ABACUS: MYSTERY OF THE BEAD

The Bead Unbaffled - An Abacus Manual

Converting Hexadecimal (base 16) to Decimal (base 10)Traditional methods for converting a hexadecimal number to it's decimal equivalent can be cumbersome and time consuming. I love this method because it is a simple division process which makes it extremely fast and very efficient. The technique requires dividing the hexadecimal number by hex A with the remainder(s) forming the decimal answer.

## This is an advanced abacus technique. It's a good idea to understand what hexadecimal numbers are and how they work. For help see the charts below. For a further explanation of the division technique used in this tutorial see Chinese Short Division techniques.

Hex A Division Table (a.k.a. Division rules)

Dividing by hex A is remarkably simple in part because the division table for hex A is easily understood. As a result no memorization is necessary. For example look at the rule 4 ÷ A = 6+4; by way of explanation multiply 4 * 16 = 64 and the relationship becomes clear. As we progress through the division table take the following examples: begin by comparing hex B, C and E with their corresponding decimal equivalents in the chart below.

## ▪ 7 ÷ A = B+2 because 7 * 16 = 112 : 11 in 112 becomes hex B and the table reads B+2

▪ 8 ÷ A = C+8 because 8 * 16 = 128 : 12 in 128 becomes hex C and the table reads C+8

▪ 9 ÷ A = E+4 because 9 * 16 = 144 : 14 in 144 becomes hex E and the table reads E+4

▪ A ÷ A = forward 1, which means carry 1 to the left.

▪ B ÷ A = forward 1 with 1 remaining.## Decimal and Hex Equivalents

DEC0 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 HEX0 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

Example 1: Convert hexadecimal 15 to its decimal equivalent. Set the problem with divisor A on the left.

Divide hexadecimal 15 by A

Step 1:A meets 1, 1+6, keep 1 on H and add 6 to rod I.

Step 2 and the answer:A meets A on rod I, forward 1 to H leaving the answer 21.Hex 15 is decimal 21.

Example 2: Convert hexadecimal 9A to its decimal equivalent. Set the problem with divisor A on the left.

Divide hexadecimal 9A by A

Step 1:A meets 9, E+4, place E on rod H add 4 to rod I.

Step 2:A meets A on rod H, forward 1 to G.

Step 3 and the answer:A meets A on rod I, forward 1 to H leaving the answer 154.Hex 9A is decimal 154.

The first two examples were fairly simple. Here's one that is a little more complex.

Example 3: Convert hexadecimal 7D3B to its decimal equivalent.But before we solve the problem by dividing by A I'd like to show how it's done using the traditional method I most often see on the internet. Comparatively this traditional method is quite a lot of work.

## Traditional Method

## Refer to the Decimal and Hex Equivalents Chart;

7D3B = 7, 13, 3, 117×16³ + 13×16² + 3×16¹ + 11×16⁰=

7*4096 + 13*256 + 3*16 + 11*1 =

28672 + 3328 + 48 + 11 = 32059Now see how simple it is to solve the problem by dividing by hex A. Because of the diagrams and step by step instructions it may seem like a lengthy process. But it really isn't. Once this technique is mastered problems like this can be solved with lightning speed.

Set the problem with divisor A on the left.

Divide hexadecimal 7D3B by A

Step 1:A meets 7, B+2, place B on rod H add 2 to I. In order to add 2 to rod I use complimentary numbers for 10. Subtract complement 8 and carry 1 to the left. With that carry what was once B on rod H becomes C. Simple abacus basics.

Step 1A B C D E F G H I J K L M . . . .0 0 A 0 0 0 0 7 D 3 B 0 0 place B -8carry +10 0 A 0 0 0 0 C 5 3 B 0 0

Step 2:A meets 5, 8, place 8 on rod I

Step 3:A meets 3, 4+8, place 4 on J and add 8 to K. In order to add 8 to rod K use complimentary numbers for 10. Subtract complement 2 and carry 1 to the left. With that carry what was 4 on J becomes 5. Again simple abacus basics.At this point what's left is the integer C85 on rods HIJ and a remainder 9 on rod K signified by the red dot.

Step3A B C D E F G H I J K L M . . . .0 0 A 0 0 0 0 C 8 3 B 0 0 place 4 -2carry +10 0 A 0 0 0 0 C 8 5 9 0 0

Now divide the integer C85 by A

Step 4:A meets A forward 1, remove A from H and forward 1 to G

Step 5:A meets 2, 3+2, place 3 on H and add 2 to I. In order to add 2 to rod I use complimentary numbers for 10. Subtract complement 8 and carry 1 to the left. With that carry what was 3 on H becomes 4. This leaves the integer 140 and the remainder 59 on JK signified by the red dots.

Step 5A B C D E F G H I J K L M . . . .0 0 A 0 0 0 1 2 8 5 9 0 0 place 3 -8carry +10 0 A 0 0 0 1 4 0 5 9 0 0

Now divide the integer 140 by A

Step 6:A meets 1, 1+6, keep 1 on G and add 6 to H. Again use complimentary numbers for 10. Subtract complement 4 and add 1 to the left. This leaves the integer 20 on GH and the remainder 059 as signified by the red dots.

Step 6A B C D E F G H I J K L M . . . .0 0 A 0 0 0 1 4 0 5 9 0 0 -4carry +10 0 A 0 0 0 2 0 0 5 9 0 0

Now divide the integer 20 by A

Step 7 and the answer:A meets 2, 3+2, place 3 on G and add 2 to rod H. With 3 on rod G and remainders on HIJK the answer is 32059.Hex 7D3B isdecimal 32059

## References:

## Hannu Hinkka; a special thank you to Hannu for sharing this powerful and exciting abacus technique with members of the Yahoo Soroban/Abacus newsgroup.

## Abacus Resources:

▪ Converting Hexadecimal and base 10

▪ Add & Subtract Hexadecimal numbers## Hex to decimal Converter

▪ Hexadecimal to Decimal converter

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Abacus: Mystery of the Bead

▪Advanced Abacus Techniques## © December, 2014

Totton Heffelfinger Toronto Ontario Canada

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