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Hands as a Calculator By William G. Mandras

Former Professor of Engineering and MathematicsAcademy of Aeronautics

Queens, NY

Table of Contents: Page

Preface 2

Section

1.0) Addition 3

1.4) Summary, Addition 7

2.0) Multiplication, Visual, Multiplier: 9 8

3.0) Multiplication, Trachtenberg, Multiplier: 5 12

4.0) Multiplication, Trachtenberg, Multiplier: 6 16

5.0) Multiplication, Tens Compliment 18

6.0) Summary, Multiplication 21

6.1) Trachtenberg Multiplication Table 22

1


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Hands as a Calculator

Hand calculations have many advantages over the pencil and paper methods in arithmetic.The students require no external device and can practice the procedures at any time or place.Rote memory of the addition and multiplication tables, facts that can not be visualized, is themost difficult memorization possible. Representing numbers with the hands provides thevisuals needed to enable a much easier memorization process to occur. A good pedagogicalapproach is to have the students verify the addition and multiplication tables using theirhands as a calculator. Through repetition of the procedures, the addition and multiplicationfacts will be memorized.

Hand calculations are especially suited for the hearing impaired students whom, of necessity, are highly skilled in using their hands in communications.

The larger sums and products being the most difficult for the students, this paper will limitthe hand diagrams to the two digit sums and products. Considering that multiplication isdefined as repeated addition, the single digit sums and products can be obtained by addition.

Hand calculations is enabled by base 10 numeration, Trachtenberg 1 multiplication, settheory, and 10s compliment multiplication.

Hand calculations are intended to augment the current elementary mathematics pedagogy.

1 Jakow Trachtenberg invented a system for multiplication such that the product of binary multiplication can beobtained by simple arithmetic operations on the multiplicand. A multiplication table is on the last page.

2


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1.0) Addition, augend 5, addend 6 to 9; sums from 11-14.Rule: Start with all fingers folded and raise each finger in synch with the count. Start the LH

and RH count at 5 and count up to the multiplier and multiplicand. Right Digit = the sumof the open fingers. The Left Digit is always 1 for sums from 10 to 18.

1.1) Addition (Continued), augend 6, addend 6 to 9; sums from 12 - 15.

3

H+) Add: 5 + 6 = 6 + 5 = 11 H+) Add: 5 + 7 = 7 + 5 = 12

RH count 6 = 1 open fingerLH count none, no open fingersAdd the open fingersRight Digit: 0 + 1 = 1Left Digit = 1Addition 5 + 6 = 11

RH count 6, 7 = 2 open fingersLH count none, no open fingersAdd the open fingersRight Digit: 0 + 2 = 2Left Digit = 1Addition 5 + 7 = 12

RH count 6, 7, 8 = 3 open fingersLH count none, no open fingersAdd the open fingersRight Digit: 0 + 3 = 3Left Digit = 1Addition 5 + 8 = 13

RH count 6, 7, 8, 9 = 4 open fingersLH count none, no open fingersAdd the open fingersRight Digit: 0 + 4 = 4Left Digit = 1Addition 5 + 9 = 14

H+) Add: 5 + 8 = 8 + 5 = 13 H+) Add: 5 + 9 = 9 + 5 = 14


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Rule: See section 1.0.4

H+) Add: 6 + 6 = 2 * 6 = 12 H+) Add: 6 + 7 = 7 + 6 = 13

H+) Add: 6 + 8 = 8 + 6 = 14 H+) Add: 6 + 9 = 9 + 6 = 15

RH count 6 = 1 open fingerLH count 6 = 1 open finger

Add the open fingersRight Digit: 1 + 1 = 2Left Digit = 1Addition 6 + 6 = 2 * 6 = 12

RH count 6, 7 = 2 open fingersLH count 6 = 1 open finger

Add the open fingersRight Digit: 1 + 2 = 3Left Digit = 1Addition 6 + 7 = 13

RH count 6, 7, 8 = 3 open fingersLH count 6 = 1 open finger


RH count 6, 7, 8, 9 = 4 open fingersLH count 6 = 1 open Finger



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1.2) Addition (Continued), augend 7, addend 7 to 9; sums from 14 - 16.Rule: See section 1.0.

