Menu reference list
Symbols
Here you'll find the standard mathematical symbols.
 These buttons are for the nicelooking versions of the symbols. Sometimes you can write directly with the keyboard; it won't be very pretty, but it will work. For example, you can use the keyboard slash
/
for fractions. You can use the keyboard parentheses()
, but the parentheses from buttons are better because they expand with the content.
Constants
Constants  

Number pi The ratio of a circumference to its diameter  π  
Number e is the only function such that and .  e  
Imaginary unit A number defined to satisfy  i  
Imaginary unit A number defined to satisfy 

Infinity Use it to calculate limits.  ∞ 
 You must use the buttons for
e
andi
. Keyboarde
andi
are just variables; they aren't the standard constants.
The four basic operations
Basic operations  

Addition  Addition  
Subtraction  Subtraction  
Multiplication  Multiplication  
Division  Division 
 You can use multiple symbols for multiplication:
*
asterisk,·
middle dot, the one in the menu, and also aspace
. That's it; a space between variables or numbers is an implicit multiplication. The*
asterisk is automatically converted to a nicer middle dot·
.
 About division, apart from the symbol in the menu, you can use also
/
slash and, of course, the fraction symbol.
Brackets
Brackets  

Parentheses Use them to control the order of the operations.  Brackets  
Vector See Linear algebra section.  Vector  
List Some results or parameters are given or requested in this form.  
Absolute value Removes the sign from a number.  Absolute value  
Norm Length of a vector.  Norm 
 Do not use curly or square brackets as parentheses. Use only proper parentheses.
: error
: ok
Bidimensional symbols
Bidimensional symbols  

Fraction  Fraction  
Square root  Square root  
Root  Root  
Power  Power  
Element of list Use it also for a vector, a matrix, an equation, etc.  Subscript 
Inequality symbols
Inequality symbols  

Greaterthan sign  Inequality  
Lessthan sign  
Greater than or equal to  
Less than or equal to  
Not equal to 
Other symbols
Decimal separator
The dot (.
) is for the decimal point, the comma (,
) is for lists, and the apostrophe ( '
) is for derivation. The decimal point is the dot, but the others don't have that function.
There is no digit grouping symbol, nor are there spaces. Spaces mean implicit multiplication. The head decimal point is not allowed; use leading zero in those cases. The trailing decimal point is allowed.
3.1416
: OK3,1416
: Error3'1416
: Error1 234 567
: Error.12
: Error0.12
: OK12.0
: OK12.
: OK
Example 

Plus Minus
Sometimes we are interested in the result of an expression when we add and subtract the same amount, as when we want to compute the roots of a degree two polynomial. The symbol allows us such and more things.
 If, for instance, we want to compute , we expect
 We can also use it as a unary operator:
Every possible sign will be computed every time we use the symbol. Therefore, if we write symbols, we will get a list of elements. Some of them may be repeated since it is a list, not a set (for instance, ).
We can use the symbol in all the basic operations (plus, minus, product, division, root, power) and some elementary functions (exponential, logarithm, trigonometric and hyperbolic functions and its inverse).
Examples 

Examples
Arithmetic
Here you'll find commands concerning integers and the rounding of decimals. You'll also find some divisibility commands, which are shared with Polynomials.
Integers and rounding
Also, see the Document settings section to see how many decimal places are shown.
Integers and rounding  

Absolute value You can write  (pipe) with the keyboard, too.  Absolute value  
Floor Round down to the next smaller integer.  Rounding  
Ceiling Round up to the next greater integer. 

