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 that and . | e | |
Imaginary unit A number defined to satisfy | i | |
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 | ||
---|---|---|
Greater-than sign | Inequality | |
Less-than sign | ||
Greater than or equal to | ||
Less than or equal to | ||
Not equal to |
Other symbols
Other symbols | ||
---|---|---|
Decimal mark | 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
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 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 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 | |
---|---|
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
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! | ∞ | ||
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()
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 basic combinatorial functions. You can calculate its 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 | ||
Permutations | Permutation | |
Combinations | Combination | |
Variations with repetitions | ||
Permutations with repetitions | Permutation | |
Combinations with repetitions | 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.
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
.
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. 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 |
Examples
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.
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
Format
You can apply some formatting options but only to texts in text lines (created by that Action).
Options |
---|
apply to characters |
Bold |
Italics |
Color |
apply to whole line |
Font family |
Font size |
Format |
---|