(Roberto Bigoni)

performances, instructions, examples

SuperCalculator is implemented in Javascript, so you can use it only if your browser allows Javascript.

SuperCalculator produces graphs and tables in pop-up windows, so you can fully exploit its resources only if your browser allows pop-ups.

SuperCalculator is intended for educational, non-professional use only; we do not take responsibility for any inaccuracies or errors.

SuperCalculator, unlike an ordinary calculator,

• performs calculations on integers, sexagesimal, rational, real and complex numbers with the desired precision;
• performs calculations on vectors and matrices of real or complex numbers;
• performs calculations on Boolean expressions;
• performs some statistical calculations;
• performs some financial mathematics calculations;
• it allows you to declare functions of one or more variables (real or complex) assigning their expression to suitable identifiers;
• it allows you to obtain tables and graphs of real functions of real variable;
• it allows you to describe and represent simple plane geometric figures;

Calculations on integer and rational numbers, represented by fractions or by repeating decimals, are always exact.

Calculations on integer numbers exploit a library of L. Baird.

Rational numbers can be input as fractions or repeating decimals. In this case the repeated digits must be written between square brackets ([...]). Example: 1.2[34].

The decimal separator in real and rational numbers is the period; commas are not allowed: commas act only as separators between the items of a list.

Real numbers can be input in exponential notation. Example: 1.234e-5. The letter e must be written lowercase, since the letter 'E', written in capital letters, represents the basis of the natural logarithms.

When calculations generate real numbers, results will be approximated to the number of digit written in the decimals field (by default 6).

If this number is too high, the computing time may become intolerable. In the field time you can set the maximum waiting time (in seconds, by default 1).

Numbers in sexagesimal format, used for angular measurements, are expressed by three groups of digits separated by the symbols °, ', " for degrees, primes and seconds. Possible expressions of degrees, primes and seconds in input can be rational numbers; in output these expressions are normalized so that the degrees are expressed by natural numbers, the primes by natural numbers <60 and the seconds by rational numbers < 60.

Example: 12°15'18"+34°56'55.34571"; Sin[30°]; a=15.45°; b=12.18°78'; a+b

The imaginary unit is written as I. The letter I must be written in capital letters.

Examples:

• 2I; 1+3I; 5-12I;
• 1/2I; 1/2+3/4I; 1.2[3]I; 1.2-3.4[56]I;

Boolean values are represented with the characters T and F, mandatory uppercase. Using the appropriate operators described below you can perform Boolean calculations.

Examples:

• (T&&F)==F
• T->F;

It is possible to make some calculations on matrices and on vectors. Vectors can be written as lists and the matrices as lists of lists.

Example: Det[{{1,2,3},{0,1/2,-3},{a,b,0}}].

The constants selector allows you to enter directly into the input field the values of the most common mathematical and physical constants.

Other constants can be stored in memory for further usage.

The operators selector allows you to enter directly into the input field the names of the most common mathematical functions and operators.

To obtain documentation on available operators, use the question mark ? followed by the name of the operator. The uppercase or lowercase format of the characters is irrelevant.

Example: ?Sin, ?sin.

Input and output

In the input field you can write expressions that must be immediately evaluated. These expressions must be typewritten using the standard operation signs (see below) and, eventually, parentheses. After, click the calculate button.

You can also write two or more statements separated by a semicolon (;).

Examples: 100!; 100!!; 100!/100!!
f[x]:=x^2-1; Graph[x,f[x],-2,2]
c=Circ[{0,0},1]; Draw[c].

In the input field you can too declare a function or assign a constant value. In these cases calculate does'nt generate an explicit result but only stores the function or the constant.

However, every activation of the execute button writes, in the output field, the expression in the input field and, below, the result of the evaluation. These two lines are numbered so you can get them using the commands In and Out.

To set a new calculation, you can click on the button new input.

The results obtained in output can be saved on the local disk by selecting the text, activating the copy function (ctrl c), opening an editor (eg. Notepad), pasting (ctrl v) and saving the file generated in the appropriate location.

Since images and tables are produced on stand-alone pages, the browser in use must be allowed to open pop-ups. The pages generated in the pop-ups can be saved with the standard procedure save page as.

In the pages with graphs, using Chrome or Firefox, the graph can be saved as an image in *.png format by clicking on it and setting the name and the location of the image.

Immediate expressions, declarations of function, assignments of constant can contain the following operation signs:

The following signs of operation act only on natural operands producing natural results. Logical operators perform bitwise calculations.

The following signs of operation act on logical (boolean) operands.

Some binary arithmetic operations may be written in prefixed form, given by the name of an operator followed by a pair of square brackets containing the two operands.

Also some logical or bitwise binary operations may be written in prefixed form, given by the name of an operator followed by a pair of square brackets containing the two operands.

