Ipercalc

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Iperoperazioni

Evoluzione delle quattro operazioni dell’aritmetica (uno, due tre, … infinito)

Ricerca a cura del socio: Gianfranco ROMERIO

in collaborazione con il ricercatore russo: Constantin RUBTSOV

Nell’ambito della loro cooperazione come coautori del rapporto

“Ackermann’s Function and New Arithmetic Operations”

Cari Amici,

Vi presento una simulazione del primo prototipo di micro-iper-calcolatore in grado di calcolare qualche iper-operazione (zerazione e tetrazione), con le relative istruzioni. Il "coso" ha un doppio funzionamento:
- a tasti;
- con input tramite operatori a "stringa di caratteri".

E' il primo prototipo, realizzato interamente dall’Amico Rubtsov (KAR), approfittando dell’autunno russo che avanza. Al momento non vi sono applicazioni pratiche, ma il vederlo funzionare trasmette ai coautori dell’articolo un'enorme soddisfazione.

Sarete delusi dalla "zerazione", che fornisce semplicemente come risultato il massimo dei due numeri inseriti, aumentato di uno, se sono diversi, e uno dei due numeri inseriti più due, se sono uguali. Cioè:

a ° b = max(a, b) + 1 per a diverso da b
a ° b = a + 2 = b + 2 per a = b.

Per ogni numero reale x, con x > a , y = a ° x fornisce il successore di x, cioè x + 1. In questo senso è l'operazione di livello immediatamente inferiore all'addizione. L’interesse teorico è che si tratta di un'altra "operazione elementare", assolutamente nuova, la cui operazione inversa genera dei numeri di tipo molto curioso, simili ai ... logaritmi dei ... numeri negativi, rappresentabili da numeri complessi a valore … multiplo.

Per chi cercasse qualche lettura amena per l’inverno, consiglio la seguente monografia di Rubtsov, scritta in russo nel lontano 1996, con un’ampia e giustificata introduzione, … cirillica, dei numeri “Delta” (Nuovi Oggetti Matematici):

La tetrazione (operazione torre) è una cosa già parzialmente nota da qualche tempo, ma molto più tosta e ... dirompente. Per ora è definita solo per super-esponenti interi. Cioè:

y = a ^ (a ^ a) = a # 3, (leggi “a-torre-tre”) simile a:
y = a . ( a . a) = a . a . a = a ^ 3 (“a elevato a tre”, le parentesi non sono necessarie).

Il “marchingegno” permette, per esempio, di calcolare direttamente la super-radice-quadrata (!!) di un numero reale qualsiasi, per esempi di 100 o di π (pi greco, abbreviato con pi), soluzioni rispettivamente delle semplici equazioni x^x= 100 e x^x= π. Si tratta dell’operazione inversa della … “torre quadrata”. Dato che “super-radice-quadrata”, in inglese si dice super-square-root, lo abbiamo abbreviato come ssqrt. Allora, inserendo gli operatori stringa nelle finestra inferiore, si ha:

ssqrt(100) = 3,59728502354042… e ssqrt(pi) = 1,85410596792103… .

Provare per credere !

Non sappiamo ancora che significato dare a y = a # 1,5, ma ci stiamo … forse … arrivando.

Grazie e a presto.

Gianfranco

Instruction Notes

concerning the « KAR-Calc » Calculator

(C. A. Rubtsov – September 2004)

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The “ KAR-Calc ” 15-bit calculator will allow to calculate the four classical basic arithmetic operations ("+", "‑", "*", "/") and, also, two new operations: “Zeration” (“ ° ”, by using the digital keyboard, and “@” in the input-output window) and its inverse operation “Deltation” or “TI-ation” (“Δ”, by using the digital keyboard and “ ~ ” in the input-output window). Actually, the calculator consists of two parts, with two different ways of operations:

1. Implementation of the elementary calculations with the help of the “mouse”;

2. Implementation of engineering calculations with the help of the alpha-numerical keyboard and by means of appropriate input-output character-string operators (in the bottom window of the calculator).

When using of the input-output character-strings, the calculator can compute mathematical expressions, taking into account the operations’ hierarchy. The complete list of character-string operators is shown hereafter. All the character-strings operators must be confirmed by Enter or Carriage Return.