5

H+) Add: 7 + 7 = 2 * 7 = 14 H+) Add: 7 + 8 = 8 + 7 = 15

H+) Add: 7 + 9 = 9 + 7 = 16

RH count 6, 7 = 2 open fingersLH count 6, 7 = 2 open fingersAdd the open fingersRight Digit: 2 + 2 = 4Left Digit = 1Addition 7 + 7 = 2 * 7 = 14

RH count 6, 7, 8 = 3 open fingersLH count 6, 7 = 2 open fingersAdd the open fingersRight Digit: 2 + 3 = 5Left Digit = 1Addition 7 + 8 = 15

RH count 6, 7, 8, 9 = 4 open fingersLH count 6, 7 = 2 open fingersAdd the open fingersRight Digit: 2 + 4 = 6Left Digit = 1Addition 7 + 9 = 16


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1.3) Addition (Continued), augend 8, 9, addend 8 to 9; sums from 16 - 18.Rule: See section 1.0.

6

H+) Add: 8 + 8 = 2 * 8 = 16 H+) Add: 8 + 9 = 9 + 8 = 17

H+) Add: 9 + 9 = 2 * 9 = 18

RH count 6, 7, 8, 9 = 4 open fingersLH count 6, 7, 8 = 3 open fingersAdd the open fingersRight Digit: 3 + 4 = 7Left Digit = 1Addition 8 + 9 = 17

RH count 6, 7, 8, 9 = 4 open fingers

LH count 6, 7, 8, 9 = 4 open fingersAdd the open fingersRight Digit: 4 + 4 = 8Left Digit = 1Addition 9 + 9 = 18

RH count 6, 7, 8 = 3 open fingersLH count 6, 7, 8 = 3 open fingersAdd the open fingersRight Digit: 3 + 3 = 6Left Digit = 1Addition 8 + 8 = 16


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1.4) Summary, Addition; Right Digit: add the open fingers, Left Digit = 1.7

9 + 9 = 18

5 + 5 = 10 5 + 6 = 11 5 + 7 = 12 5 + 8 = 13 5 + 9 = 14

6 + 7 = 13 6 + 8 = 146 + 9 = 156 + 6 = 12

7 + 7 = 14 7 + 8 = 15 7 + 9 = 16

8 + 8 = 16 8 + 9 = 17


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2.0) Multiplication, Visual: Multiplier = 2, 3; Multiplicand = 9Rule: Start with two open hands. Label the fingers from 1-left to 10-right.

8

Fold the finger that corresponds to multiplier 2.

Multiplication: 2 * 9 = 18

Left set of open fingers is the Left Digit = 1Right set of open fingers is the Right Digit = 8





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2.1) Multiplication, special case visual: Multiplier = 4, 5; Multiplicand = 9Rule: Start with two open hands. Label the fingers from 1-left to 10-right.

9








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2.2) Multiplication, special case visual: Multiplier = 6,7; Multiplicand = 9Rule: Start with two open hands. Label the fingers from 1-left to 10-right.

10








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2.3) Multiplication, special case visual: Multiplier = 8, 9; Multiplicand = 9Rule: Start with two open hands. Label the fingers from 1-left to 10-right.

11








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3.0) Multiplication, Trachtenberg multiplier 5; multiplicand = 2, 3Rule: Start with all fingers folded. Open one right hand finger in synch with the count and

alternate to left hand while counting up to the multiplicand. If the final open finger setsare equal, the Right Digit = 0; if the sets are not equal, the Right Digit = 5.

12

Sets are equal, close right handOpen fingers are the L/R digitsMultiplication: 5 * 2 = 10

RH count: 1LH count: 2Right Digit = 0, Sets are equalLeft Digit = 1, LH open fingersMultiplication: 5 * 2 = 10

H*) Multiply: 5 * 2 = 2 * 5 = 10

Visual

Sets are not equal , open right handOpen fingers are the L/R digitsMultiplication: 5 * 3 = 15

RH count: 1, 3LH count: 2.Right Digit = 5, Sets not equalLeft Digit = 1, LH open fingersMultiplication: 5 * 3 = 15

H*) Multiply: 5 * 3 = 3 * 5 = 15

Visual


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3.1) Multiplication (Continued), Trachtenberg multiplier 5; multiplicand = 4, 5Rule: See section 3.0.