Round to nearest integer, and for tiebreaking round half up.  
The sign for a number. Can be 1, 0 or 1.  Sign  
The greater of two numbers, or of a list.  
The smaller of two numbers, or of a list.  
Generate a pseudorandom number between the two given ones (including both). Also, choose randomly from a list.  Pseudorandomness 
Divisibility
Divisibility  

The numerator of a fraction  Fraction  
The denominator of a fraction  
The quotient of the integer division of the first number (dividend) by the second (divisor)  Quotient  
Remainder of the integer division of the first number (dividend) by the second (divisor); also called modulus in many textbooks  Remainder  
Greatest common divisor  gcd  
Least common multiple  lcm  
Prime factorization of an integer  Factorization  
Tests to determine whether a number is prime. This is a predicate: a command that returns only true or false .  Prime number 
Examples
Polynomials
Polynomials are simpler types of functions. However, they're so important that they have their own naming system.
Because you can divide polynomials, they share divisibility commands with the Arithmetic section.
Here you'll also find commands for complex numbers. Complex numbers were invented as a way to solve all the polynomials.
Polynomials
Polynomials  

The degree of a polynomial  Polynomial  
How many terms a polynomial has  
This is one term from a polynomial. The term number is the second parameter. The terms are ordered by descending grades. Therefore, term number 1 is always the leading term.  
The content of a polynomial. That is, gcd of their coefficients.  
Rearrange a polynomial with multiple variables arranged around the variable in the second parameter.  
Finds the roots of a polynomial or, in other words, the values of x that make it 0. 
 The command
roots(p)
does the same thansolve(p=0)
or apply theCalc
action top=0
, but the results are shown in different forms. See the example.  You can also find roots in the Complex field, if you use as a second parameter the
C
from the Logic and Sets section. See the example.
Divisibility
Divisibility  

The numerator of a rational fraction  Rational fraction  
The denominator of a rational fraction  
The quotient of the division of the first polynomial (dividend) by the second (divisor)  Polynomial  
The remainder of the division of the first polynomial (dividend) by the second (divisor)  
The greatest common divisor  
The least common multiple  
Factorization in irreducible polynomials  
This tests whether a polynomial is irreducible. This is a predicate: a command that returns only true or false . 
Complex numbers
Complex numbers  

Imaginary unit  i  
The real part of a complex number  Complex number  
The imaginary part of a complex number, which is a real number  
The modulus of a complex number  
The argument of a complex number, in the range (π,+π]  
This converts a complex number from binomial form to polar form, and also the other way around(!). The polar form is a list formatted as {norm,argument}.  
This is the conjugate of a complex number. Shift the sign of the imaginary part. 
Examples
Statistics
Sets of data
Data sets must be entered as a commaseparated list, being enclosed by curly brackets {}
.
Sets of data  

List 
Single set
These commands summarize a set of data. Somehow, they're able to measure its center or its variability. Because there are multiple definitions for that, there are also multiple measures.
Description of a single sample  

Mean, arithmetic mean, average  Mean  
This is used to summarize measures with different units (lenght, cost, weight,...) of the same object. It has no sense alone, but it is useful for comparisons among multiple objects.  
This is used for ratios and rates, as in the context of speed.  
A measure of variability that is convenient for calculations  Variance  
A measure of variability that has the same physical units of the data  Standard deviation  
A central measure, alternative to mean, that is more robust, meaning it isn't affected by the extreme data generally known as outliers.  Median  
These are values that divide the data once ordered into four groups of the same size. They're used to measure variability. See the formula reference section for details.  Quartile  
Most frequent value in data. It can be a set, if there are ties.  Mode 
Two sets
These commands measure the relationship between data pairs.
 Data set must be entered as a list of pairs. The list must be enclosed by curly brackets
{}
. The pairs must be enclosed by regular parentheses()
. Additionally, you can plot these paired data sets.
Relationship between two sets of paired data  

This is the base for the correlation coefficient. It has the same sign.  
Pearson correlation coefficient. It determines whether there is a linear relationship between the paired data.  Correlation  
Gives the equation of the line that better fits the cloud of data. Finds the best .  Regression line  
Fits the data to a power function. Finds the best .  Nonlinear regression  
Fits the data to a exponential function. Finds the best . 

