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
bin x y
bout d 1

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

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 = = 6476
0 and positive numbers:
Negative numbers: , 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.