13


RH count: 1, 3LH count: 2, 4.Right Digit = 0, Sets are equalLeft Digit = 2, LH open fingersMultiplication: 5 * 4 = 20

H*) Multiply: 5 * 4 = 4 * 5 = 20

Visual

Sets are not equal, open right handOpen fingers are the L/R digitsMultiplication: 5 * 5 = 25

RH count: 1, 3, 5LH count: 2, 4.Right Digit = 5, Sets not equalLeft Digit = 2, LH open fingersMultiplication: 5 * 5 = 25

H*) Multiply: 5 * 5 = 25

Visual


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14


RH count: 1, 3, 5LH count: 2, 4, 6Right Digit = 0, Sets are equalLeft Digit = 3, LH open fingersMultiplication: 5 * 6 = 30

H*) Multiply: 5 * 6 = 6 * 5 = 30

Visual


RH count: 1, 3, 5, 7;LH count: 2, 4, 6.Right Digit = 5, Sets not equalLeft Digit = 3, LH open fingersMultiplication: 5 * 7 = 35

H*) Multiply: 5 * 7 = 7 * 5 = 35

Visual


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15


RH count: 1, 3, 5, 7LH count: 2, 4, 6, 8Right Digit = 0, Sets are equalLeft Digit = 4, LH open fingersMultiplication: 5 * 8 = 40

H*) Multiply: 5 * 8 = 8 * 5 = 40

Visual

RH count: 1, 3, 5, 7, 9LH count: 2, 4, 6, 8Right Digit = 5, Sets not equalLeft Digit = 4, LH open fingersMultiplication: 5 * 9 = 45


H*) Multiply: 5 * 9 = 9 * 5 = 45

Visual


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4.0) Multiplication, Trachtenberg multiplier 6; multiplicand = 2 to 5Rule Start with all fingers folded. Open one right hand finger in synch with the count andalternate to left hand while counting up to the multiplicand. If the final open finger sets are equal,the Right Digit = Multiplicand; if the sets are not equal, the Right Digit = Multiplicand + 5.

16

RH count: 1LH count: 2 Sets are equalRight Digit = 2Left Digit = 1, LH open fingersMultiplication: 6 * 2 = 12

RH count: 1, 3LH count: 2 Sets not equalRight Digit = 3 + 5 = 8Left Digit = 1, LH open fingersMultiplication: 6 * 3 = 18

H*) Multiply: 6 * 2 = 2 * 6 = 12 H*) Multiply: 6 * 3 = 3 * 6 = 18

RH count: 1, 3LH count: 2, 4 Sets are equalRight Digit = 4Left Digit = 2, LH open fingersMultiplication: 6 * 4 = 24

RH count: 1, 3, 5LH count: 2, 4 Sets not equalRight Digit: 5 + 5 = 10 = 0, carry 1LH open fingers (2) + carry (1) = 3Left Digit = 3Multiplication: 6 * 5 = 30



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4.1) Multiplication (Continued), Trachtenberg multiplier 6; multiplicand = 6 to 9Rule: See section 4.0.

17

RH count: 1, 3, 5LH count: 2, 4, 6 Sets are equalRight Digit = 6Left Digit = 3, LH open fingersMultiplication: 6 * 6 = 36

RH count: 1, 3, 5, 7LH count: 2, 4, 6 Sets not equalRight Digit: 7 + 5 = 12 = 2, carry 1LH open fingers (3) + carry (1) = 4Left Digit = 4Multiplication: 6 * 7 = 42

H*) Multiply: 6 * 6 = 36 H*) Multiply: 6 * 7 = 7 * 6 = 42


RH count: 1, 3, 5, 7

LH count: 2, 4, 6, 8; Sets are equalRight Digit = 8Left Digit = 4, LH open fingersMultiplication: 6 * 8 = 48

RH count: 1, 3, 5, 7, 9LH count: 2, 4, 6, 8; Sets not equalRight Digit: 9 + 5 = 14 = 4, carry 1LH open fingers (4) + carry (1) = 5Left Digit = 5Multiplication: 6 * 9 = 54


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5.0) Multiplication, Tens compliment, multiplier 6; Multiplicand = 6, 7, 8, 9Rule: Start with all fingers folded and open one finger in synch with the count. Start LH countwith the multiplier and count to 10. Start RH count with the multiplicand and count to 10.Right Digit: multiply the open fingers. Left Digit: add the closed fingers.