Strings Action

pi returns number “π”: 3,14159258... ;

e returns number “e”: 2,718281828459045… ;

cos (x) calculates the cosine of a given angle (arc) x;

sin (x) calculates the sine of a given angle (arc) x;

tg (x) calculates the tangent of a given angle (arc) x;

ctg (x) calculates the cotangent of a given angle (arc) x;

arccos (x) calculates the arc-cosine of a given number x;

arcsin (x) calculates the arc-sine of a given number x;

arctg (x) calculates the arc-tangent of a given number x;

arcctg (x) calculates the arc-cotangent of a given number x;

atan (x, y) calculates the arc-tangent for a point with given coordinates x and y;

ch (x) calculates the hyperbolic cosine of a given value (hyperbolic arc, sector) x;

sh (x) calculates the hyperbolic sine of a given value (hyperbolic arc, sector) x;

th (x) calculates the hyperbolic tangent of a given value (hyperbolic arc, sector) x;

cth (x) calculates the hyperbolic cotangent of a given value (hyperbolic arc, sector) x;

arcch (x) calculates the hyperbolic arc-cosine of a given number x;

arcsh (x) calculates the hyperbolic arc-sine of a given number x;

arcth (x) calculates the hyperbolic arc-tangent of a given number x;

arccth (x) calculates the hyperbolic arc-cotangent of a given number x;

log (a, x) calculates the logarithm, to the base of a, of a given number x;

lg (x) calculates the logarithm, to the base of 10, of a given number x;

ln (x) calculates the logarithm, to the base of e, of a given number x;

exp (x) calculates the exponential function for a given value of variable x;

sqr (x) returns the square of a given number x;

sqrt (x) returns the square root of a given number x;

ssqrt (x) returns the super-square-root of a given number x;

abs (x) returns the absolute value of a given number x;

sign (x) returns the value (±1) of the sign of a given number x;

round (x, y) returns the rounded off value of number x, up to precision y;

frac (x, y) returns the fractional part of a given number x;

trunc (x, y) returns the whole part of a given number x;

min (x1, x2..., xn) ' evaluates the minimum value of the set of listed numbers xi;

max (x1, x2..., xn) ' evaluates the maximum value of the set of listed numbers xi;

c returns the speed of light in vacuum (299’792’458 m/s).

Trigonometric functions are computed both in radians "R" and in grades "G", chosen according to the position of an appropriate switch.

At the input, it is possible to indicate an inverse value of a number, as suggested by the author, by:

______________________

Example 1. For calculating the value of expression:

please input:

2 + :5 * 6 or: 2 + (:5) * 6

and, then, press "Enter", in the keyboard. The result will be shown both in the upper indicator window and in the lower input-string window.

Remarks. In inputting operator strings, it is forbidden to use a succession of symbols belonging to different operation hierarchies. For example, “*-“. The brackets must always be used.

Example 2. For calculating the value of expression :

(where c is the speed of light in vacuum),

please input:

max (pi, 7@9.8*exp (:ssqrt (9~5-9)), min (c, 2*pi~ssqrt (e)))

and, then, push "Enter". The result is shown both in the indicator and in the input-output string windows.

Remarks. When evaluating major expressions and before the beginning of their execution, it is recommended to save these expressions into the clipboard. For selecting the expressions, it is sufficient to bring the pointer of the mouse over the input string and double click with the left button. For saving the selected string in the clipboard, please press Ctrl + C on the keyboard. For recalling the expression from the clipboard, please press Ctrl + V.

___________________

Description of the Calculator Keys

(C. A. Rubtsov – September 2004)

________________

The calculator keys react only to any single click operated by the left button of the mouse.

Keys Action

0-9 digit-by-digit input of numbers;

00 input of 2 "0" symbols (times 100);

000 input of 3 "0" symbols (times 1000);

, input of the decimal point;

/-/ change of sign of a number;

+ addition of the contents of the indicator register X to the contents of register Y;

- subtraction, from the contents of register Y, of the contents of the indicator register X;

multiplication of the contents of the indicator register X with the contents of register Y;

/ division of the contents of register Y by the contents of the indicator register X;

implementation of the “Zeration” operation, between the contents of the indicator register X and

the contents of register Y;

implementation of the inverse of the "Zeration" operation (Deltation, TI-ation) from the contents

of register Y and the contents of the indicator register X;

Sqrt calculation of the square root of the contents of the indicator register X;

x^2 raising to the 2nd power of the contents of the indicator register X;

YX interchanging of contents between registers X and Y;

<= deleting the last input symbol;

C clear, reset of the contents of registers X, Y or after an error message;

C memory clear, reset of the memory register M;

MS memory save, saving of contents of the indicator register X in the memory register M;

MR memory recall, copy of the contents of memory register M in the indicator register X;

M- subtraction from the storage memory M of the contents of the indicator register X;

M + addition of the contents of the indicator register X to the storage memory M. The results is stored

in the storage memory M;

x! calculation of the factorial of a number (from 0 up to 17) and storage of the result in the indicator register X .

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Ackermann’s Function and New Arithmetical Operations

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