Fits the data to a logarithmic function. Finds the best . 
Formula reference
Formula reference  

Probability distributions
It is also possible to use the most common probability distributions. Currently, the following ones are available
Probability distributions  

In the uniform distribution, all intervals of the same length on the distribution's support are equally probable.  Uniform variable  
The normal distribution is determined by its mean and its standard deviation . It is widely used in natural science among other fields.  Normal variable  
The exponential distribution is the probability distribution that describes the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.  Exponential variable  
The distribution with degrees of freedom is the distribution of a sum of the squares of independent standard normal random variables.  Chisquared variable  
Student's distribution arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.  Student tvariable  
The Bernoulli distribution is the probability distribution of a random variable which takes the value with probability and the value 0 with probability .  Bernoulli variable  
The binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of independent experiments, each asking a yes–no question, i.e. each ruled by the same Bernoulli distribution.  Binomial variable  
The geometric distribution with parameter is the discrete probability distribution of the number failures before the first success, each try ruled by a Bernoulli variable with parameter .  Goemetric variable  
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event.  Poisson variable  
The distribution is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA), e.g., test.  F variable 
Moreover, we can get a random number following such distributions, obtain its distribution and density function and find the quantile of given probability.
Probability functions  

It retrieves a random number following a given distribution.  
Distribution function of a random variable at a given point. In some cases, the analytical expression is given. , cumulative distribution function (CDF) 

Density function of a random variable at a given point. In some cases, the analytical expression is given. , probability density function (PDF). 

Quantile function of a random variable for a given probability. , inverse function of CDF. 
Statistical data representation
Examples
Functions
The most common functions are the polynomial and rational ones, both of which are called algebraic functions. They're covered in the Polynomials section. This section is about other groups of socalled transcendental functions. Their exact values can't be calculated with basic arithmetic alone. You can calculate approximations, but it's impractical to do so by hand. It's better to use an electronic calculator oras was done in the old daysa table or book of values.
Trigonometric functions
Trigonometric functions started in the basic geometry of triangles, but now they comprise a complex field that's a central subject of calculus.
Trigonometric functions have two modes of work: in degrees
and in radians
. See the Document settings section.
Trigonometric functions  

Number pi This is useful when working with radians.  π  
Angle degree  Degree  
Direct  
Sine, related to the side opposite the angle  Trigonometric function  
Cosine, related to the side adjacent to the angle  
Tangent, sin/cos  
Reciprocal  
Cosecant, 1/sin  Trigonometric function  
Secant, 1/cos  
Cotangent, 1/tan  
Inverse  
This is one of the many angles whose sine is the given number. It's the one in [π/2,π/2].  Trigonometric function  
This is one of the many angles whose cosine is the given number. It's the one in [0,π]. 

This is one of the many angles whose tangent is the given number. It's the one in (π/2,π/2). 
You can use the Simplify
action to force nontrivial simplifications over trigonometric expressions. Moreover, the Verify
action can test for trigonometric identities.
Logarithms and exponentials
Exponential and logarithmic functions are very important in calculus. The logarithm is used for some physical measures, such as the units pH
in chemistry and dB
in acoustical physics.
Logarithms and exponential  

Number e This is the basis of the Napierian logarithm.  e  
Logarithm with base  Logarithm  
Exponential, e powered to the given number  Exponentiation  
Natural or Napierian logarithm  Logarithm  
Logarithm 
You must enter e
with the button. You can't simply type e
with the keyboard, because then it's just a variable called e
but not the number.
The base of a logarithm can be set as a subindex of the function log().
If no base is set, just log()
means the decimal logarithm or, in other words, the base 10 logarithm.
Also, ln()
means the natural logarithm, which is the base e
logarithm.
You can use the Simplify
action to force nontrivial simplifications over logarithmic and exponential expressions. Also, the Verify
action can test for identities.
Hyperbolic functions
Examples
Calculus
Here you'll find lots of buttons and commands that are relevant to calculus, functions and successions.
Calculus actions  

