Integer Representation and Binary Arithmetic

Integer Representation

Bit-- Binary Digit

1 byte = 8 bits
1 word = N bytes, take N to be 2 (e.g., 16 bit machine)

Integer takes up 2 bytes; can be signed or unsigned.

Unsigned Integers

Can represent whole numbers from 0 to 65,535

(0 to 2¹⁶ - 1).

In binary, this is from

0_²to 1111111111111111^₂

Internally, binary representation of decimal value as 16 bits.

Signed Integers

Need to reserve one bit for the sign.
Three ways:

Sign-Magnitude
1's Complement
2's Complement

Sign-Magnitude

Uses most significant bit of the word to represent the sign.

0 - Positive
1 - Negative.

Rest of the number is encoded in magnitude part

 37 = 00000000 00100101 
-37 = 10000000 00100101

 6712 = 00011010 00111000 
-6712 = 10011010 00111000

Can represent numbers from -32,767 to 32,767.

-2¹⁵+1 .. 2¹⁵-1

But, two representations for zero:

 0 = 00000000 00000000
-0 = 10000000 00000000

Arithmetic can be cumbersome.

1's Complement

Negative number is stored as bit-wise complement of corresponding positive number.
Leftmost bit of positive number is 0. That of negative number is 1.

 196 = 00000000 11000100
-196 = 11111111 00111011

Can represent numbers from -32,767 to 32,767.

-2¹⁵+1 .. 2¹⁵-1

Arithmetic is easier than sign-magnitude.
But, still have two representations for zero:

 0 = 00000000 00000000
-0 = 11111111 11111111

2's Complement

Modern Method
Positive number represented in same way as other two methods
Negative number obtained by taking 1's Complement of positive number and adding 1.

    6713 = 00011000 00011101 
1's Comp = 11100111 11100010 
2's Comp = 11100111 11100011

Word integer can represent numbers from -32,768 to 32,767.

-2¹⁵ .. 2¹⁵-1

Byte integer can represent numbers from -128 to 127.

-2⁷.. 2⁷-1

One version of zero:

00000000 00000000

Conversion of Byte Integer to Word

Sign Extension
Copy sign bit of the byte into all the bits of the upper byte of the word.

 37 = 00100101 -> 00000000 00100101 
-37 = 11011011 -> 11111111 11011011

cbw

converts the signed byte in AL to a word in AX

Conversion of Word Integer to Byte

Remove upper byte of word. Retain only the lower byte.
Meaningful only if original number can be represented by a byte.

Integer Arithmetic (1's Comp and 2's Comp)

Addition: Simply add the two binary representations.
Subtraction: Find negative of one number, add to the second.

Addition in 1's Comp

Add binary representations of the two numbers.
If there is a carry, add it back in on the right side.

    51    00110011
+ (-37) + 11011010
        ----------
        1 00001101
               + 1
        ----------
    14    00001110

Addition in 2's Comp

Add binary representations of the two numbers.
Disregard the carry.

    51    00110011
+ (-37) + 11011011
         ----------
    14  1 00001110

Subtraction in 2's Comp

Overflow

If two numbers have different signs, their sum will never overflow.
If they have the same sign, they might overflow.
Overflow has occurred if sign of result is different than sign of addends.