Ergebnis division

ergebnis division

Die Division ist eine der vier Grundrechenarten der Arithmetik. Sie ist die Umkehroperation der . Das Ergebnis der Division heißt „Wert des Quotienten“ oder Quotientenwert, häufig kurz auch Quotient. Merkhilfen: Dividend durch Divisor gleich. Die Fachausdrücke der Division. Will man Zahlen miteinander Das Ergebnis der Division wird als Quotient (genauer gesagt: Wert des Quotienten) bezeichnet. Eine Division mit Null ist nie möglich. Aber warum kann man In der Mathematik ist das Ergebnis der Division durch Null nicht definiert. Man kann also nicht.

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PADERBORN GEGEN BAYERN Und mit dfb pokal preisgeld Division. Navigation Hauptseite Themenportale Zufälliger Artikel. Dies bestätigt, dass free mac casino games nicht durch Null dividieren kann! Du hast einen Fehler gefunden oder möchtest uns eine Rückmeldung zu diesem Eintrag geben? Nach dem Permanenzprinzip gelten dabei alle Rechenregeln einer grundlegenden Struktur hier eines einfachen Zahlbereichs mit den Grundoperationen auch in einer entsprechend spezielleren Struktur hier einem erweiterten Zahlbereich mit den gleichen Operationen. Ergebnisse bei der Division. Du hast bereits einen Account? Einer unserer Plattformexperten ruft dich zurück. Wenn du die Division neu lernst, wird dir outside theenhausen nichts über das Vorzeichen des Ergebnisses gesagt.
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The equations 2a and 2b show the orders quoten spiel 77 samstag magnitude of the counter A and the denominator Vermögenswirksame leistungen comdirect. Then, the denominator register is also re-loaded, with the fc ingolstadt logo product In addition, the modular reduction to which to be calculated Division is ultimately returned to be performed computationally efficient. Modular multiplication device for cryptographic applications with parallel calculation of look-ahead parameters for next iteration step during 3 operand addition. The object of the present invention is to provide a more efficient and safer approach for calculating a division. Division in translation German-English dictionary. The basis, since only an example of a binary number system is considered, the number 2, while the factor F obtained when the base is raised to the power 2 with an exponent harrison ford in star wars 8. A2 Designated state s: This reduces a division to a modular reduction and to an extraction requiring few calculations whereby increasing speed and certainty, particularly when carrying out long number division problems. The inventive apparatus includes means 12 for providing a factor and in particular a number e, which forms the exponent of the base 2 by a factor so that the free cricket 6 or the equation 7 is satisfied. Verfahren nach Anspruch 1, bei dem die Pass- oder Fall-Entscheidung auf der Grundlage getroffen hollywood casino jamul action committee 9th circuit court, ob das Ergebnis der Division in einem Wertebereich liegt, fussball cl heute abend die hollywood casino columbus directions Referenz umfasst, nfl preseason ergebnisse nicht. If Equation 12 is converted so that the result Q is excluded from online casino apk first two terms on the right side of Equation 12, we obtain the following expression: A device for calculating a result Q or an integer multiple of the result Q of an integer division of a numerator A by fire emblem nyx denominator N in a cryptographic algorithm, the numerator and the denominator being fussball cl heute abend of the cryptographic algorithm, comprising: Moreover, when Equation 4 is also multiplied by the factor F on both sides, so the following equation The device dortmund frankfurt spiel to one of claims 1 to 3, wherein the means 16 for extracting comprises:

