Review Questions, Binary Integer Arithmetic; © 1999 Charles Abzug

Review Questions:

Binary Integer Arithmetic

© 1998 Charles Abzug

Indicate the decimal value associated with each of the given bit sequences, when the bit sequence is interpreted as an integer number in each of the following representations:

Unsigned Numbers
Signed-Magnitude Numbers
Ones'-Complement Representation
Two's-Complement Representation
Saturation Arithmetic

101100111110₂
010010101100₂
110101110101₂
011100111010₂
101010101100₂
110000111010₂
111001111110₂
100110010101₂
001010011010₂
100011001100₂

Describe the algorithm for the addition of two Signed-Magnitude binary integers.

Fill in the following table, showing the specified values for integer arithmetic of 10-bit numbers:

Number Representation	Range of Numbers Represented	How many zeroes?	Carry Bit represents Carry-Out from which bit in the Sum Register?	How do you go about finding the representation of the negative of a number?
Unsigned Numbers
Signed-Magnitude
Ones'-Complement
Two's-Complement
Saturation Arithmetic

Instructions for the Next Several Problems: Add each of the following pairs of binary numbers five times, following the rules of:

Unsigned Numbers
Signed-Magnitude Numbers
Ones'-Complement Representation
Two's-Complement Representation
Saturation Arithmetic

In each case:

show the carry in for each bit;
show the bit sequence that would appear in the Sum Register;
indicate what is the decimal value represented by the bit sequence of the Augend;
indicate what is the decimal value represented by the bit sequence of the Addend;
indicate what is the decimal value represented by the bit sequence that would appear in the Sum Register;
indicate whether a 0 or a 1 would appear in the Carry Bit of the Status Register;
indicate the value (0 or 1) that would appear in the Overflow Bit of the Status Register; and
state whether or not the value of the bit sequence appearing in the Sum Register is the correct sum of the values of the numbers represented in the Augend and Addend Registers.

UNSIGNED INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SATURATION ARITHMETIC
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

UNSIGNED NUMBERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	1	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	1	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	1	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	1	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SATURATION ARITHMETIC
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	0	1	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

UNSIGNED NUMBERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SATURATION ARITHMETIC
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	0	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

UNSIGNED NUMBERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SATURATION ARITHMETIC
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	0	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

UNSIGNED NUMBERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	1	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	1	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	1	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	1	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

SATURATION ARITHMETIC
Carry/Borrow Bits:									Decimal Values:	Z bit:
Augend Register:	1	1	1	0	1	1	1	1		N bit:
Addend Register:	1	1	0	0	1	1	1	0		C bit:
Sum Register:										V bit:

Answers to Selected Questions:

Question 1:

101100111110₂

Unsigned Number: 0 + 2 + 4 + 8 + 16 + 32 + 0 + 0 + 256 + 512 + 0 + 2,048 = 2,878

Signed-Magnitude: negative sign bit, magnitude 0 + 2 + 4 + 8 + 16 + 32 + 0 + 0 + 256 + 512 + 0, thus representing -830

Ones'-Complement: a negative number, whose complement is : 010011000001₂ or 1,217. Thus, the value of the original bit sequence is -1,217.

Two's Complement: a negative number, which when first complemented and then incremented comes out to 010011000010₂ or 1,218. Thus, the value of the original bit sequence is -1,218.

Saturation Arithmetic: Same value as for Unsigned Numbers: 2,878.

010010101100₂

Since the MSB is zero, this number represents the same value in all of the representation schemes listed: 0 + 0 + 4 + 8 + 0 + 32 + 0 + 128 + 0 + 0 + 1,024 = 1,196

110101110101₂

Unsigned Number: 1 + 0 + 4 + 0 + 16 + 32 + 64 + 0 + 256 + 0 + 1,024 + 2,048 = 3,445

Signed-Magnitude: negative sign bit, magnitude 1 + 0 + 4 + 0 + 16 + 32 + 64 + 0 + 256 + 0 + 1,024, thus representing -1,397

Ones' Complement: a negative number, whose complement is 001010001010₂ or 740. Thus, the value of the original bit sequence is -650.

Two's Complement: a negative number which when first complemented and then incremented comes out to 001010001011₂ or 651. Thus, the value of the original bit sequence is -651.

Saturation Arithmetic: Same value as for Unsigned Numbers: 3,445.

011100111010₂

Since the MSB is zero, this number represents the same value in all of the representation schemes listed: 0 + 2 + 0 + 8 + 16 + 32 + 0 + 0 + 256 + 512 + 1,024 = 1,850

101010101100₂

Unsigned Number: 0 + 0 + 4 + 8 + 0 + 32 + 0 + 128 + 0 + 512 + 0 + 2,048 = 2,732

Signed-Magnitude: negative sign bit, magnitude 0 + 0 + 4 + 8 + 0 + 32 + 0 + 128 + 0 + 512 + 0, thus representing -684.

Ones' Complement: a negative number, whose complement is 010101010011₂ or 1,363. Thus, the value of the original bit sequence is -1,363.