18

LH count: 7, 8, 9, 10 = 4 Open fingersRH count: 7, 8, 9, 10 = 4 Open fingersMultiply open fingers: 4 * 4 = 16Right Digit = 6, carry 1Add closed fingers: 1 + 1 = 2Left Digit: 2 + 1 (carry) = 3Multiplication: 6 * 6 = 36

*

+

LH count: 7, 8, 9, 10 = 4 Open fingersRH count: 8, 9, 10 = 3 Open fingersMultiply open fingers: 4 * 3 = 12Right Digit = 2, carry 1Add closed fingers: 1 + 2 = 3Left Digit: 3 + 1 (carry) = 4Multiplication: 6 * 7 = 42

*

+



LH count: 7, 8, 9, 10 = 4 Open fingersRH count: 9, 10 = 2 Open fingersMultiply open fingers: 4 * 2 = 8Right Digit = 8Add closed fingers: 1 + 3 = 4Left Digit = 4Multiplication: 6 * 8 = 48

LH count: 7, 8, 9, 10 = 4 Open fingersRH count: 10 = 1 Open fingerMultiply open fingers: 4 * 1 = 4Right Digit = 4Add closed fingers: 1 + 4 = 5Left Digit = 5Multiplication: 6 * 9 = 54

*

+

*

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5.1) Multiplication (Continued), Tens compliment, multiplier 7; Multiplicand = 7, 8, 9Rule: See section 5.0.

19

H * ) M u l t ip l y : 7 * 7 = 4 9 H * ) M u l t i p l y : 7 *

LH count: 8, 9, 10 = 3 Open fingersRH count: 8, 9, 10 = 3 Open fingersMultiply open fingers: 3 * 3 = 9Right Digit = 9Add closed fingers: 2 + 2 = 4Left Digit = 4Multiplication: 7 * 7 = 49

*

+

H * ) M u l t i p l y : 7 * 9 = 9 * 7

LH count: 8, 9, 10 = 3 Open fingersRH count: 10 = 1 Open fingerMultiply open fingers: 3 * 1 = 3Right Digit = 3Add closed fingers: 2 + 4 = 6Left Digit = 6Multiplication: 7 * 9 = 63

*

+

LH count: 8, 9, 10 = 3 Open fingersRH count: 9, 10 = 2 Open fingersMultiply open fingers: 3 * 2 = 6Right Digit = 6Add closed fingers: 2 + 3 = 5Left Digit = 5Multiplication: 7 * 8 = 56

*

+


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5.2) Multiplication (Continued), Tens compliment, multiplier 8, 9; Multiplicand = 8, 9Rule: See section 5.0.

20


LH count: 9, 10 = 2 Open fingersRH count: 9, 10 = 2 Open fingersMultiply open fingers: 2 * 2 = 4Right Digit = 4Add closed fingers: 3 + 3 = 6Left Digit = 6Multiplication: 8 * 8 = 64

*

+

LH count: 9, 10 = 2 Open fingersRH count: 10 = 1 Open fingerMultiply open fingers: 2 * 1 = 2Right Digit = 2Add closed fingers: 3 + 4 = 7Left Digit = 7Multiplication: 8 * 9 = 72

*

+

H*) Multiply: 9 * 9 = 81

LH: 9 + ? = 10, ? = 1 Open fingerRH count: 10 = 1 Open fingerMultiply open fingers: 1 * 1 = 1Right Digit = 1Add closed fingers: 4 + 4 = 8Left Digit = 8Multiplication: 9 * 9 = 81

*

+


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6.0) Summary, Multiplication:Key: T2, T5, T6 = Trachtenberg: 2, 5, & 6. 10C = Tens compliment. V9 = Visual nine.

Underlined italic numbers (6), are obtained by addition as shown.

21

81

8

12

16

10 12 14 16 18

15 18 21 24 27

20 24 28 32 36

25 30 35 40 45

36 42 48 54

49 56 63

64 72

28 + 4 = 3224 + 4 = 28 T5 T6

T5 T618 + 3 = 21 21 + 3 = 24 V9

V9

V9

V9

5

6

7

8

9

4 5 6 7 8 9

2

3

4

T2 T2 T5 T2 V9

8 + 4 = 12

12 + 4 = 16

T6

T6 T6 T6 V9

T5 T5 T5 T5 V9

10C 10C

10C

V9


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6.1) Trachtenberg 2 Multiplication Table: N * M, N = 2 to 9

Note: All rules apply to the multiplicand (M)

2 Jakow Trachtenberg invented the system that yields the products of binary multiplication through simple arithmeticoperations on the multiplicand.

22

7 2M, (+5 if M is odd) (1/2)M & Drop fraction

M, if even

M + 5, if odd6

(1/2)M & Drop fraction

50, if M is even5, if M is odd (1/2)M & Drop fraction

4 10 - M, (+5, if M is odd) (1/2)M - 1 & Drop fraction

3 2(10 - M), (+5, if M is odd) (1/2)M - 2 & Drop fraction

2 M + M

Mult.by

Right Digit Left Digit

9 10 - M M - 1

8 2(10 - M) M - 2

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