Derive  Derivative  
Integrate  Integral 
A derivation or integration will use the first alphabetical variable by default. You can change that by configuring these actions, clicking on the action icon in the line.
Calculus buttons  

Derivative You can also use the apostrophe (') from the keyboard.  Derivative  
An integral with differential  Integral  
A definite integral with differential Be aware that the sign of the function matters. The definite integral isn't always the area.  Integral  
Limit You can use infinity here.  Limit  
Limit right  Limit of a function  
Limit left  
To infinity . . . and beyond!  ∞  
Piecewise function  Piecewise function  
Function application  
Expression with a restricted domain  Restriction  
Summation with underandover scripts  Sum  
Summation with under scripts  
The product with underandover scripts  Product  
The product with under scripts 
Function application must be used only when the function is not yet defined (such as in ODEs). Do not use it to define or to use a function. Instead, use simple parentheses. See the example below.
Calculus commands  

Domain of a function  Function  
Taylor's polynomial  Taylor's polynomial  
Ordinary Differential Equations  
This fills the plane with vectors defined by a function. Use it to visualize gradients, forces, derivatives in a phase plane, etc.  Vector field  
This fills the plane with curves that follow the given vector field, i.e., curves as the solution to the ODE associated to a vector field. Use it to get an overview of the stability of the field.  Integral curve  
This draws a particular integral curve, which begins at a given point. 
Examples
Linear algebra
Find here operations of vectors and matrices.
Vectors, which use brackets, are written horizontally. You can write them with the button of the men, or directly with the keyboard.
Matrices are best written with the button of the menu. However, they can also be written with the keyboard as a vector of multiple samedimension vectors, as in many programming languages.
Once a matrix is created, you can still modify its layout. You can, for example, insert or remove columns and rows. There are buttons for that in the menu. Usually they're disabled, but they become enabled when the cursor enters a matrix.
Vectors are automatically seen as matrices by some commands. You needn't be concerned about the conversion. The usual operations are aware of vectors and matrices. For example, the common product symbol means different things when used between a scalar and a vector, two vectors, a vector and a matrix, or two matrices.
Linear algebra  

Makers  
Vector  Vector  
Matrix  Matrix  
Determinant  Determinant 
Buttons for vectors  
Scalar product, dot product  Dot product  
Vector product, cross product  Cross product  
Norm  Norm  
Element of vector 
Buttons about matrices  
Determinant  Determinant  
Inverse  Matrix  
Transpose  Transpose  
Identity matrix  Matrix  
Element of matrix 
Commands  
Dimension of a vector  Vector  
Dimensions of a matrix; first files, then rows  Matrix  
Rank of a matrix; max number of linearly independent rows or columns  Rank  
A matrix whose rows are a base of the kernel  Kernel  
A matrix whose rows are a base of the image  Image  
A list of eigenvalues, repeated as many times as their multiplicity  Eigenvalues  
A matrix whose rows are eigenvectors, ordered matching the eigenvalues result list  Eigenvectors  
The Jordan normal form of the matrix, if it exists. It gives the lower triangular form but not the upper.  Jordan normal form  
Angle between two vectors. 
 For the
kernel()
,image()
andeigenvectors()
commands, the result is a matrix whose columns are the vectors that form a base. Note that, because there are always many bases, there are many other correct results. You can get a particular vector from the result R using R^{T}_{1}, R^{T}_{2}, R^{T}_{3},...
Matrix layout modifiers  

Insert column at left  
Insert column at right  
Remove column  
Insert row above  
Insert row below  
Remove row 
Examples
Combinatorics
Here you'll find the elementary combinatorial functions. You can calculate their values. You can also apply these functions to a list and see the entire collection. Please be careful, though. The results can easily be too big.
Combinatorics  