The priority in the evaluation of an expression is given by parentheses, possibly nested.

Expressions may contain the following constants.

(Physical constants have precision limited to their experimental values.)

In the input field you can call the following functions that require only one real or complex argument (exceptionally two in powers, radices and logarithms). If the argument is represented by x, it must be real; if the argument is represented by z it can be real or complex. You can obtain tables or graphs only for real function of real variable.

These functions can be directly integrated, graphed or tabulated.

Example:

Graph[x,Sin[x],0,2P]
Graph[x,Erf[x],-2,2,0.1]
Table[x,Sqrt[x],0,4]
Coords[x,Cosh[x],-1,1,0.1]

You can also call the following functions with real arguments.

The graphs of these functions cannot be directly produced. To get their graphs, you must declare a function giving it your identifier and writing one or more of the upper names in the second term of the declaration.

Example:

db[n]:=BinomialD[0.5,10,n]; Table[n,db[n],0,10,1]

There are also the following operators on real numbers.

The following operators act on Sexagesimal arguments.

For statistical calculations you can use the following operators that act on a list of real numbers.

Furthermore you can use the following operators that act on lists of pairs of real numbers.

The list can be represented by a previously declared constant.

Example:

data = 12,14,13,15,11,12,12,14
Mean[data]

The following operator act on a rational numbers.

The following operators act on one or more natural numbers.

Binomial can compute also pairs of real or complex numbers.

The following operators act on one o more previously declared functions. The function identifier or a list of such identifiers must be the first argument. The second and the third argument give the analyzed range.

The following operators allow you to get graphs of real functions of one real variable or some geometric plane figures.

The following operators act on one or more numerical vectors.

The following operators act on square matrices.

Using the standard operation signs one can make additions, multiplications, subtractions, divisions, powers to integer exponent of matrices.

Example.

• m1={1,2,3}{2,3,4}{3,4,1}
• m1^2
• m2={5,2,1}{1,-1,3}{-1/2,0,1}
• m1*m2
• m1[0]+m1[1]
• m1[0][2]^2

The following operators act on the content of lists.

The following operators generate outputs of financial mathematics.

The following operators rewrite the input strings and the output values for further calculations.

The following operators give calendarial informations.

Geometry

Mathematical expressions

Mathematical expressions must be inputted by typewritten strings, using the keys of a standard keyboard. They can contain numbers given immediately or represented by identifiers of constants, operations signs, built-in functions or user declared functions, parentheses eventually nested and well balanced.

Examples.

• PI*8^2
• (2+3)/4
• (Sin[PI/3])^2
• f1[2]+f2[Ln[2]]
• ((10!/9!)+2)/5

Grouping marks

The grouping marks are:

• parentheses: in the mathematical expressions, eventually nested, rule the priority of calculation.
• square brackets: are the mark for functions and operators and may contain their arguments;
  square brackets enclose the period of a rational number: example 1.2[3];
  square brackets enclose the index of an item of a list: example EigenVectors[m][0].
• braces : enclose the items of a list; the items must be separated by commas.

Storage of a constant value

To store a constant value one must write an arbitrary identifier followed by = and by the value.

Example.

• a = 12
• b = 1/2
• halfln2 = Ln[2]/2
• data = {2,2,3,3,4,4,4,4,5,5,5,4,4,3,2,1}
• matrix1 = {{2,2,3,3},{4,4,4,4},{5,5,5,4},{4,3,2,1}} (but matrix1 = {2,2,3,3}{4,4,4,4}{5,5,5,4}{4,3,2,1} works fine)

Identifiers are case sensitive.

Identifiers can be reassigned.

Declaration of a function

To declare a real function of one or more variables, one must write an arbitrary identifier followed by brackets enclosing one or more variables. The brackets must be followed by := (Pascal like) and by the expression of the function in terms of these variables.

Examples.

• f[x] := x^2-1
• f1[x] := Sin[2x]+2Cos[x]
• f2[x,y] := Sqrt[x^2+y^2]

Parametric functions can be declared as functions of a single variable expressed by a list of two or more expressions. The list must contain only names of functions

Example.

• f[x] := {Cos[x],Sin[x]}
• • g1[x] := x^2
  • g2[x] := x^3
  • g[x] := {g1[x],g2[x]}

The explicit functions of a real variable can be the subject of various procedures that require at least four arguments: the variable name, the function header if explicitly declared, the beginning and the end of the analysis interval.

A function that is not explicitly declared can be implicitly declared within the procedure.

By default the interval is divided into the number of points entered in the field points. A fifth optional argument can set the interval scan step.

• Search for zeros. Use the operator Zeros with four arguments.