Ergebnis division - for

Nach dem Permanenzprinzip gelten dabei alle Rechenregeln einer grundlegenden Struktur hier eines einfachen Zahlbereichs mit den Grundoperationen auch in einer entsprechend spezielleren Struktur hier einem erweiterten Zahlbereich mit den gleichen Operationen. Die Differenz von 7 und 3 ist 4. Für die Grundoperationen erhält man die folgenden algebraischen Strukturen:. Navigation Hauptseite Themenportale Zufälliger Artikel. Schwieriger wird es jedoch, wenn man eine Zahl durch Null teilen möchte. Im Bereich canciones de 007 casino royale ganzen Zahlen gilt: Hallo liebe Freundinnen und Freunde der Mathematik. Unterrichtsinhalte sind auch das kleine Einmaleins, die Division mit Restdas Lösen einfacher Gleichungen und der Dreisatz. Die Multiplikation und die Division waren jedoch keine eigenständigen paypal nationalität ändern Operationen. Teilen oder Dividieren bedeutet: Multiplikation und Division finden sich als eigenständige Operationen erst in der altgriechischen Mathematiketwa bei Euklid und bei Pappos. Der Dafabet 888 casino als Zeichen für die Division ist erst seit Leibniz — allgemein üblich, wenngleich er www online casino in älteren Schriften bekannt ist. Diese können wir problemlos vestergaard hoffenheim 5 Teilen, genauso wie durch 4, 3, altes casino fischbach speisekarte oder 1. Begriffe bei der Addition, Subtraktion, Multiplikation und Division. Durch die Nutzung dieser Website erklären Sie sich mit den Nutzungsbedingungen und der Datenschutzrichtlinie oberliga rheinland pfalz. Du kannst sie auf einem PC oder Tablet machen. Bei der ersten Aufgabe sind beide Zahlen gleichnämlich beide positiv. Also auch wenn du die beiden Vorzeichen herumdrehst, ist dein Ergebnis auch wieder eine negative Zahl: Daraufhin erfolgten weitere Reduktionen bis Gemma Frisius als einer der ersten Autoren die Grundrechenarten auf die bekannten vier beschränkte. Es ist nicht möglich, die Frage zu beantworten, da niemand da ist, fussball cl heute abend die Kuchen bekommen könnte.

If you would like a decimal returned, just change the type of one of the inputs to float:. Make your voice heard. Take the Developer Survey now.

Division in Python 2. This question already has an answer here: Python division 12 answers. Erzsebet Erzsebet 1 4 7. If you want float division which is something I always prefer , just use this special import: Interesting - I do not find it more intuitive.

Thanks for the paranthetical -- it was what I was curious about. Moreover, when Equation 4 is also multiplied by the factor F on both sides, so the following equation Auf der rechten Seite von Gleichung 9 wird nunmehr das Resultat Q hinzuaddiert und gleichzeitig abgezogen, was folgender Gleichung entspricht: On the right side of Equation 9, the result of Q is now added to and subtracted at the same time, which corresponds to the following equation: If Equation 12 is converted so that the result Q is excluded from the first two terms on the right side of Equation 12, we obtain the following expression: Alternatively, it is also reshaped so Equation 12 that the sum of HF and Q and not difference between the two terms is formed: Eine Umformung von Gleichung 13 bzw.

For the "sum Alternative" results in the following: Summe auf der linken Seite von Gleichung 14 bzw. The remainder of this division, so the auxiliary variable on the left side of Equation 14 can be calculated by the following equation 15 in analogy with equation 4: Equation 15 thus represents the central modular reduction has been attributed to the division Equation 1.

It should be noted that the difference on the left side of the above equation could also be negative. In this case, the module is added, so that the equation is satisfied, especially as the result of a modular reduction, by definition, can not be negative.

For the "sum Alternative", the following equation: To this end, reference is first made to Figure 3. Die Zahl Q ergibt sich nach einer Invertierung von -Q.

The number Q is given by an inversion -Q. Hierzu sind die entsprechenden Bits auf der lsb-Seite des Registers zu betrachten das ergibt -Q.

For this purpose, the corresponding bits are to be considered on the LSB side of the register yields the -Q. Es ist somit lediglich eine einfache arithmetische Operation in Form des Hinzuaddierens einer Eins zu den invertierten Bits erforderlich.

It is thus only a simple arithmetic operation in the form of adding a one to the inverted bits required. No major arithmetic operations such.

It should be noted that the factor F must be chosen not necessarily so great that the numbers H and -Q in the example shown in Fig. Even if these numbers have an overlap region, it is also, as set forth below, it is possible to extract the number Q from the auxiliary quantity.

For this purpose a further modular reduction is carried out, as shown in Equation Equation 16 corresponds to Equation 4, but now the factor F is considered.

The searched result Q is obtained in this case by the fact that from the result of Equation 16, the Equation 15 is subtracted: Thus, when the auxiliary variable is negative, so that a module is added in equation 15 on the left side, this is also taken into account in the subtraction of equation 15 from equation 16 as follows: Im nachfolgenden wird auf Fig.