Two's Complement: a negative number which when first complemented and then incremented comes out to 010101010100₂ or . Thus, the value of the original bit sequence is -.1,364

Saturation Arithmetic: Same value as for Unsigned Numbers: 2,732.

110000111010₂

Unsigned Number: 0 + 2 + 0 + 8 + 16 + 32 + 0 + 0 + 0 + 0 + 1,024 + 2,048 = 3,130

Signed-Magnitude: negative sign bit, magnitude 0 + 2 + 0 + 8 + 16 + 32 + 0 + 0 + 0 + 0 + 1,024, thus representing -1,082.

Ones' Complement: a negative number, whose complement is 001111000101₂ or 965. Thus, the value of the original bit sequence is -965.

Two's Complement: a negative number which when first complemented and then incremented comes out to 001111000110₂ or 966. Thus, the value of the original bit sequence is -966.

Saturation Arithmetic: Same value as for Unsigned Numbers: 3,130.

111001111110₂

Unsigned Number: 0 + 2 + 4 + 8 + 16 + 32 + 64 + 0 + 0 + 512 + 1,024 + 2,048 = 3,710

Signed-Magnitude: negative sign bit, magnitude 0 + 2 + 4 + 8 + 16 + 32 + 64 + 0 + 0 + 512 + 1,024, thus representing -1,662.

Ones' Complement: a negative number, whose complement is 000110000001₂ or 385. Thus, the value of the original bit sequence is -385.

Two's Complement: a negative number which when first complemented and then incremented comes out to 000110000010₂ or 386. Thus, the value of the original bit sequence is -386.

Saturation Arithmetic: Same value as for Unsigned Numbers: 3,710.

100110010101₂

Unsigned Number: 1 + 0 + 4 + 0 + 16 + 0 + 0 + 128 + 256 + 0 + 0 + 2,048 = 2,453

Signed-Magnitude: negative sign bit, magnitude 1 + 0 + 4 + 0 + 16 + 0 + 0 + 128 + 256 + 0 + 0, thus representing -405.

Ones' Complement: a negative number, whose complement is 011001101010₂ or 1,642. Thus, the value of the original bit sequence is -1,642.

Two's Complement: a negative number which when first complemented and then incremented comes out to 011001101011₂ or 1,643. Thus, the value of the original bit sequence is -1,643.

Saturation Arithmetic: Same value as for Unsigned Numbers: 2,453.

001010011010₂

Since the MSB is zero, this number represents the same value in all of the representation schemes listed: 0 + 2 + 0 + 8 + 16 + 0 + 0 + 128 + 0 512 = 666

100011001100₂

Unsigned Number: 0 + 0 + 4 + 8 + 0 + 0 + 64 + 128 + 0 + 0 + 0 + 2,048 = 2,252

Signed-Magnitude: negative sign bit, magnitude = 0 + 0 + 4 + 8 + 0 + 0 + 64 + 128 + 0 + 0 + 0, thus representing -204.

Ones' Complement: a negative number, whose complement is 011100110011₂ or 1,843. Thus, the value of the original bit sequence is -1,843.

Two's Complement: a negative number which when first complemented and then incremented comes out to 011100110100₂ or 1,844. Thus, the value of the original bit sequence is -1,844.

Saturation Arithmetic: Same value as for Unsigned Numbers: 2,252.

Question 4:

UNSIGNED INTEGERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	0	0	1	0	1	1	1	1	+47	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	0	1	1	1	1	1	0	1	+125	V bit:	0

Correct sum

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	0	0	1	0	1	1	1	1	+47	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	0	1	1	1	1	1	0	1	+125	V bit:	0

Correct sum

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	0	0	1	0	1	1	1	1	+47	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	0	1	1	1	1	1	0	1	+125	V bit:	0

Correct sum

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	0	0	1	0	1	1	1	1	+47	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	0	1	1	1	1	1	0	1	+125	V bit:	0

Correct sum

SATURATION ARITHMETIC
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	0	0	1	0	1	1	1	1	+47	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	0	1	1	1	1	1	0	1	+125	V bit:	0

Correct sum

Question 5:

UNSIGNED NUMBERS
Carry/Borrow Bits:	1	0	0	1	1	1	0	1	Decimal Values:	Z bit:	0
Augend Register:	0	1	1	0	1	1	1	1	+111	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	1	0	1	1	1	1	0	1	+189	V bit:	0

Correct sum

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:	X	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	0	1	1	0	1	1	1	1	+111	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	1
Sum Register:	0	0	1	1	1	1	0	1	+61	V bit:	1

Not the correct sum (Overflow)

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:	1	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	0	1	1	0	1	1	1	1	+111	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	1	0	1	1	1	1	0	1	-66	V bit:	1

Not the correct sum (Overflow)

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:	1	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	0	1	1	0	1	1	1	1	+111	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	1	0	1	1	1	1	0	1	+61	V bit:	1

Not the correct sum (Overflow)

SATURATION ARITHMETIC
Carry/Borrow Bits:	1	0	0	1	1	1	0	1	Decimal Values:	Z bit:	0
Augend Register:	0	1	1	0	1	1	1	1	+111	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	1	0	1	1	1	1	0	1	+189	V bit:	0

