Menu reference list Symbols 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 Imaginary unit A number defined to satisfy 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 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 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 Greater-than sign Inequality 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 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 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 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. Examples Complex numbers Statistics 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 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. , 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 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 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 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. Hyperbolic functions Hyperbolic functions Hyperbolic function Examples Trigonometric functions Trigonometric reciprocal functions Trigonometric inverse functions Logarithmic and exponential functions Hyperbolic functions 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! ∞ 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 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 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 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() 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 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 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 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 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 Buttons about logic 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 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. 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 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 Greek 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 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 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 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 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 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 Plotter settings You can change some plotter settings by clicking on the settings button at the right corner of the plotter top-bar. 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. Plot settings Examples Programming 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 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 Table of Contents Symbols Arithmetic Polynomials Statistics Functions Calculus Linear algebra Combinatorics Logic and sets Solve Greek Units of measure Graphics Programming Format