  Examples: f[x]:=x^2-1; Zeros[x,f[x],-2,2]
Zeros[x,x^2-1,-2,2]; Zeros[x,Sin[2x],0,2P]

• Search for maxima. Use the operator Maxima with four arguments.

  Examples: f[x]:=x^2-1; Maxima[x,f[x],-2,2]
Maxima[x,x^2-1,-2,2]; Maxima[x,Sin[2x],0,2P]

• Search for minima. Use the operator Minima with four arguments.

  Examples: f[x]:=x^2-1; Minima[x,f[x],-2,2]
Minima[x,x^2-1,-2,2]; Minima[x,Sin[2x],0,2P]

• Production of a graph. Use the operator Graph with four or five arguments.

  Examples: f[x]:=x^2-1; Graph[x,f[x],-2,2]
Graph[x,x^2-1,-2,2]; Graph[x,Sin[2x],0,2P,0.1]

• Execution of all previous procedures. Use the operator Analysis with four or five arguments.

  Examples: f[x]:=x^2-1; Analysis[x,f[x],-2,2]
Analysis[x,x^2-1,-2,2]; Analysis[x,Sin[2x],0,2P,0.1]

• Simultaneous production of the graphs of two or more functions. Use the operator Graph with four or five arguments.

  Examples: Graph[x,{Sin[x],Cos[x]},0,2P]

• Production of tables of one or more functions.

  Example.

  Table[x,Sin[x],0,2P,0.1]

  You can simultaneously tabulate multiple functions by entering their names in a list enclosed by braces.

  Example.

  Table[x,{Sin[x],Cos[x]},0,2P]

Graphs and tables are produced in pop-up windows, so they are only viewable if your browser allows pop-ups.

Graphs of functions

Example.

Graph[x,Sin[x],-2P,2P]

The rectangle containing the graph can be saved as an image in * .png format on a storage device of your system by clicking on it with the right mouse button. In the following examples we show only these images.

The buttons next to the rectangle containing the graph allow you to modify its appearance to improve its graphic rendering or to highlight some features.

For example, by clicking repeatedly on the button , the graph is enlarged.

By repeatedly clicking the button , you increase the unit on the y axis.

You can plot several functions together, inserting their names into a list delimited by braces.

Example.

Graph[x,{Sin[x],Cos[x]},-2P,2P]

You can get the graph of a previously declared function.

Example.

qsin[x]:=Sin[x]^2; Graph[x,qsin[x],-2P,2P]

or, more simply

Graph[x,Sqr[Sin[x]],-2P,2P]

Also in this case you can plot several functions together.

Graph[x,{x^2-1,1-x^2},-2,2]

You can define a parametric function in the following way

f[x]:={x^2,x}

and obtain its graph

Graph[x,f[x],-2,2]

Geometric drawings

Simple geometric shapes are coded as lists of at least two pairs of coordinates. Once this list has been defined, the drawing is made with the operator Draw.

Example: a segment

Draw[{-1,-1},{1,1}]

Example: a polygonal line

Draw[{-2,0},{-1,1},{0,0},{1,1},{2,0}]

You can represent regular polygons, circumferences, ellipses, arcs of curves defined by functions. To achieve these results, you must operate as follows:

• generate the description of the drawing, assigning the output of one of the following operators to a constant identifier:

  • PoliV
  • Circ
  • Ellipse
  • Translate
  • Rotate
• get the drawing with the operator Draw with argument given by one or more identifiers.

Examples.

penta=PolyV[{0,0},1,5]; Draw[penta]

c=Circ[{0,0},1]; qu=PolyV[{0,0},1,4]; Draw[c,qu]

To represent arcs of curves described by functions, proceed as follows:

• Produce a list of points with the operator Coords applied to the function identifier, to the defined interval and, if necessary, to the increment step and assign the output of this operator to a constant;

  para=Coords[x,Sqr[x],-1,1,0.1]

• The identifier of the arc of parabola can be one of the arguments of Draw.

Example

para=Coords[x,Sqr[x],-1,1,0.1]; c=Circ[{0,0},1]; Draw[para,c]

The description of a drawing can be translated or rotated; eg:

tri=PolyV[{0,0},1,3]; trasl=Translate[tri,{2,1}]; rot=Rotate[tri,45°]; Draw[tri,trasl,rot]

Histograms

You can represent two-dimensional histograms using the operator Histogram with argument given by list of pairs {number, number} or {string, number}. The list must be preceded by a string with the title.

Example.

f[k] := Binomial[10,k]; Histogram["binomial coefficients",Coords[k,f[k],0,10,1]]

t = {"London",15},{"Berlin",10},{"Paris",18},{"Madrid",20},{"Rome",18}; Histogram["temperatures",t]

last revision: May 2018