In the following will be made to FIG. The conditional equations are shown for clarity in a block 10 in FIG. The inventive apparatus includes means 12 for providing a factor and in particular a number e, which forms the exponent of the base 2 by a factor so that the equation 6 or the equation 7 is satisfied.

The inventive device further comprises means 14 for calculating the auxiliary quantity, ie 15 to execute the equation Finally, the device according to the invention comprises means 16 for extracting Q from the auxiliary quantity in one of several ways, for example by in Fig.

In the following reference is made to Fig. Optionally, a fifth result register can be used, or the counter register, the denominator register or the third register for the first auxiliary variable can be used as a result register, when it is desired.

In a step 20 the value of e is first selected according to the equation. Then, the denominator register is also re-loaded, with the second product In a step 26, a modular reduction is then calculated according to equation sixteenth Dann, nach dem Berechnen im Schritt 26 wird der Nenner um 1 inkrementiert Schritt 28 , um in einem Schritt 30 die zentrale Reduktionsgleichung 15 zu berechnen.

After calculating the difference in step 32 is checked in a step 34 whether the result is negative. If this is the case, then the module is added step 36 in order to obtain the result Q of the division step It should be noted that the embodiment of the present invention shown in Fig.

Bei dem in Fig. As set forth below, the inventive concept can also be used without increased effort to not only calculate the result of a division, but the integer multiple of the division.

Particularly, in the case where x is an integer multiple of 2, and a binary system is present, this can be achieved by register shifting to the right by an appropriate number of locations.

Wenn Gleichung 19 in Analogie zu Gleichung 16 und 17 ausgewertet wird, so wird ebenfalls das x-fache von Q erhalten. If equation is evaluated in analogy to equation 16 and 17 19, the x-fold of Q is also obtained.

Eine weitere Alternative zum Extrahieren von Q bzw. Another alternative for extracting Q or a multiple of Q of Equation 19 is to use to evaluate the following equation 20, whereby Equation 20 generally corresponds to equation 19, where now, however, the integer y is to be chosen different from x.

Wenn nunmehr Gleichung 20 von Gleichung 19 subtrahiert wird, ergibt sich folgende Gleichung Now, if Equation 20 is subtracted from Equation 19, the following equation 21 on the left hand side of equation 21 is now not obtained the result Q, but an integral multiple of the result Q, namely the difference between y and x.

Q can be obtained from Equation 21 again by a division by y - x is performed. The present invention is thus suitable because of their flexibility, security and performance especially for cryptographic algorithms and for cryptographic coprocessors on which typically a safe and efficient implementation of the modular reduction in circuit implemented.

A device for calculating a result Q or an integer multiple of the result Q of an integer division of a numerator A by a denominator N in a cryptographic algorithm, the numerator and the denominator being parameters of the cryptographic algorithm, comprising: The device according to claim 1, wherein the means 12 for providing is formed to determine the factor in such a way that a base raised to the power of an auxiliary number equals the factor.

The device according to claim 2, wherein the base is 2 so that the multiplication by the factor corresponds to a shift by a number of positions in a register, wherein the number of positions equals the auxiliary number.

The device according to one of claims 1 to 3, wherein the means 16 for extracting comprises: The device according to one of the preceding claims, wherein the division is an integer division.

The device according to claim 3, wherein the auxiliary number is selected according to the following equation: The device according to claim 5, wherein the means 16 for extracting is arranged to decide 34 whether the result is negative and, in this case, to add to the result the modulus provided in the means 14 for modularly reducing for being used.

The device according to claim 5, further comprising: The device according to claim 1, wherein the means 14 for modularly reducing is formed to use a modulus equaling a sum of a product of the denominator and the factor and of an integer, wherein the integer is greater than or equal to 1, and wherein the means 16 for extracting is arranged to perform a modular reduction using a modulus equaling a sum of a product of the denominator and the factor and of another integer, wherein the other integer is unequal to the integer so that the result of the division is obtained by the means 16 for extracting when a difference of the other integer and the integer equals 1, or the result, multiplied by an integer multiplier, is obtained, wherein the integer multiplier equals the difference of the other integer and the integer.

The device according to claim 1, embodied as a crypto coprocessor of a cryptographic device. The device according to one of claims 1 to 3, wherein the integer is negative so that the result can be obtained without inversion.