Variations or kpermutations of n  Variations  
Permutations  Permutation  
Combinations  Combination  
Variations with repetition or ntuples of msets  Variations with repetition  
Permutations with repetition  Permutation  
Combinations with repetition  Combination  
Binomial coefficient  Binomial coefficient  
Factorial You can also write ! with the keyboard.  Factorial 
Examples
Logic and sets
You can verify
whether a statement is true or false. Use it to test identities, for example.
There are logical (Boolean) operators, which you can use to combine statements.
You can make statements about sets. There are two types of them:
 finite sets: a list of elements inside curly brackets, such as
{1,2,3}
.  standard number sets: represented by standard symbols, for example
R
.
You can make operations with the finite sets, but not with the others.
You can also work with intervals and do operations with them
Logic and sets  

Actions  
Verify  Proposition  
Buttons about logic  
Equal to  Inequality  
Not equal to  
Lessthan sign  
Greaterthan sign  
Less than or equal to  
Greater than or equal to  
Logical and  and  
Logical or  or  
Buttons about sets  
List  
Element of  Set  
Contains as member  
Union  
Intersection  
Set minus  
Natural numbers  Number  
Integer numbers  
Rational numbers  
Real numbers  
Complex numbers 
Examples
Solve
This section contains commands to find solutions to equations, inequalities and their corresponding systems.
You can perform the Calc
action over equations, inequalities and systems of them, and they'll be solved by default. Alternatively, you can use the solve
command. The results are the same, but they have different forms.
There isn't a solution for every equation in the real numbers. You can also use solve
in complex numbers.
Not all equations have algorithms to find the solutions. If the system can't find all the exact solutions, you can try numerical_solve
to find one approximate solution. The command numerical_solve
doesn't find all solutions. Instead, it finds one each time.
The Calc
action first tries internally the complete solve
, and if it fails then it uses numerical_solve
.
We can also store the equation's solutions in a set and then acces each solution by using the commands you can see in the last example below.
Solve  

Find all solutions: all values that satisfy the equation.  Equation  
Find one approximate value that satisfies the equation. An iterative method is used, and you can set the initial value.  Newton's method  
Solve inequalities and their corresponding systems.  Inequality  
Evaluate the first parameter (expression) by replacing the second (variable) by the third (value) and performing the operations. 
Examples
Greek
It's common to use Greek letters in formulas, and this section contains them all. See Greek alphabet for background information.
You can use Greek letters for the names of the variables. You could even use Chinese, Japanese or Russian letters, for example. You'd need an appropriate keyboard for that, or you could copy and paste the symbols from a web page, for example. See Unicode for background information.
* Do not use this π for 3.1416, use the one in the Symbols
section. The π here is only text, so it has no value.
Lowercase  

Alpha  
Beta  
Gamma  
Delta  
Epsilon  
Zeta  
Eta  
Theta  
Iota  
Kappa  
Lambda  
Mu  
Nu  
Xi  
Omicron  
Pi  
Rho  
Final sigma  
Sigma  
Tau  
Upsilon  
Phi  
Chi  
Psi  
Omega 
Uppercase  

Capital alpha  
Capital beta  
Capital gamma  
Capital delta  
Capital epsilon  
Capital zeta  
Capital eta  
Capital theta  
Capital iota  
Capital kappa  
Capital lambda  
Capital mu  
Capital nu  
Capital xi  
Capital omicron  
Capital pi  
Capital rho  
Capital sigma  
Capital tau  
Capital upsilon  
Capital phi  
Capital chi  
Capital psi  
Capital omega 
Examples
Units of measure
You can do calculations using units. We follow the metric system, also called the International System of Units, or SI for short. Find out all the units and metric prefixes supported. See Metric system for background information.
 You must write the units with the buttons of this section. You can't write them with the keyboard. An
m
from here is ameter
, but anm
from the keyboard is just a variable.
 Usually a space between symbols means
product
, but between quantities it meanssum
. If you want to multiply quantities, you must write the product symbol in between.
At the top of the section there is a selector of the SI prefixes for the units below.
The result of an operation between quantities has its unit selected automatically. You can force the unit of a quantity by using the Convert
command.
You can obtain a quantity by multiplying a number and a unit. Using those commands, can split a quantity into its coefficient and unit.
Commands  

