Wakerly Section 2.4 and further

1. Wakerly Section 2.4 and further Addition and Subtraction of Nondecimal Numbers

2. Addition and Subtraction • Use same technique as decimal • Except that the addition and subtraction tables are different • Already seen addition table • Truth table for Sum and Cout function

3. Subtraction table

4. Examples • 191+141 (Let’s first convert these to binary as an exercise.) • 210-109

5. Addition and Subtraction of Octal and Hexadecimal Numbers • Not really too different • But the addition and subtraction tables must be developed.

6. Section 2.5: Rep. of Negative Numbers • More accurately: representation of signed numbers • Signed-magnitude representation • Radix-complement representation • 2’s-complement representation • Diminished radix-complement representation • Ones’ complement representation • Excess representations

7. Signed-magnitude representation • Also called, “sign-and-magnitude representation” • A number consists of a magnitude and a symbol representing the sign • Usually 0 means positive, 1 negative • Sign bit • Usually the entire number is represented with 1 sign bit to the left, followed by a number of magnitude bits

8. Machine arithmetic with signed-magnitude representation • Takes several steps to add a pair of numbers • Examine signs of the addends • If same, add magnitudes and give the result the same sign as the operands • If different, must… • Compare magnitude of the two operands • Subtract smaller number from larger • Give the result the sign of the larger operand • For this reason the signed-mag rep is not as popular as one might think because of its “naturalness”

9. Complement number systems • Negates a number by taking its complement instead of negating the sign • Exact meaning of taking its complement is defined in various ways – will see • Not natural for humans, but better for machine arithmetic • Will describe 2 complement number systems • Radix complement – very popular in real computers • Diminished radix-complement – not very useful, may skip or not spend much time on it

10. Radix-complement number representation • Must first decide how many bits to represent the number – say n. • Complement of a number = rn – number • Example: 4-bit decimal: • Original number = 3524 • 10’s complement = 10000-3524 = 6476 • 0 and positive numbers: 0000-4999 • Negative numbers: 5000-9999, where 9999 is ‘minus 1.’

11. Two’s-complement representation • Just radix-complement when radix = 2 • Used a lot in computers and other digital arithmetic circuits • 0 and positive numbers: leftmost bit = 0 • Negative numbers: leftmost bit = 1 • To find a number’s complement – just flip all the bits and add 1 • See graphical view – Fig. 2.3, p. 40.

12. Two’s-Comp Addition and Subtraction Rules • Starting from 1000 (-8) on up, each successive 2’s comp number all the way to 0111 (+7) can be obtained by adding 1 to the previous one, ignoring any carries beyond the 4th bit position • Since addition is just an extension of ordinary counting, 2’s comp numbers can be added by ordinary binary addition! • No different cases based on operands’ signs! • Overflow possible • Occurs if result is out of range • To detect – happens if operands are the same sign but sum is a different sign of that of the operands

13. Binary multiplication • Grammar school method for decimal: add a list of shifted multiplicands computed according to the digits of the multiplier • Same method can be used in binary • For two unsigned operands, the only possible values of the multiplier digits are 0 and 1 • Thus it’s trivial to form the shifted multiplicands

14. Binary multiplication in binary on a machine • More convenient to add each shifted multiplicand as it is created to a partial product • Will do an example. • In general when we multiply an n-bit number by an m-bit number, the result requires at most n+m bits to express • The shift-and-add algorithm requires m partial products and additions to obtain result, but the 1st addition is trivial (adding to 0)

15. Binary code for decimal numbers • Any encoding needs at least 4 bits/decimal digit • BCD (8421), a weighted code • Packed BCD • 2421 code • Self-complementing: the code for the 9s’ comp of any digit may be obtained by complementing the individual bits of the digit’s code word • Excess 3 • Not a weighted code, but is also self-complementing • Since code follows standard binary counting sequence, standard binary counters can easily be made to count in excess-3

16. Biquinary code • Uses more than 4 bits • First 2 bits indicate whether the number is in the range 0-4 or 5-0 • One-hot • Last 5 bits indicate which of the five numbers in the selected range is represented • Also one-hot • Advantage: error-detection property. If any 1 bit in a code word is accidentally changed to the opposite value, the resulting code word doesn’t represent a decimal digit at all – flagged as error.