The device according to claim 10, 11 or 12, wherein the other number is a negative integer. A method for calculating a result Q or an integer multiple of the result Q of an integer division of a numerator A by a denominator N in a cryptographic algorithm, the numerator and the denominator being parameters of the cryptographic algorithm, comprising the following steps: DE DEC1 en Cryptographic device and cryptographic method of calculating a result of a modular multiplication.

Modular multiplication process for cryptography uses multiplicand in three bit segments in an multiplication addition operation. System, method and apparatus for an incremental modular process including modular multiplication and modular eduction.

Remainder calculating device, table generating device, and multiplication remainder calculating device. A device for locking a flat, preferably circular disc-shaped substrate on the substrate plate of a vacuum coating system.

Scheme for computing Montgomery division and Montgomery inverse realizing fast implementation. Scheme for carrying out modular calculations based on redundant binary calculation.

Method and apparatus for implementing short-word division techniques in a multiple modulus conversion context. Remainder calculation method, multiplication remainder calculation method, remainder calculator, multiplication remainder calculator and recording medium.

Arithmetic method and apparatus and crypto processing apparatus for performing multiple types of cryptography.

Data processing method for modulo calculation of division remainder has whole number represented by partial data words with calculation of argument via recursive steps.

Hardware implementation for modular multiplication using a plurality of almost entirely identical processor elements. Method and apparatus for performing a modular exponentiation in a cryptographic processor.

Modular multiplication device for cryptographic applications with parallel calculation of look-ahead parameters for next iteration step during 3 operand addition.

Data parallel processing method by dividing data into data groups and connecting directly to respective processing units, bypassing communication unit.

A calculating device, method for performing an operation with an encrypted operand, carry-select adder and cryptographic processor.

Ergebnis Division Video

Grundrechenarten: Addition, Subtraktion, Multiplikation, Division

division ergebnis - the

Nicht nur, dass das Ergebnis kleiner ist als beide einzelnen Zahlen, sondern sie bestimmen auch das Vorzeichen. Nehmen wir an, zwei Menschen haben Lotto gespielt. Eine Division durch null mit Festkommazahlen löst auf praktisch allen Rechnern einen Laufzeitfehler eine Ausnahme vom Typ Division durch null engl. In allgemeineren Strukturen mit nichtkommutativer Multiplikation muss man zwischen Linksdivision und Rechtsdivision unterscheiden. Dividierst du einen positiven Dividend durch einen negativen Divisor, so ist dein Quotient dieses Mal eine negative Zahl. Diese können wir problemlos durch 5 Teilen, genauso wie durch 4, 3, 2 oder 1. Auch wenn du einen negativen Dividend durch einen positiven Divisor teilst, bleibt dein Ergebnis eine negative Zahl. Tja, und damit haben wir schon alles beieinander. Da der Kernel in Zusammenarbeit mit der Laufzeitumgebung der Programmiersprache die fehlerbehandelnde Laufzeitumgebung zur Verfügung stellt, kann eine Division durch null im Kernel selbst ggf. Man kann also nicht durch Null dividieren! Die Videos hier auf sofatutor verstehe buchmacher vergleich ehrlich gesagt besser, als wenn mein Lehrer wm viertelfinale mir steht. Dies bedeutet, die Casino mit paybox einzahlung durch Null ergibt keine Lösung. In allgemeineren Strukturen mit nichtkommutativer Multiplikation muss man zwischen Linksdivision und Rechtsdivision unterscheiden. Bei mehreren aufeinanderfolgenden Divisionen in einer Zeile wird die Reihenfolge von mma bedeutung nach rechts abgearbeitet; die Elk übersetzung ist fussball cl heute abend linksassoziativ [2] [3] [4] [5] [6]: Von Experten erstellt und angepasst an die Lehrpläne der Bundesländer. Und bei der Multiplikation? Subtraktion und Division führt man als abgeleitete mathematische Operationen der Grundoperationen ein. Dies bestätigt, dass man nicht durch Null dividieren kann! Der Operator für die Division ist das Geteiltzeichen: Weshalb man nicht durch Null Dividieren kann Die Division ist nicht nur eine umgekehrte Multiplikation, sondern auch eine Art wiederholte Subtraktion. Teilen oder Dividieren bedeutet: Bei Heron und Diophantos kamen zu den bekannten vier Rechenoperationen das Quadrieren und das Quadratwurzelziehen als weitere Grundrechenarten hinzu. Begriffe bei der Addition, Subtraktion, Multiplikation und Division lernst du in der 5. Beispiel aus einer Konditorei: Als Rechenzeichen für die Division werden: Die Multiplikation und die Division waren jedoch keine eigenständigen arithmetischen Operationen.

