Here you'll find the standard mathematical symbols.

• These buttons are for the nice-looking 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
Infinity
Use it to calculate limits.
• You must use the buttons for e and i. Keyboard e and i are just variables; they aren't the standard constants.

### The four basic operations

Basic operations
Subtraction Subtraction
Multiplication Multiplication
Division Division
• You can use multiple symbols for multiplication: * asterisk, · middle dot, the one in the menu, and also a space. 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
Less-than 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: OK
• 3,1416: Error
• 3'1416: Error
• 1 234 567: Error
• .12: Error
• 0.12: OK
• 12.0: OK
• 12.: 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

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 tie-breaking 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 pseudo-random 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 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 than solve(p=0) or apply the Calc action to p=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.

Complex numbers

### Sets of data

Data sets must be entered as a comma-separated 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
$\left\{\begin{array}{lc}{X}_{k}& n=2k-1\\ \frac{{X}_{k}+{X}_{k+1}}{2}& n=2k\end{array}\right\$
$\begin{array}{l}\text{quartile}\left(0,X\right)={X}_{1}\\ \text{quartile}\left(1,X\right)=\left\{\begin{array}{lc}\text{median}\left(\left\{{X}_{1},\dots ,{X}_{k}\right\}\right)& n=2k-1\\ \text{median}\left(\left\{{X}_{1},\dots ,{X}_{k}\right\}\right)& n=2k\end{array}\right\\\ \text{quartile}\left(2,X\right)=\text{median}\left(X\right)\\ \text{quartile}\left(3,X\right)=\left\{\begin{array}{lc}\text{median}\left(\left\{{X}_{k},\dots ,{X}_{n}\right\}\right)& n=2k-1\\ \text{median}\left(\left\{{X}_{k+1},\dots ,{X}_{n}\right\}\right)& n=2k\end{array}\right\\\ \text{quartile}\left(4,X\right)={X}_{n}\end{array}$

### 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. Chi-squared 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 t-variable
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.
Density function of a random variable at a given point. In some cases, the analytical expression is given.
Quantile function of a random variable for a given probability.

### Statistical data representation

 Histograms are often used to visually represent continuous data. As opposed to histograms, bar charts are useful for displaying discrete data, as well as categorical data. Pie charts are another common tool for categorical data, when we want to show the proportion that each category occupies out of the whole. Boxplots are very useful for a concise representation of a data set. In one boxplot we see the median, the interquantile range (IQR), as well as outliers.

### Examples

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 so-called 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 or--as was done in the old days--a 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 non-trivial 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 non-trivial simplifications over logarithmic and exponential expressions. Also, the Verify action can test for identities.

### Examples

Trigonometric functions
Trigonometric reciprocal functions
Trigonometric inverse functions
Logarithmic and exponential functions
Hyperbolic functions

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!
Piece-wise function Piecewise function
Function application
Expression with a restricted domain Restriction
Summation with under-and-over scripts Sum
Summation with under scripts
The product with under-and-over 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

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 same-dimension 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
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() and eigenvectors() 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 RT1, RT2, RT3,...
Matrix layout modifiers
Insert column at left
Insert column at right
Remove column
Insert row above
Insert row below
Remove row

### Examples

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 k-permutations of n Variations
Permutations Permutation
Combinations Combination
Variations with repetition or n-tuples of m-sets 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

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

Intervals
The interval , that is all the numbers between and :
The interval , that is all the numbers between and including :
The interval , that is all the numbers between and including :
The interval , that is all the numbers between and including both and :
Logic and sets
Actions
Verify Proposition
Equal to Inequality
Not equal to
Less-than sign
Greater-than sign
Less than or equal to
Greater than or equal to
Logical and and
Logical or or
List
Element of Set
Contains as member
Union
Intersection
Set minus
Natural numbers Number
Integer numbers
Rational numbers
Real numbers
Complex numbers

### Examples

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.

Writing a system of equations is simple: separate each equation by commas; write them inside braces and separate them by commas; or put all of them inside braces, each one in a new line (Shift+Enter).
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

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

Greek

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 a meter, but an m from the keyboard is just a variable.
• Usually a space between symbols means product, but between quantities it means sum. 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
Hour
Minute
Second
Liter
Newton
Hertz
Pascal
Watt
Joule
Coulomb
Volt
Ohm
Siemens
Weber
Bar
Henry
Tesla
Lux
Lumen
Gray
Becquerel
Sievert
Katal

Units of measure

### 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

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 bottom-left to see them. In the bigger version there are sliders to change the point of view. Moreover, the bigger version of the 2-D plotter has a crosshair and shows, at the bottom-right, 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.

Graphics
Actions
Plot
Plot 3-D
Commands
Place the boundaries as parameters, and you'll get a region object that's ready to plot.

### 2-D

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.

### 3-D

In the cartesian space you can plot:

• functions, of two variables
• linear equations, of three variables, which are planes
• lists of them

### Examples

Plotter settings
Plot settings

Here you'll find the usual commands for programming. These include conditional statements, loops, begin-end 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

Conditional statements
if statement. If the condition is hold, then performs the action inside the block.
else statement. It should be preceded by an if. If the condition specified in the if statement is not hold, then runs the commands inside the else's block.
It should be preceded by an if. If the condition specified in the previous if statement is not hold and the current condition is satisfied, then performs the action inside the block.

### Loop statements

Loop statements
for statement. Write easily a loop that needs to be run a specific number of times.
while statement. Repeats the code block until the condition is not satisfied. Be sure that the condition does not hold in some cases, otherwise the code will run infinitely.
repeat statement. While the condition does not hold, repeats the code block. Again, be sure that the condition is satisfied sometime.

### Begin, local and return

Begin, local and return
This block is extremely useful when one defines its own functions. It allows performing different actions inside one block and define local variables.
return statement. Returns a value in a user function.
Allows defining local variables: variables defined just in one code block.

### Examples

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