``` HOW TO USE the ASIAN ABACUS for all: Add, Subtract, Multiply, Divide ===================================================================== The CHINESE ABACUS has two beads in the upper rows and five beads in the lower rows. One upper bead is equivalent to "5" and one lower bead is equivalent to "1". For representation of the decimals 0...9 only one "5" and four "1" beads are necessary in an Abacus. ... BUT - The full Abacus can be used for HEXA-DECIMAL Calculations The columns represent the digits of a number (incl. a decimal point). To enter a number shift the beads towards the inner wooden bar... Example: 1,234,567.89 US\$ is represented: +---------------------------+ | o o o o o o o o . . . . . | | . . . . . . . . . . . . . | | . . . . . . . . o o o o o | |===========================| | . . . . o o o o . o o o o | | . . . . . o o o . . o o o | | o o o o . . o o o . . o o | | o o o o o . . o o o . . o | | o o o o o o . . o o o . . | | o o o o o o o . o o o o . | +---------------------------+ neutral 1 2 3 4 5 6 7 8 9 The power of calculating with an Abacus is in the "Secrets": "SECRETS" for ADDITION: "SECRETS" for SUBTRACTION: *********************** ************************** 1 = lower 5, cancel 4 1 = cancel 5, return 4 2 = lower 5, cancel 3 2 = cancel 5, return 3 3 = lower 5, cancel 2 3 = cancel 5, return 2 4 = lower 5, cancel 1 4 = cancel 5, return 1 1 = cancel 9, forward 10 *) 1 = cancel 10, return 9 2 = cancel 8, forward 10 2 = cancel 10, return 8 3 = cancel 7, forward 10 3 = cancel 10, return 7 4 = cancel 6, forward 10 4 = cancel 10, return 6 5 = cancel 5, forward 10 5 = cancel 10, return 5 6 = cancel 4, forward 10 6 = cancel 10, return 4 7 = cancel 3, forward 10 7 = cancel 10, return 3 8 = cancel 2, forward 10 8 = cancel 10, return 2 9 = cancel 1, forward 10 9 = cancel 10, return 1 6 = raise 1, cancel 5, forward 10 6 = cancel 10, return 5, cancel 1 7 = raise 2, cancel 5, forward 10 7 = cancel 10, return 5, cancel 2 8 = raise 3, cancel 5, forward 10 8 = cancel 10, return 5, cancel 3 9 = raise 4, cancel 5, forward 10 9 = cancel 10, return 5, cancel 4 *) "forward 10" means raise 1 bead in the left column. "cancel 10" means cancel 1 bead in the left column. MULTIPLICATION with an ABACUS: ****************************** EXAMPLE: 23 * 456 = 10488 Demonstration "With Paper & Pencil - AS WITH AN ABACUS": -------------------------------------------------------- The TRICK is: Accumulation of elementary multiplications ... Remember INDENTATION! ( ex.: 2*6 means 20*6 ) 23 * 456 = ===================== 3*6 = 18 S1 = 18 2*6 = 12 S2 = 138 --------------------- 3*5 = 15 S3 = 288 2*5 = 10 S4 = 1288 --------------------- 3*4 = 12 S5 = 2488 2*4 = 8 S6 = 10488 --------------------- TOTAL = 10488 ( Sx = Subtotal ) ===== DEMONSTRATION with an ABACUS: ----------------------------- To START: Set the multiplier (23) far left Let the same space (00) on the far right Set the multiplicand (456) beside Set the indicator (^) in between Multiplier+Multiplicand+SPACE 2 3 4 5 6 +-v-v-------------v-v-v-----+ | o o o o o o o o o . . o o | | . . . . . . . . . . . . . | | . . . . . . . . . o o . . | |===========================| | o o . . . . . . o . o . . | | o o . . . . . . o . . . . | | . o o o o o o o o o . o o | | . . o o o o o o o o o o o | | o . o o o o o o . o o o o | | o o o o o o o o . o o o o | +---------------------------+ ^Indicator 1st STEP: 3*6=18 Set 18 in the space on the right side 2 3 4 5 6 1 8 +-----------------------v-v-+ | o o o o o o o o o . o o . | | . . . . . . . . . . . . . | | . . . . . . . . . o o . o | |===========================| | o o . . . . . . o . o o o | | o o . . . . . . o . . . o | | . o o o o o o o o o . . o | | . . o o o o o o o o o o . | | o . o o o o o o . o o o . | | o o o o o o o o . o o o o | +-----------------------v-v-+ ^1 8 ( = S1 ) 2nd STEP: 2*6=12 Cancel the last digit (6) of the multiplicand, move the indicator and add 12 2 3 4 5 1 2 +---------------------v-v---+ | o o o o o o o o o . o o . | | . . . . . . . . . . . . . | | . . . . . . . . . o . . o | |===========================| | o o . . . . . . o . o o o | | o o . . . . . . o . . o o | | . o o o o o o o o o . o o | | . . o o o o o o o o . . . | | o . o o o o o o . o o . . | | o o o o o o o o . o o o o | +---------------------v-v-v-+ ^1 3 8 ( = S2 ) 3rd STEP: 3*5=15 Add 15 2 3 4 5 1 5 +---------------------v-v---+ | o o o o o o o o o . o . . | | . . . . . . . . . . . . . | | . . . . . . . . . o . o o | |===========================| | o o . . . . . . o . o o o | | o o . . . . . . o . o o o | | . o o o o o o o o o . o o | | o . o o o o o o o o o . . | | o . o o o o o o . o o . . | | o o o o o o o o . o o o o | +---------------------v-v-v-+ ^2 8 8 ( = S3 ) 4th STEP: 2*5=10 Cancel the next digit (5) of the multiplicand, move the indicator and add 10 2 3 4 1 0 +-------------------v-v-----+ | o o o o o o o o o o o . . | | . . . . . . . . . . . . . | | . . . . . . . . . . . o o | |===========================| | o o . . . . . . o o o o o | | o o . . . . . . o . o o o | | . o o o o o o o o . . o o | | . . o o o o o o o o o . . | | o . o o o o o o . o o . . | | o o o o o o o o . o o o o | +-------------------v-v-v-v-+ ^1 2 8 8 ( = S4 ) 5th STEP: 3*4=12 Add 12 2 3 4 1 2 +-------------------v-v-----+ | o o o o o o o o o o o . . | | . . . . . . . . . . . . . | | . . . . . . . . . . . o o | |===========================| | o o . . . . . . o o o o o | | o o . . . . . . o o o o o | | . o o o o o o o o . o o o | | . . o o o o o o o . o . . | | o . o o o o o o . o . . . | | o o o o o o o o . o . o o | +-------------------v-v-v-v-+ ^2 4 8 8 ( = S5 ) 6th STEP: 2*4=8 Cancel the next digit (4) of the multiplicand, move the indicator and add 8 2 3 8 +-------------------v-------+ | o o o o o o o o o o o . . | | . . . . . . . . . . . . . | | . . . . . . . . . . . o o | |===========================| | o o . . . . . . o . o o o | | o o . . . . . . . . o o o | | . o o o o o o o . o o o o | | . . o o o o o o o o o . . | | o . o o o o o o o o . . . | | o o o o o o o o o o . o o | +-----------------v-v-v-v-v-+ ^1 0 4 8 8 ( = TOTAL ) ========= The multiplicand is exhausted, the RESULT is readable on the right. EXERCISE: 345 * 6789 = 2342205 Try it with ABACUS! ============================================================ 5*9 = 45 S01 = 45 4*9 = 36 S02 = 405 3*9 = 27 S03 = 3105 --------------------- 5*8 = 40 S04 = 3505 4*8 = 32 S05 = 6705 3*8 = 24 S06 = 30705 --------------------- 4*7 = 35 S07 = 34205 4*7 = 28 S08 = 62205 3*7 = 21 S09 = 272205 --------------------- 5*6 = 30 S10 = 302205 4*6 = 24 S11 = 542205 3*6 = 18 S12 = 2342205 --------------------- TOTAL = 2342205 ( Sxy = Subtotal ) ======= DIVISION with an ABACUS: ************************ The method depends on the type of divisor: (1) "Short Division", when the divisor is only one digit EXAMPLE: 87654 : 3 = 29218 (2) "Long Division", when the divisor has more than one digit EXAMPLE: 654 : 32 = 20.4375 Demonstration EXAMPLE (1) "With Paper & Pencil": ------------------------------------------------ 87654 : 3 = 29218 - 6 (=3*2), set 2 --- 27 (Remainder=2), pull down 7 - 27 (=3*9), set 9 ---- 06 (Remainder=0), pull down 6 - 6 (=3*2), set 2 --- 05 (Remainder=0), pull down 5 - 3 (=3*1), set 1 --- 24 (Remainder=2), pull down 4 - 24 (=3*8), set 8 --- 0 (Remainder=0); Dividend exhausted, END ! DEMONSTRATION EXAMPLE (1) with an ABACUS: ----------------------------------------- To START: Set the divisor (3) left Set the dividend (87654) right, let space for decimals Set the Dividend-Indicator (^D) after its 1st digit Divisor Dividend 3 8 7 6 5 4 +-v-----------v-v-v-v-v-----+ | o o o o o o . . . . o o o | | . . . . . . . . . . . . . | | . . . . . . o o o o . . . | |===========================| | o . . . . . o o o . o . . | | o . . . . . o o . . o . . | | o o o o o o o . . o o o o | | . o o o o o . . o o o o o | | . o o o o o . o o o . o o | | o o o o o o o o o o . o o | +---------------------------+ ^D 1st STEP: 8:3=2 ; Set 2 left between divisor and dividend Set the Result-Indicator (^R) behind Subtract 3*2=6 from 