Bi-quinary coded decimal

Bi-quinary coded decimal is a numeral encoding scheme used in many abacuses and in some early computers, notably the Colossus.^[2] The term bi-quinary indicates that the code comprises both a two-state (bi) and a five-state (quinary) component. The encoding resembles that used by many abacuses, with four beads indicating the five values either from 0 through 4 or from 5 through 9 and another bead indicating which of those ranges (which can alternatively be thought of as +5).

Biquinary code example^[1]

Reflected biquinary code

Japanese abacus. The right side represents 1,234,567,890 in bi-quinary: each column is one digit, with the lower beads representing "ones" and the upper beads "fives".

Several human languages, most notably Fula and Wolof also use biquinary systems. For example, the Fula word for 6, jowi e go'o, literally means five [plus] one. Roman numerals use a symbolic, rather than positional, bi-quinary base, even though Latin is completely decimal.

The Korean finger counting system Chisanbop uses a bi-quinary system, where each finger represents a one and a thumb represents a five, allowing one to count from 0 to 99 with two hands.

One advantage of one bi-quinary encoding scheme on digital computers is that it must have two bits set (one in the binary field and one in the quinary field), providing a built-in checksum to verify if the number is valid or not. (Stuck bits happened frequently with computers using mechanical relays.)

Examples

Several different representations of bi-quinary coded decimal have been used by different machines. The two-state component is encoded as one or two bits, and the five-state component is encoded using three to five bits. Some examples are:

• Roman and Chinese abacuses
• Stibitz^[3] relay calculators at Bell Labs from Model II onwards
• FACOM 128 relay calculators at Fujitsu

IBM 650

The IBM 650 uses seven bits: two bi bits (0 and 5) and five quinary bits (0, 1, 2, 3, 4), with error checking.

Exactly one bi bit and one quinary bit is set in a valid digit. The bi-quinary encoding of the internal workings of the machine are evident in the arrangement of its lights – the bi bits form the top of a T for each digit, and the quinary bits form the vertical stem.

Value	05-01234 bits[1]	IBM 650 front panel while running, with active bits just discernible Close-up of IBM 650 indicators while running, with active bits visible
0	10-10000
1	10-01000
2	10-00100
3	10-00010
4	10-00001
5	01-10000
6	01-01000
7	01-00100
8	01-00010
9	01-00001

Remington Rand 409

The Remington Rand 409 has five bits: one quinary bit (tube) for each of 1, 3, 5, and 7 - only one of these would be on at the time. The fifth bi bit represented 9 if none of the others were on; otherwise it added 1 to the value represented by the other quinary bit. The machine was sold in the two models UNIVAC 60 and UNIVAC 120.

Value	1357-9 bits
0	0000-0
1	1000-0
2	1000-1
3	0100-0
4	0100-1
5	0010-0
6	0010-1
7	0001-0
8	0001-1
9	0000-1

UNIVAC Solid State

The UNIVAC Solid State uses four bits: one bi bit (5), three binary coded quinary bits (4 2 1)^{[4][5][6][7][8][9]} and one parity check bit

Value	p-5-421 bits
0	1-0-000
1	0-0-001
2	0-0-010
3	1-0-011
4	0-0-100
5	0-1-000
6	1-1-001
7	1-1-010
8	0-1-011
9	1-1-100

UNIVAC LARC

The UNIVAC LARC has four bits:^[9] one bi bit (5), three Johnson counter-coded quinary bits and one parity check bit.

Value	p-5-qqq bits
0	1-0-000
1	0-0-001
2	1-0-011
3	0-0-111
4	1-0-110
5	0-1-000
6	1-1-001
7	0-1-011
8	1-1-111
9	0-1-110

See also

• Binary-coded decimal
• Binary number
• Chisanbop
• Finger binary
• Quinary
• Two-out-of-five code
• FACOM 128