performances, instructions, examples
(Java applet - Roberto Bigoni - October 2011)

SuperCalculator, unlike an ordinary calculator, can perform calculations on integer, rational, real and complex numbers.

Calculations on integer and rational numbers are always exact.

Integers can be input as binary or hexadecimal: in this case the sequence of digits must end with 'b' or 'h'. Example: 1011000111011111101b; 12ABCEF01h.

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 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.

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

Obviously, if this number is too high, the computing time may become intolerable.

The imaginary unit is written as I or i: Examples 1+I; 2/3-3/2i; 1.2+3.4i; (2+3I)^2; Sqrt[1-I].

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}}].

These objects can also be input using the button matrix in the U.I.

Square matrices (if not singular) and vectors can be used together to obtain solutions of a systems of linear equations.

It is possible to define functions of one or more variables, assigning their expression to arbitrary identifiers.

One can get tables and graphs of real functions.

One can describe and draw simple plane geometric figures.

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

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 execute button or push the keys Alt-Enter.

You can also write two or more statements separated by a semicolon (;).
Examples: 100!; 100!!; 100!/100!!
f[x]:=x^2-1; Graph[f,-2,2]
c=Circle[0,0,1]; Draw[c].

In the input field you can too declare a function or assign a constant value. In these cases execute 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 or the pair of keys Alt+End.

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

The following operation signs work only on integer operands.

The following operation signs work on logical operands.

The priority in the evaluation of an expression is given by parentheses, eventually 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[Sin,0,2PI]
Graph[Erf,-2,2,0.1]
Table[Sqrt,0,4]
List[Cosh,-1,1,0.1]
Integrate[Gauss,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:

lp2[n]:=LegendreP[n,2]; Table[lp2,0,10,1]

There are also the following operators on real numbers.

The following operators act on list 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 number giving its decimal approximation to the given accuracy.

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.

Some input fields allow the setting of the numbers of points to calculate in tables and graphs, the maximum and the minimum values, the colours in the graphs.

The following operators act on one or more numerical vectors.

The vector product can be performed only on two-dimensional o three-dimensional vectors.

Vectors can be multiplied or divided by scalars. If they have the same dimension, they can be added or subtracted.

It is possible to extract a component of a vector.

Examples.

• v1={1,2,3}
• v2={1,-1,3}
• v1*v2
• v1**v2
• 3*v1-4*v2[m1]
• v1[0]+v1[1]

The following operators act on square matrices.

The following operators act on hermitian 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}
• Inverse[m1]
• 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.

Trigonometry

The operator Triangle allows you to solve a triangle given three of its elements; at least one of them must be a side; angles are input as radians; unknown elements are input as 0.

The last two functions require four arguments expressed by sexagesimal numbers in the following order:

• latitude and longitude of the first point;
• latitude and longitude of the second point.

SOUTH latitudes and WEST longitudes require a minus sign.

Distances are calculated as the crow flies along a path on the surface of a theoretical geoid using the Vincenty's method, as applied in a page of Chris Veness.

The following operators allow us to produce and represent some geometric plane figures.

Mathematical expressions

The 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 a 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,Sin}

• g[x] := Sqrt[x^1+1]

  f1[x] := {g,Tan}

Declared real functions of one real variable may be analyzed by the following operators.

• Tabulation of some values (default 51) of a function of a single variable by the operator Table applied to three arguments: the function identifier and the bounds of the range analyzed.
• Approximation of the zeroes by the operator Zeroes applied to three arguments: the function identifier and the bounds of the range analyzed. This operator may be successful only if the function is continue and its signs at the bounds of the range are opposite.
• Approximation of the most significant point of the graph of a function of a single variable (zeroes, extremes, inflection points) by the operator Analysis applied to three arguments: the function identifier and the bounds of the range analyzed.
• Production of graphs of one or more functions by the operator Graph applied to three arguments: the function identifier (or the list of identifiers) and the bounds of the range analyzed.

Functions graphs

It is possible to get the graph of a function only if the function has an identifier assigned by the user; that is, you can directly graph only the built-in functions.

For example, if you want the graph of the circular sine, you can write Graph[Sin,-2PI,2PI], but if you want the graph of another circular sine you must before declare a function like this: sine1[x] := 2Sin[x/2+1] and, after, use the operator Graph in this way: Graph[sine1,-2PI,2PI].

It is possible to graph more functions together. For example, you can graph together sine and cosine in this way Graph[{Sin,Cos},-2PI,2PI].

It is possible to get the graph of a parametric function of two real variables. For example, after the declaration of the function f[x]:={Cos,Sin}, the statement Graph[f,0,2PI] makes a circle of radius 1.

Example.

Given the functions f1[x] := Cos[x+PI/4] and f2[x] := Cos[x-PI/4]:

• we can graph them separately;

  Graph[f1,-2PI,2PI]

  

  Graph[f2,-2PI,2PI]

  

• we can graph them together;

  Graph[{f1,f2},-2PI,2PI]

  

• we can define a parametric function

  f[x] := {f1,f2}

  and get its graph

  Graph[f,-2PI,2PI]

  

geometric figures

We can draw some plane geometric figures, like segments, polygons, circles, ellipses, arcs of curves, in a Cartesian plane. In order to do this, we must take the following steps:

• we assign an identifier to the result of one of the following operators:

  • Segment
  • Polygonal
  • Polygon
  • Circle
  • Ellipse
  • Translate
  • Rotate
• we generate the drawings using the operator Draw.

Examples.

penta=Polygon[0,0,1,5]; Draw[penta]

c=Circle[0,0,1]; square=Polygon[0,0,1,4]; Draw[c,square]

To obtain arcs of curves given by real valued functions, we must take the following steps:

• we create a list of points using the operator List; example:

  lpara=List[Sqr,-1,1,0.1]

• then we create a drawable object using the operator Polygonal; example:

  para=Polygonal[lpara]

Now this object can be an argument for the operator Draw.

Example.

lpara=List[Sqr,-1,1,0.1]; para=Polygonal[lpara]; c=Circle[0,0,1]; Draw[para,c]

The description of a drawing can be translated or rotated

tri=Polygon[0,0,1,3]; transl=Translate[tri,2,1]; rot=Rotate[tri,45°]; Draw[tri,transl,rot]

last revision: October 2011