Convert the quantity in the first parameter to the unit of the second parameter. If there is no second parameter, it will be converted to the SI default unit.  
Coefficient of a quantity  
Unit of a quantity 
Units
SI Prefixes  

n  nano  0.000 000 001 
µ  micro  0.000 001 
m  mili  0.001 
c  centi  0.01 
d  deci  0.1 
da  deca  10 
h  hecto  100 
k  kilo  1000 
M  mega  1 000 000 
G  giga  1 000 000 000 
Units  

Meter  
Gram  
Second  
Ampere  
Kelvin  
Mol  
Candela  
Angle degree  
Angle minute  
Angle second  
Radian  
Steradiant  
Hour  
Minute  
Second  
Liter  
Newton  
Hertz  
Pascal  
Watt  
Joule  
Coulomb  
Volt  
Ohm  
Farad  
Siemens  
Weber  
Bar  
Henry  
Tesla  
Lux  
Lumen  
Gray  
Becquerel  
Sievert  
Katal  
Percent  
Permil 
Examples
Currencies
Beside units, we can use currencies and do basic arithmetic with them but it is not possible to convert one unit into another. We should use the currency symbol provided in the tab.
Currencies  

Dollar  
Euro  
Pound  
Franc  
Krone  
Bitcoin  
Ruble  
Rupee  
Won  
Yen 
Currencies 

Graphics
The Sheet can have an area for plotters on the right.
Plotters are containers for graphs of functions. They can be heavily configured. You can see that plotters have background
, axis
and grid
. They also have center
and scale
with respect to the view. Each plotter has a name on top as well as a button to configure its properties.
You can change the point of view in real time by dragging the mouse, or by rolling the mouse wheel, over the plotter.
There are bigger versions of plotters, too. Click on the arrow at the bottomleft to see them. In the bigger version there are sliders to change the point of view. Moreover, the bigger version of the 2D plotter has a crosshair and shows, at the bottomright, the coordinates of the selected point. This is useful for obtaining information about the drawn functions.
You can put things in the plotter by using the Plot
action over a formula. If there is no plotter in the sheet, a new one will be created. Otherwise, the graphical object will be placed in an existing plotter.
Graphical objects can also be heavily configured. Each graphical object has label
, color
and width,
and maybe border
, interior
and transparency
. These properties can be configured by clicking the Plot
action icon located next to the formula. You can also move one graphical object from one plotter to another through the use of this configuration popup.
See Graph for background.
2D
You can plot in the cartesian plane:
 functions, of one variable
 equations, of two variables, that are implicit functions
 inequations, of two variables, that are regions
 lists of them
 regions, defined by command
region()
You can also plot elements of ODEs, as explained in the Calculus section.
3D
In the cartesian space you can plot:
 functions, of two variables
 linear equations, of three variables, which are planes
 lists of them
Plotter settings
You can change some plotter settings by clicking on the settings button at the right corner of the plotter topbar. The options you can modify are the followings.
General plotter format 

Horizontal axis style 

Vertical axis style 

Plot settings
You can also change the settings of a particular plot by clicking on the graph icon next to the function which is being represented. The options you can modify are the followings.
Examples
Programming
Here you'll find the usual commands for programming. These include conditional statements, loops, beginend block, local variables and return.
Range notation
We can create a range of numbers using the syntax: start..end
or start..end..step
. It is also possible to couple ranges.
Range notation 

Concatenation
We can concatenate two strings, lists, or vectors. Moreover, we can add a column or a row to a matrix easily.
Concatenation 

Conditional statements
Loop statements
Begin, local and return
Examples
Format
You can apply some formatting options but only to texts in text lines (created by that Text action).
Options 

apply to characters 
Bold 
Italics 
Color 
apply to whole line 
Font family 
Font size 
Format 
