P R O C E D U R E S A N D T R I C K S O F C A L C U L A T I N G ======================================================================= Instruction Manuals (eg. CURTA & WALTHER) are sources of ancient wisdom C O N T E N T ************* (A) NEGATIVE RESULTS (B) SHORTENED METHOD OF MULTIPLICATION (C) SHORTENED METHOD OF CALCULATING DISCOUNT & NET SUM: (D) MULTIPLE DIVISIONS WITH THE SAME DIVISOR: MAKE MULTIPLICATIONS! (E) DIVISION (Additive Method; Division by Multiplication) (F) CHAIN MULTIPLICATIONS WITH BACK-TRANSFER (G) CHAIN MULTIPLICATIONS WITHOUT BACK-TRANSFER (H) MULTIPLICATION INVOLVING ADDITION OF PRODUCTS (I) CUBES WITHOUT BACK-TRANSFER (K) CALCULATING SQUARE ROOTS - Methode TOEPLER - Methode HERRMANN (L) TEST-PATTERNS FOR CALCULATORS ( LINK to "FAQs" ) (M) PHOENIX-ZAHL Experiment ( LINK to "WALTHER-WSR-160" ) (A) NEGATIVE RESULTS ******************** NEGATIVE RESULTS are displayed in the arithmetic unit as the COMPLEMENT of the next higher 10, 100, 1000, ... EXAMPLE: -12 = 99...9988 -------- To get the "TRUE VALUE", two mechanical procedures are available ... (a) with BackTransfer: ---------------------- Make the BACK TRANSFER 1 SUBTRACTIVE TURN shows on the right side the "true value" ( ignore the 99..99 on the left side !) (b) without BackTransfer: ------------------------- Set the COMPLEMENT RESULT into Input Unit Make a 1st subtractive turn ( result should be ZERO ) A 2nd SUBTRACTIVE TURN shows on the right side the "true value" ( ignore the 99..99 on the left side !) (B) SHORTENED METHOD OF MULTIPLICATION ************************************** Operations that work, when the machine has a COUNTER WITH 10s-CARRY EXAMPLE(1): 13,974 * 9 = 125,766 ----------- Instead of multiplying 13,974 by 9, let us calculate 13,974 * (10 - 1), or (13,974 * 10) - 13,974 ; thus accomplishing the calculation in two turns instead of nine: Set carriage to 2nd position. Enter 13,974 and make one positive turn. This additive turn produces the multiplication by 10. Move the carriage to the 1st position and make one negative turn. By means of these two turns, the calculation is finished. The counter shows 9 and the result = in the arithmetic unit shows the PRODUCT = 125,766 ======= EXAMPLE(2): 345,67 * 89 = 30764,63 ----------- Instead of 17, the calculation will be done in 3 steps: o Carriage to 3rd position o Enter 345,67 o One positive turn ( = Multiplication with 100 ), counter shows 100 o Carriage to 2nd position. One subtractive turn, counter shows 90 o Carriage to 1st position. One subtractive turn, counter shows 89 == The arithmetic unit shows the PRODUCT = 30764,63 ======== (C) SHORTENED METHOD OF CALCULATING DISCOUNT & NET SUM: ******************************************************* EXAMPLE: Item Price = 7,683.00 $ -------- - 3 % = 230.49 $ ----------------------- Net Sum = 7,452.51 $ ========== Simultanious multiplication of 2 different numbers by the same factor: Set 3(%) at far left and 97(%) at far right of the input unit. Multiply these numbers with the item price (= shown in the counter). The arithmetic unit will show both(!), the DISCOUNT & NET SUM. (D) MULTIPLE DIVISIONS WITH THE SAME DIVISOR: MAKE MULTIPLICATIONS! ******************************************************************* At 1st calculate the reciprocal of the divisor 11.7 ==>> 0.0854701 At 2nd use it as a factor: 1633.0 : 11.7 0.0854701 * 1633.0 = 139.573 314.5 : 11.7 ===>>> 0.0854701 * 341.5 = 29.188 ... ... 67.8 : 11.7 0.0854701 * 67.8 = 5.795 (E) DIVISION (Additive Method; Division by Multiplication) ********************************************************** EXAMPLE: 123,456 : 789 = 156.47148 -------- In this 2nd method of division, the divisor is entered into input unit. The dividend will be build up (from left to right) in the arithmetic unit and the quotient comes in the counter; all 3 numbers are visible! Shift the carriage to the far right. Enter 789. Make 1 additive turn with the crank (a 2nd add.turn get 1578, too big! Make 1 sub.turn). Shift carriage left and make 5 add.turns to get 11835. Shift carriage left and make 6 add.turns to get 123084. Shift carriage left and make 4 add. turns to get 1233996. Shift carriage left and make 7 add. turns to get 12345483. Shift carriage left and make 1 add.turn to get 123455619. Shift carriage left and make 5 add.turns to get 12345601350 (too big?). Make 1 sub.turn to get 12345593460. Shift carriage left and make 8 add.turns to get 12345599772; an additional add.turn get 12345600561. The former number is clother to 123456, so a sub.turn will bring it back in the arithmetic unit. Estimating the decimal marker, the QUOTIENT = 156.47148 is visible in the counter. ========= (F) CHAIN MULTIPLICATIONS WITH BACK-TRANSFER ******************************************** BRUNSVIGA-13RM or WALTHER-WSR160 have build-in "Back-Transfer", means that an intermediate result in the arithmetic unit can be brought back into input unit (ref. to the manuals) without new entering! EXAMPLE(1): 123 * 45 * 67 = 370,845 (1st Product 123 * 45 = 5,535) ----------- ======= EXAMPLE(2): 345 * 6.78 $ = 2,339.10 $ ----------- - 35 % = 818.68 $ ------------------------- Net Sum = 1,520.42 $ ========== Multiply as usual 345 * 6.78 and back-transfer the intermediate product 2339.10 into input unit. Multiply with 35(%). The intermediate result 818.69 is in arithmetic unit and the multiplier 35 is in the counter. Calculate the NET SUM in transfering 35 into 65 WITH ONLY 3 ADD.TURNS on 2nd position. This means: Build the difference 35% to 100% (G) CHAIN MULTIPLICATIONS WITHOUT BACK-TRANSFER *********************************************** EXAMPLE: 123 * 45 * 67 = 370,845 (1st Product 123 * 45 = 5,535) -------- The multiplication 123 * 45 is performed in the normal way. 5,535 is shown in the arithmetic unit. Clear the input register. Set it to 5,535 and make a backward turn, causing the arithmetic unit to zero (=Proof!). Set counter to zero. Multiply with 67 and get 370,845. ======= (H) MULTIPLICATION INVOLVING ADDITION OF PRODUCTS ************************************************* EXAMPLE: 123 * 45 + 67 * 89 = 11,498 (1st Product 123 * 45 = 5,535) -------- The multiplication 123 * 45 is performed in the normal way. 5,535 is shown in the arithmetic unit. Clear the input register and the counter. Multiply 67 * 89 (short-cut!) and the arithmetic unit shows 11,498. ====== (I) CUBES WITHOUT BACK-TRANSFER ******************************* EXAMPLE: 327 |