08 (Remainder=2), move ^D 3 2 2 7 6 5 4 +---------v---v-------------+ | o o o o o o o . . . o o o | | . . . . . . . . . . . . . | | . . . . . . . o o o . . . | |===========================| | o . . . o . o o o . o . . | | o . . . o . o o . . o . . | | o o o o . o . . . o o o o | | . o o o o o . . o o o o o | | . o o o o o o o o o . o o | | o o o o o o o o o o . o o | +---------------------------+ ^R ^D 2nd STEP: 27:3=9 ; Set 9 left, move ^R Subtract 3*9=27 from 27 (Remainder=0), move ^D 3 2 9 0 0 6 5 4 +-----------v-v-v-----------+ | o o o o o . o o . . o o o | | . . . . . . . . . . . . . | | . . . . . o . . o o . . . | |===========================| | o . . . o o . . o . o . . | | o . . . o o . . . . o . . | | o o o o . o o o . o o o o | | . o o o o o o o o o o o o | | . o o o o . o o o o . o o | | o o o o o . o o o o . o o | +---------------------------+ ^R ^D 3rd STEP: 6:3=2 ; Set 2 left, move ^R Subtract 3*2=6 from 06 (Remainder=0), move ^D 3 2 9 2 0 5 4 +-------------v---v---------+ | o o o o o . o o o . o o o | | . . . . . . . . . . . . . | | . . . . . o . . . o . . . | |===========================| | o . . . o o o . . . o . . | | o . . . o o o . . . o . . | | o o o o . o . o o o o o o | | . o o o o o . o o o o o o | | . o o o o . o o o o . o o | | o o o o o . o o o o . o o | +---------------------------+ ^R ^D 4th STEP: 5:3=1 ; Set 1 left, move ^R Subtract 3*1=3 from 05 (Remainder=2), move ^D 3 2 9 2 1 2 4 +---------------v---v-------+ | o o o o o . o o o o o o o | | . . . . . . . . . . . . . | | . . . . . o . . . . . . . | |===========================| | o . . . o o o o . o o . . | | o . . . o o o . . o o . . | | o o o o . o . . o . o o o | | . o o o o o . o o . o o o | | . o o o o . o o o o . o o | | o o o o o . o o o o . o o | +---------------------------+ ^R ^D 5th STEP: 24:3=8 ; Set 8 left, move ^R Subtract 3*8=24 from 24 (Remainder=0), END ! Divisor RESULT: Dividend exhausted 3 2 9 2 1 8 +-----------------v-v-v-----+ | o o o o o . o o o o o o o | | . . . . . . . . . . . . . | | . . . . . o . . o . . . . | |===========================| | o . . . o o o o o . . . . | | o . . . o o o . o . . . . | | o o o o . o . . o o o o o | | . o o o o o . o . o o o o | | . o o o o . o o . o o o o | | o o o o o . o o o o o o o | +---------------------------+ ^R EXERCISE: 22 : 7 = 3.1428571 Remainder 3 Try it with ABACUS! ================================================================ 22 - 21 (*3) ---- 10 - 7 (*1) --- 30 - 28 (*4) ---- 20 - 14 (*2) ---- 60 - 56 (*8) ---- 40 - 35 (*5) ---- 50 - 49 (*7) ---- 10 - 7 (*1) --- 3 Demonstration EXAMPLE (2A) "With Paper & Pencil - AS USUAL": ------------------------------------------------------------ 654 : 32 = 20.4375 - 64 (*2) ---- 140 pull down 4, (*0), set DECIMAL POINT, pull down ZEROs - 128 (*4) ----- 120 - 96 (*3) ----- 240 - 224 (*7) ----- 160 - 160 (*5) ----- 0 Remainder=0, END ! Demonstration EXAMPLE (2B) "With Paper & Pencil - AS WITH AN ABACUS": --------------------------------------------------------------------- The TRICK is: Subtraction of elementary multiplications ... 654 : 32 = 20.4375 - 6 (=3*2), set 2 1st digit --- 05 (Remainder=0), pull down 5 - 4 (=2*2) --- 14 (Remainder=1), pull down 4 14 14:32=0 , set 0 2nd digit Dividend exhausted, set DECIMAL POINT, pull down ZEROs 140 - 12 (=3*4), set 4 3rd digit ---- 20 (Remainder=2), pull down 0 - 8 (=2*4) --- 12 (Remainder=12), pull down 0 120 - 9 (=3*3), set 3 (!!! NOT 4 !!!) 4th digit --- 30 (Remainder=3), pull down 0 - 6 (=2*3) --- 24 (Remainder=24), pull down 0 240 - 21 (=3*7), set 7 5th digit ---- 30 (Remainder=3), pull down 0 - 14 (=2*7) ---- 16 (Remainder=16), pull down 0 160 - 15 (=3*5), set 5 6th digit ---- 10 (Remainder=1), pull down 0 - 10 (=2*5) ---- 0 (Remainder=0), END ! DEMONSTRATION EXAMPLE (2B) with an ABACUS: ------------------------------------------ Let as an E X E R C I S E for YOURSELF ... impressum: ********************************************************************* © C.HAMANN http://public.beuth-hochschule.de/~hamann 10/01/05 ```