oldest go. Alt+P turns pointer and drawing off and on.

6.4The curtain

Write the exercise and its solution before the lesson. In class, put the cursor on the last line to show and press Alt+S: the viewers see the sheet up to there, and nothing below. In the editor a dashed “curtain” line marks the place, and the hidden lines are lighter.

• Alt+R reveals the next line: the steps of a chain come one at a time. Empty lines and notes go along.
• Moving the cursor never reveals anything: read the hidden part as a script. What you type above the curtain is seen at once; below, it stays hidden.
• Once everything is revealed (“curtain · all revealed”), what you write at the end is seen as you type.
• Alt+S again removes it, asking first if lines are still hidden. A reload of the editor keeps it where it was.
• In the palette, also “Curtain: reveal up to this line” and “Curtain: cover from this line”.
• Exports and published sheets always contain the whole sheet.

7The live

The live sends the sheet to the students' own devices: sharper than a shared screen, and each of them can scroll back on their own. It needs an authorised account; students need none.

Start. Alt+L. The first time a window opens to log in, with Google or with a code by email (the code comes from Cloudflare: look in the spam if it does not arrive). The link for the students is copied by itself: it is always the same for your account, on any computer, so you can send it once. “Copy the live link” copies it again.

While it runs. The bar shows “live”, a green dot and how many are watching, and “rec” while it is recorded (9.2). Changing sheet (Ctrl+P) changes it for everyone.

For the students. The link opens the sheet as you write it. If they scroll back, the sheet stops following you until they press “↓ live” (or End). Before you start, they see “Waiting…”; if their connection drops, “reconnecting…”, and it resumes by itself.

Stop. Alt+L again: those still connected keep the last version, whoever opens the link later waits for the next live. If you close the editor without stopping, the live comes back by itself when you reopen it, even days later: stop it to work in private.

The login lasts 30 days. When it expires the editor says “login expired: Alt+L to log back in”. “Log out” (palette) ends it at once.

8Sheets

8.1Opening, naming, deleting

Ctrl+P lists your sheets, the most recent first: type to filter, Enter opens. Alt+N makes a new one, in the same place as the open one. A name can have subfolders, up to two: analysis1/limits; it cannot start with a dot or contain \ : * ? " < > |. “Rename sheet” and “Delete sheet” are in the palette; deleting cannot be undone.

8.2Saving

There is no need to save: the editor does it by itself a moment after you stop typing (“saved” in the bar). Ctrl+S saves at once. Every few keys the text is also kept in the browser as a safety net: if the browser closes before saving, the next time a bar offers to restore it. Each sheet reopens with its cursor and scroll where you left them.

8.3Changes from elsewhere

If a sheet changes outside this window (in another tab, on another computer, in a text editor), the editor reloads it in silence when you have nothing unsaved (“reloaded from disk”), and otherwise asks with a bar which version to keep (2.2).

8.4In the cloud

For authorised teachers, “Log in and use the cloud” adds the cloud's sheets to the same list: Ctrl+P shows each one's place when there are both, and the bar shows “(cloud)” after a cloud sheet's name. “Move to the cloud” and “Move to the computer” (or “Move to this browser”) move the open sheet; “Move all sheets to the cloud” moves them all. A move never overwrites: a sheet with the same name in the other place stops it.

Without a network the open cloud sheet keeps working, with “not saved: cloud unreachable” in the bar: it is saved by itself when the network is back, and leaving the page asks first. The server keeps no backup of the sheets: keep a copy of the important ones. Up to 2000 sheets of 512 KB each.

8.5Backup and import

“Download all sheets” downloads a .zip with every sheet of one place (when there are two, one command each). With the sheets in the browser's storage, the editor reminds you after a week without a backup. “Import sheets or backups” takes .math files or .zip backups, “Import a folder” a whole folder; both write into the place of the open sheet and ask once about sheets that exist already. That is also how sheets move between browsers.

“Change the sheets' folder” switches to another folder; “Keep the sheets in this browser too” adds the browser's storage next to the cloud.

8.6Logging out

“Log out” (with your email) saves the open cloud sheet, stops the live and removes every trace of the cloud from this browser: on a shared computer, the next person finds none of your sheets.

9After the lesson

9.1Exports

Alt+E exports the whole sheet, without notes: PDF (the print dialog opens: choose “Save as PDF”; every --- starts a new page), HTML (one file, to open in any browser), TeX (a LaTeX document that compiles). “Copy the LaTeX of the current line” copies the LaTeX of the line under the cursor.

9.2Recorded lessons

Every live is recorded, sheet by sheet, exactly as the students saw it: never your notes, nor what was under the curtain, nor the pointer. In the palette, “Lessons and published sheets” lists the lessons, the open sheet's first: copy the link, watch it, or delete it (the link then stops working).

The link replays the lesson as it was written, with long pauses shortened. Space plays and pauses, the arrows jump to your previous or next pause, Home and End go to the ends, the button on the right changes the speed (1×, 2×, 4×). A lesson in which the sheet never changed is not kept.

9.3Published sheets

“Publish the sheet” saves a frozen copy of the sheet, as it is now, and copies its link: later changes do not touch it. Publishing again makes another copy, with another link. Published sheets are in the same list as the lessons, where they can be withdrawn. Anyone with a link can open it: send it only to whoever should see it.

10Every shortcut

On the Mac Cmd stands for Ctrl, and Option for Alt. If the browser uses Alt and a letter for its menus, add Shift: Alt+Shift+V does what Alt+V does. Letters are read by their key, whatever the keyboard layout.

11Every command

All of them in the palette (Ctrl+K); some appear only when they can act (the cloud's once logged in, the curtain's once it is on).

12Languages

The interface is in English and in Italian: it follows the browser's language, and the palette's “Language: Italiano” switches it. The language also sets the decimal comma of a new browser and the babel of the TeX export. These pages have their own link at the top.

13Access and privacy

The live and the cloud are for authorised teachers and tutors, free: ask on the home page. You log in with the address you gave, through a Google account or a one-time code by email. Students never need an account, and get no cookies.

Your sheets stay on your computer, unless you keep them in the cloud; how every piece of data is handled is on the privacy page.