Multiple ways you may achieve this as the following examples:. If you would like a decimal returned, just change the type of one of the inputs to float:.

Make your voice heard. Take the Developer Survey now. Division in Python 2. This question already has an answer here: Python division 12 answers.

Erzsebet Erzsebet 1 4 7. If you want float division which is something I always prefer , just use this special import: Interesting - I do not find it more intuitive.

In particular, in cryptoprocessors the modular reduction is a frequently used operation are implemented for efficient algorithms typically hardware in a crypto coprocessor.

Die modulare Reduktion kann daher schnell und effizient ausgerechnet werden. The modular reduction can therefore be calculated quickly and efficiently.

Depending on the hardware implementation, the extraction of the result of the division from the auxiliary quantity can either be read directly from a long-number register when a register of sufficient length is present.

Alternatively, a further modular reduction and a subtraction can be carried out for extracting the result, in which case the calculation amount also remains in the frame, as well as the other modular reduction is carried out quickly and safely using on cryptocoprocessors already present efficient reduction circuits.

The inventive concept provides a significant acceleration of the division while increasing security. The Division invention requires on the same processor only six cycles per bit, which corresponds to an acceleration to 4.

The Division inventive concept is also safe against SPA attacks since the current or time consumption is independent of the particular bit pattern of the numbers processed, so the numerator and the denominator.

Preferred embodiments of the present invention are explained below with reference to the accompanying drawings. Before discussing in detail to the figures, a derivation of the Division inventive concept will first be given, which is based on a modular reduction of a first product from the counter and the factor, wherein the modulus is equal to a sum of a second product of the denominator and the factor and is an integer.

Without loss of generality, it is assumed that both the numerator and the denominator A N are binary numbers such that: The equations 2a and 2b show the orders of magnitude of the counter A and the denominator N.

Equation 1 can be rewritten as follows: The value of H of Equation 3 is calculated as follows: The result of the division of A and Q N thus represents the result of the so-called DIV operation, while the remainder H is obtained by the modular reduction of the counter A by the denominator N as a module: It should be noted that each floating-point can be attributed to an integer division, namely z.

Typically, floating-point are returned within an arithmetic logic unit to integer divisions. According to the invention a factor F is now introduced, which is defined in the case of a binary number system as follows: The basis, since only an example of a binary number system is considered, the number 2, while the factor F obtained when the base is raised to the power 2 with an exponent e.

According to the invention, the factor F shall satisfy the condition: The factor is thus determined so that the product of the factor F and the denominator of the division Equation 1 of the division is greater than the desired result Q.

It should be noted that the exact result Q of the division need not be known for this consideration, just this is supposed to be calculated.

Only the order of Q must be known in order to dimension the factor F correctly. Thus, it is, however, typically not a problem, the magnitude of the result of the division from the counter and to evaluate the denominator, since Equation 6 comprises only a greater condition, so that a correct execution of the algorithm is guaranteed whenever the factor F chosen to be very large becomes.

However, it is preferable to choose a factor rather small, since the factor determines the length of registers required for the calculation of the division.

If the factor is selected to be very large, very long registers are required, while shorter Register sufficient if the factor F is selected smaller.

The following equation 8 gives for the binary case equation 5 , a preferred dimensioning of the variable e to: Equation 8 contains only information about the counter A equation 2a and information about the denominator N equation 2b.

If e is dimensioned as shown in Equation 8, the condition for the factor of Equation 6 is always fulfilled. Wenn Gleichung 3 mit dem Faktor F multipliziert wird, so ergibt sich folgende Gleichung: If Equation 3 is multiplied by the factor F, then the following equation: Moreover, when Equation 4 is also multiplied by the factor F on both sides, so the following equation Auf der rechten Seite von Gleichung 9 wird nunmehr das Resultat Q hinzuaddiert und gleichzeitig abgezogen, was folgender Gleichung entspricht: On the right side of Equation 9, the result of Q is now added to and subtracted at the same time, which corresponds to the following equation: If Equation 12 is converted so that the result Q is excluded from the first two terms on the right side of Equation 12, we obtain the following expression: Alternatively, it is also reshaped so Equation 12 that the sum of HF and Q and not difference between the two terms is formed: Eine Umformung von Gleichung 13 bzw.

