1. Integers’ Representation. Binary Addition. Two's Complement. Unsigned number representation Binary Addition, Subtraction. Overflow of unsigned numbers. Negative integers’ representation. Sign Magnitude. Complement number systems One’s complement Two’s complement. Addition. Subtraction. Overflow. Textbook: Ch4, pages 210 - 225. Central Connecticut State University, MIPS Tutorial. Chapter 8. Wakerly: Chapter 2. NUMBER SYSTEMS AND CODES

2. The representation of the numbers in the computers should satisfy the following requirements:  Be comfortable for keeping them in the memory (take less hardware for storing).  Be comfortable for calculations (need less hardware and act faster). Number representation requirements The operations we do with the numbers in the computers are:  Addition  Subtraction  Multiplication  Division  Comparison

3. Number representation examples We can represent at most 4 unsigned numbers with these 4 patterns. This way: 0 0 - 0 0 1 - 1 1 0 - 2 1 1 - 3 Or this way: 0 0 - 3 0 1 - 0 1 0 - 2 1 1 - 1 For transmission through the noisy channels we use some redundancy to be able to recover the information after transmitting. 0 0 0 - 0 0 0 1 - 0 0 1 0 - 0 0 1 1 - 1 1 0 0 - 0 1 0 1 - 1 1 1 0 - 1 1 1 1 - 1 We use only 2 patterns from 8 for representing 2 numbers. Then if at the receiver side we got “110” then we can assume that it was “1” when it was sent. Number of possible patterns of N bits = 2 N How to assign these patters to the numbers we want to represent ? It depends on our choice based on requirements of the task we want to implement.

4.  The representation of the decimal positive numbers in natural binary positional form in the computers is called “Unsigned” representation. Unsigned number representation 0 0 0 - 0 0 0 1 - 1 0 1 0 - 2 0 1 1 - 3 1 0 0 - 4 1 0 1 - 5 1 1 0 - 6 1 1 1 - 7 This is the unsigned representation of positive numbers 0-7 on 3 bits.

5. Binary Addition Algorithm Binary Addition Table the carry into the column | 1 1 1 1 0 0 0 0 <--- 1 0 1 0 1 0 1 0 oprnd1 1 1 0 1 1 1 0 0 oprnd2 --- --- --- --- --- --- --- --- 11 10 10 10 10 01 01 00 ^ | the carry out of the column Definition: Two bit patterns representing two integers are manipulated to create a third pattern which represents the sum of the integers. Definition: Two bit patterns representing two integers are manipulated to create a third pattern which represents the sum of the integers. To add two bits:  Count the number of ones in a column and write the result in binary.  The right bit of the result is placed under the column of bits.  The left bit is called the "carry out of the column". To add two bits:  Count the number of ones in a column and write the result in binary.  The right bit of the result is placed under the column of bits.  The left bit is called the "carry out of the column".

6. To add two N-bit (representations of) integers:  Proceed from right-to-left, column-by-column, until you reach the leftmost column.  For each column, perform 1-bit addition.  Write the carry-out of each column above the column to its left. The bit is the left column's carry-in. Binary addition of N bits An addition example with 4 bits 0 10 110 0110 0110 0110 0110 0110 0110 6 + 0111 ==> 0111 ==> 0111 ==> 0111 ==> 0111 7 ---- ---- ---- ---- ---- -- 1 01 101 1101 13 01111 110 0110 1110 = 110 10 + 0001 0111 = 23 10 --------- ----- 1000 0101 = 133 10 01111 110 0110 1110 = 110 10 + 0001 0111 = 23 10 --------- ----- 1000 0101 = 133 10

7.  Usually the operands and the result have a fixed number of bits (usually 8, 16, 32, or 64). These are the sizes that processors use to represent integers.  If the bits in the leftmost columns of the sum are zero, include them in the answer to keep the result the same bit-length as the operands.  Compute the carry-out of the leftmost column, but don't write it as part of the answer (because there is no room.) Binary addition details 1111 0111 1001 ---- 0000 0111 0001 ---- 1000 0011 0001 ---- 0100

8. Unsigned overflow  When the bit patterns are regarded as representing positive integers (unsigned binary representation), a carry-out of 1 from the leftmost column means the sum does not fit into the fixed number of bits.  This is called Overflow.  When the bit patterns are regarded as representing positive and negative integers (as described in the last pages of this chapter), then a carry-out of 1 from the leftmost column  is not necessarily overflow.  When the bit patterns are regarded as representing positive integers (unsigned binary representation), a carry-out of 1 from the leftmost column means the sum does not fit into the fixed number of bits.  This is called Overflow.  When the bit patterns are regarded as representing positive and negative integers (as described in the last pages of this chapter), then a carry-out of 1 from the leftmost column  is not necessarily overflow. 1111 0111 1001 ---- 0000 0111 0001 ---- 1000 Correct Unsigned Binary Addition The result is CORRECT only if the CARRY OUT of the high order column is ZERO.

9. Unsigned subtraction An Example with 4 bits 0 1 10 - - - 1 0 0 1 9 minuend 0 0 1 1 - 3 subtrahend ---------- --- 0 1 1 0 6difference The borrowing of the values is done from the previous columns similar to 10 based arithmetic. Just instead of 10 the 2 is borrowed (10 binary). Pros and Cons Unsigned numbers represent only positive integers Addition is simple. Subtraction is complex.