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ABACUS: MYSTERY OF THE BEAD The Bead Unbaffled - An Abacus Manual

Hexadecimal Numbers - Addition and Subtraction

Hex and Decimal Equivalents

DEC	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
HEX	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F	10	11	12	13	14	15	16	17	18	19	1A	1B	1C	1D	1E	1F

Hexadecimal Values on a 2:5 Bead Suan Pan

When doing problems involving hexadecimals on a suan pan one must be aware of an important distinction. While hex A through hex F are all shown with two heaven beads touching the beam, there still exists the hex numbers 10, 11, 12, 13... etc.  Compare the two values A & 10; while both show variations of 10 on the abacus, when added to another number each will yield a different answer. From the beginning it's important to distinguish between hex values that contain the expressions A through F and those using purely numeral expressions.

Complementary numbers

When calculating decimal numerals (base 10) abacus operators make use of complementary numbers. Use the complementary numbers for hex (base 16) in the same way. In hex the complementary numbers add up to 16.

Complementary Numbers Hexadecimal

F & 1

E & 2

D & 3

C & 4

B & 5

A & 6

9 & 7

8 & 8

Addition

This is an advanced technique. It's important to have a good understanding of Takashi Kojima's methods for solving problems of addition.

Adding 1 through 5 to Hex A provides no obstacles.

Example 1a: A + 6 = 10

Step 1: Add 6 to A Add 6 to A. There aren't enough beads. Solution: The complementary number for hex 6 is A. Subtract A and carry one bead to the left leaving the answer.  Hex A + 6 = 10.

Example 1b: 10 + 6 = 16

Step 1: Add 6 to 10 On the other hand adding hex 10 + 6 is a straight forward example. Add 6 to 10 leaving the answer 16.

 

Example 2: D3 + 1A = ED

Step 1: Add 1 to D  &  A to 3 Another straight forward example. There are no complementary number operations. Simply add 1 to D and A to to 3 leaving the answer. D3 + 1A = ED.

 

Example 3: F4 + 89 = 17D

Step 1: Add 8 to F4 Set F4 onto the frame. Add 8 to F. There are no beads available. Solution: The complementary number for 8 is 8. Subtract 8 and carry 1 to the left leaving 174.

Step 2: Add 9 to 4 Add 9 to 4. Make sure to use the top two heaven beads. Bring them down and subtract one earth bead leaving the answer. F4 + 89 = 17D. Note: As this example shows, always make use of the heaven beads where possible.

 

Example 4: EA + AF = 199

Step 1: Add A to E Set EA onto the frame. Add A to E. There are not enough beads available. Solution: The complementary number for A is 6. Subtract 6 and carry 1 to the left leaving 18A on the frame.

Step 2: Add F to A Add F to A. There are not enough beads available Solution: Subtract the complementary 1; do this by  moving up one heaven bead and four earth beads. Carry 1 to the left leaving the answer. EA + AF = 199.

 

Example 4: EC + BD = 1A9

Step 1: Add B to E Set EC onto the frame. Add B to E. There are not enough beads available. Solution: The complementary number for B is 5. Subtract 5 and carry 1 to the left leaving 19C on the frame.

Step 2: Add D to C Add D to C. There are not enough beads available Solution: Subtract the complementary 3; do this by  moving up one heaven bead and two earth beads. Carry 1 to the left. Do this by bringing down the heaven bead and subtracting 4 leaving the answer. EC + BD = 1A9.

 

Subtraction

This is an advanced technique. It's important to have a good understanding of Takashi Kojima's methods for solving problems of subtraction.

Example 1a: A - 6 = 4

Step 1: Subtract 6 from A Subtract 6 from A. Make use of the heaven beads. Subtract A and make up the difference by adding four earth beads to the frame. This leaves the answer. A - 6 = 4

Example 1b: 10 - 6 = A

Step 1: Subtract 6 from 10 Subtract 6 from 10. There are no beads on the unit rod. Solution: The complementary number for 6 is A. Subtract 1 from the tens rod and add A to the unit rod leaving the answer. 10 - 6 = A

 

Example 2: 63 - 5B = 8

Step 1: Subtract 5 from 6 Set 63 onto the frame. Subtract 5 from 6 on the tens rod  leaving 13.

Step 2: Subtract B from 3 Subtract B from 3 on the units rod. There aren't enough beads. Solution: Subtract 1 from the tens rod and add the complementary 5 leaving the answer. 63 - 5B = 8

 

Example 3: D4 - A9 = 2B

Step 1: Subtract A from D Set D4 onto the frame. Subtract A from D leaving 34.

Step 2: Subtract 9 from 4 Subtract 9 from 4 on the units rod. There aren't enough beads. Solution: Subtract 1 from the tens rod and add the complementary 7. Do this by adding two heaven beads and subtracting 3 leaving the answer. D4 - A9 = 2B Note: As this example shows, always make use of the heaven beads where possible.

To really get a handle on this I suggest making your own examples. Here's a very useful website where you can check to see if your answers are correct. The site uses javascript so make sure it's enabled on your browser.

REFERENCES:

Addition And Subtraction of Hex Numbers Calculator

 

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▪ Advanced Abacus Techniques

© November, 2007
Totton Heffelfinger   Toronto Ontario  Canada
Email
totton[at]idirect[dot]com