For the "sum Alternative" results in the following: Summe auf der linken Seite von Gleichung 14 bzw. The remainder of this division, so the auxiliary variable on the left side of Equation 14 can be calculated by the following equation 15 in analogy with equation 4: Equation 15 thus represents the central modular reduction has been attributed to the division Equation 1.

It should be noted that the difference on the left side of the above equation could also be negative. In this case, the module is added, so that the equation is satisfied, especially as the result of a modular reduction, by definition, can not be negative.

For the "sum Alternative", the following equation: To this end, reference is first made to Figure 3. Die Zahl Q ergibt sich nach einer Invertierung von -Q.

The number Q is given by an inversion -Q. Hierzu sind die entsprechenden Bits auf der lsb-Seite des Registers zu betrachten das ergibt -Q.

For this purpose, the corresponding bits are to be considered on the LSB side of the register yields the -Q. Es ist somit lediglich eine einfache arithmetische Operation in Form des Hinzuaddierens einer Eins zu den invertierten Bits erforderlich.

It is thus only a simple arithmetic operation in the form of adding a one to the inverted bits required. No major arithmetic operations such.

It should be noted that the factor F must be chosen not necessarily so great that the numbers H and -Q in the example shown in Fig.

Even if these numbers have an overlap region, it is also, as set forth below, it is possible to extract the number Q from the auxiliary quantity.

For this purpose a further modular reduction is carried out, as shown in Equation Equation 16 corresponds to Equation 4, but now the factor F is considered.

The searched result Q is obtained in this case by the fact that from the result of Equation 16, the Equation 15 is subtracted: Thus, when the auxiliary variable is negative, so that a module is added in equation 15 on the left side, this is also taken into account in the subtraction of equation 15 from equation 16 as follows: Im nachfolgenden wird auf Fig.

In the following will be made to FIG. The conditional equations are shown for clarity in a block 10 in FIG.

The inventive apparatus includes means 12 for providing a factor and in particular a number e, which forms the exponent of the base 2 by a factor so that the equation 6 or the equation 7 is satisfied.

The inventive device further comprises means 14 for calculating the auxiliary quantity, ie 15 to execute the equation Finally, the device according to the invention comprises means 16 for extracting Q from the auxiliary quantity in one of several ways, for example by in Fig.

In the following reference is made to Fig. Optionally, a fifth result register can be used, or the counter register, the denominator register or the third register for the first auxiliary variable can be used as a result register, when it is desired.

In a step 20 the value of e is first selected according to the equation. Then, the denominator register is also re-loaded, with the second product In a step 26, a modular reduction is then calculated according to equation sixteenth Dann, nach dem Berechnen im Schritt 26 wird der Nenner um 1 inkrementiert Schritt 28 , um in einem Schritt 30 die zentrale Reduktionsgleichung 15 zu berechnen.

After calculating the difference in step 32 is checked in a step 34 whether the result is negative. If this is the case, then the module is added step 36 in order to obtain the result Q of the division step It should be noted that the embodiment of the present invention shown in Fig.

Bei dem in Fig. As set forth below, the inventive concept can also be used without increased effort to not only calculate the result of a division, but the integer multiple of the division.

Particularly, in the case where x is an integer multiple of 2, and a binary system is present, this can be achieved by register shifting to the right by an appropriate number of locations.

Wenn Gleichung 19 in Analogie zu Gleichung 16 und 17 ausgewertet wird, so wird ebenfalls das x-fache von Q erhalten. If equation is evaluated in analogy to equation 16 and 17 19, the x-fold of Q is also obtained.

Eine weitere Alternative zum Extrahieren von Q bzw. Another alternative for extracting Q or a multiple of Q of Equation 19 is to use to evaluate the following equation 20, whereby Equation 20 generally corresponds to equation 19, where now, however, the integer y is to be chosen different from x.

Wenn nunmehr Gleichung 20 von Gleichung 19 subtrahiert wird, ergibt sich folgende Gleichung Now, if Equation 20 is subtracted from Equation 19, the following equation 21 on the left hand side of equation 21 is now not obtained the result Q, but an integral multiple of the result Q, namely the difference between y and x.

Q can be obtained from Equation 21 again by a division by y - x is performed.

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