Correct sum

Question 6:

UNSIGNED NUMBERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	1	Decimal Values:	Z bit:	0
Augend Register:	1	0	1	0	1	1	1	1	+175	N bit:	0
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	1	1	1	1	1	1	0	1	+253	V bit:	0

Correct sum

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:	X	1	1	1	1	1	1	O	Decimal Values:	Z bit:	0
Augend Register:	1	0	1	0	1	1	1	1	-47	N bit:	1
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	0	0	0	1	1	1	1	1	+31	V bit:	0

Correct sum

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	1	0	1	0	1	1	1	1	-80	N bit:	1
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	1	1	1	1	1	1	0	1	-2	V bit:	0

Correct sum

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	1	Decimal Values:	Z bit:	0
Augend Register:	1	0	1	0	1	1	1	1	-81	N bit:	1
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	0
Sum Register:	1	1	1	1	1	1	0	1	-3	V bit:	0

Correct sum

SATURATION ARITHMETIC
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	X
Augend Register:	1	0	1	0	1	1	1	1	+175	N bit:	X
Addend Register:	0	1	0	0	1	1	1	0	+78	C bit:	X
Sum Register:	1	1	1	1	1	1	0	1	+253	V bit:	X

Correct sum

Question 7:

UNSIGNED NUMBERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	1	0	1	0	1	1	1	1	+175	N bit:	0
Addend Register:	1	1	0	0	1	1	1	0	+206	C bit:	1
Sum Register:	0	1	1	1	1	1	0	1	+125	V bit:	1

Not the correct sum (Overflow)

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:	X	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	1	0	1	0	1	1	1	1	-47	N bit:	1
Addend Register:	1	1	0	0	1	1	1	0	-78	C bit:	0
Sum Register:	1	1	1	1	1	1	0	1	-125	V bit:	0

Correct sum

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	1	0	1	0	1	1	1	1	-80	N bit:	0
Addend Register:	1	1	0	0	1	1	1	0	-49	C bit:	1
Sum Register:	0	1	1	1	1	1	0	1	+125	V bit:	1

Not the correct sum (Overflow)

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	1	0	1	0	1	1	1	1	-81	N bit:	0
Addend Register:	1	1	0	0	1	1	1	0	-50	C bit:	1
Sum Register:	0	1	1	1	1	1	0	1	+125	V bit:	1

Not the correct sum (Overflow)

SATURATION ARITHMETIC
Carry/Borrow Bits:	0	0	0	1	1	1	0	0	Decimal Values:	Z bit:	X
Augend Register:	1	0	1	0	1	1	1	1	+175	N bit:	X
Addend Register:	1	1	0	0	1	1	1	0	+206	C bit:	X
Sum Register:	1	1	1	1	1	1	1	1	+255	V bit:	X

Correct sum, in accordance with the rules of Saturation Arithmetic!

Question 8:

UNSIGNED NUMBERS
Carry/Borrow Bits:	1	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	1	1	1	0	1	1	1	1	+239	N bit:	0
Addend Register:	1	1	0	0	1	1	1	0	+206	C bit:	1
Sum Register:	1	0	1	1	1	1	0	1	+189	V bit:	1

Not the correct sum

SIGNED-MAGNITUDE INTEGERS
Carry/Borrow Bits:	1	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	1	1	1	0	1	1	1	1	-111	N bit:	1
Addend Register:	1	1	0	0	1	1	1	0	-78	C bit:	1
Sum Register:	1	0	1	1	1	1	0	1	-61	V bit:	1

Not the correct sum

ONES'-COMPLEMENT INTEGERS
Carry/Borrow Bits:	1	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	1	1	1	0	1	1	1	1	-16	N bit:	1
Addend Register:	1	1	0	0	1	1	1	0	-49	C bit:	1
Sum Register:	1	0	1	1	1	1	0	1	-66	V bit:	0

An additional operation (End-Around Carry) needs to be performed before the value in the sum register becomes equal to the true sum of the numbers (Ones'-Complement arithmetic).

TWO'S-COMPLEMENT INTEGERS
Carry/Borrow Bits:	1	0	0	1	1	1	0	0	Decimal Values:	Z bit:	0
Augend Register:	1	1	1	0	1	1	1	1	-17	N bit:	1
Addend Register:	1	1	0	0	1	1	1	0	-50	C bit:	1
Sum Register:	1	0	1	1	1	1	0	1	-67	V bit:	0

Correct sum

SATURATION ARITHMETIC
Carry/Borrow Bits:	1	0	0	1	1	1	0	0	Decimal Values:	Z bit:	X
Augend Register:	1	1	1	0	1	1	1	1	+239	N bit:	X
Addend Register:	1	1	0	0	1	1	1	0	+206	C bit:	X
Sum Register:	1	1	1	1	1	1	1	1	+255	V bit:	X

Correct sum, in accordance with the rules of Saturation Arithmetic!

Last revised 03 Jan 2000