COMPUTER ARITHMETIC
A basic operation in all digital computers is the addition or subtraction of two numbers. Arithmetic operations occur at the machine instruction level. They are implemented, along with basic logic functions such as AND,OR, NOT, and exclusive –OR (XOR), in the arithmetic logic unit(ALU) subsystem of the processor. We use logic Circuits to implement arithmetic operations, The time needed to perform and addition operation affects the processor’s performance. Multiply and divide operations, which require more complex circuitry than either addition or subtraction operations, also affect performance. We present some of the techniques used in modern computers to perform arithmetic operations at high speed.
Compared with arithmetic operations, logic operations are simple to implement using combinational circuitry. They require only independent Boolean operations on individual bit positions of the operands, whereas carry/borrow lateral signals are required in arithmetic operations.
ADDITION AND SUBTRACTION OF SIGNED NUMBERS
There will arise instances when we need to express numbers that are less than zero. These numbers are called signed numbers and consist of positive(+) and
negative(-) numbers. Positive numbers are greater than zero and negative numbers are less than zero.
Positive and negative whole numbers are called integers while signed fractions and decimals are called rational numbers. It is not necessary to write the + for a positive number unless you want to draw attention to the fact that it is positive. The negative sign must always in used for a negative number.
A number line for integers continues indefinitely in both the negative and positive directions. Numbers get smaller as we proceed to the left and larger to the right. The opposite of a number is the number the same distance from zero but in the opposite direction.
In order to perform operations with signed numbers, we need to define the absolute value of a number. The absolute value of a number, symbolized by placing the number between 2 vertical bars (l l) is defined to be the distance that number is located from zero on a number line without regard to the direction.
• |3| = 3
• |-5| = 5
When you add two numbers with the same signs add the absolute values, and write the sum (the answer) with the sign of the numbers. If the sign is positive, it is commonly omitted.
5 + 16 = 21
-12 + -15 = -27
Binary Addition Basic Rules for Binary Addition
0+0 = 0 0 plus 0 equals 0
0+1 = 1 0 plus 1 equals 1
1+0 = 1 1 plus 0 equals 1
1+1 = 10 1 plus 1 equals 0
with a carry of 1 (binary 2)
The technique of addition for binary numbers is similar to that for decimal numbers, except that a 1 is carried to the next column after two 1s are added.
Example: Add the numbers 310 and 110 in binary form.
Solution
The numbers, in binary form, are 11 and 01.
11
01
100
In the right-hand column, 1 + 1 = 0 with a carry of 1 to the next column.
In the next column, 1+0+1 = 0 with a carry of 1 to the next column.
In the left-hand column, 1 + 0 + 0 = 1. Thus, in binary, 11 + 01 = 100 = 410.
Binary Subtraction Basic Rules for Binary Subtraction
0 − 0 = 0 0 minus 0 equals 0
1 − 1 = 0 1 minus 1 equals 0
1 − 0 = 1 1 minus 0 equals 1
10 − 1 = 1 10 minus 1 equals 1
Example: Subtract 310 = 11 from 510 = 101 in binary form.
Solution
The subtraction procedure is shown below.
1 0 1
− 0 1 1
0
1 0 1
− 0 1 1
0
1 0 1
− 0 1 1
1 0
1 0 1
− 0 1 1
0 1 0
Starting from the left, the first array is the subtraction in the right hand column. In the second array, a 1 is borrowed from the third column for the middle column at the top and paid back at the bottom of the third column. The third array is the subtraction 10 − 1 = 1 in the middle column. The final array is the subtraction 1 − 1 = 0 and the final answer is thus 10 = 210.
