Introduction to Computer Systems and Software Lecture 2 of 2 Simon Coupland

Representation of Data Within the Computer Contents:  Decimal and Binary Integer Numbers  Binary Addition  Signed Binary Numbers  Overflow  Hexadecimal Numbers  Number Conversion  Real Numbers  Character Encoding

Introduction A few terms:  A bit – a single Binary digIT, 0 or 1  A byte – eight bits  A word – one or more bytes  Integer – whole number  Real number – a number with decimal points  Binary – Base 2 numbers  Octal – Base 8 numbers  Decimal – Base 10 numbers (everyday numbers)  Hexadecimal – Base 16 numbers

Decimal and Binary Numbers We all use decimal numbers Base 10 numbers Example: 124 Digit Digit Value Digit Value10M1M100k10k1k Example

Decimal and Binary Numbers Computers use binary numbers Base 2 numbers: Example: 124 Bit Bit Value Example

Decimal and Binary Numbers More binary numbers: = = = = = = 0

Binary Addition When adding binary numbers we use binary logic Binary Addition Truth Table 1: ABA + BCarry

Binary Addition Binary Addition Truth Table 2: ABCarry (in)A + BCarry (out)

Binary Addition Binary Addition Example: =

Binary Addition Binary Addition Example: = 0 1

Binary Addition Binary Addition Example: = 10

Binary Addition Binary Addition Example: = 110

Binary Addition Binary Addition Example: = 1110

Binary Addition Binary Addition Example: =

Binary Addition Binary Addition Example: =

Binary Addition Binary Addition Example: =

Binary Addition Binary Addition Example: =

Binary Addition Binary Addition Q1: =

Binary Addition Binary Addition Q1: =

Binary Addition Binary Addition Q2: =

Binary Addition Binary Addition Q2: =

Binary Addition Binary Addition Q3: =

Binary Addition Binary Addition Q3: =

Negative Binary Numbers Sign-true Magnitude Left most bit holds sign Example: -10 Bit Bit Valuesign Bit Value+/ Example

Negative Binary Numbers Ones complement All 1’s and 0’s are switched When negative, result = value Example: -10 Bit Bit Value Example

Negative Binary Numbers Twos complement Left most bit holds sign When negative, result = result Example: -10 Bit Bit ValueSign Bit Value+/ Example

Negative Binary Numbers Conversion to Twos complement  Ones complement the byte/word  Add 1 Example: Ones complement =

Why Use Twos Complement? Because addition rules still work: = = -98

Overflow Overflow is when the number of bits is too small to store the result of an arithmetic operation Example (twos complement) : = = -45

Overflow Overflow can be easily detected for signed binary numbers Errors can then be corrected Adding two positive numbers should give a positive result Adding two negative numbers should give a negative result Adding a positive and negative number together can never result in overflow. Why?

Hexadecimal Numbers Writing code with long binary numbers would be cumbersome and error prone A hexadecimal digit can take 16 values (0-F) One hex digit can represent a four bit word Examples: DecimalHexadecimalBinary A A

Number Conversion Binary to Hexadecimal:  From the least significant (rightmost) bit split the binary number into groups of four bits.  Each 4 bits has a hexadecimal equivalent  Example: E

Number Conversion Hexadecimal to Binary:  Convert each hexadecimal digit into its 4 bit binary equivalent  Join the all 4 bit words together  Example: A42F

Number Conversion Question Hexadecimal to Binary:  Convert D36B into binary

Number Conversion Question Hexadecimal to Binary:  Convert D36B into binary D 3 6 B

Number Conversion Question Binary to Hexadecimal:  Convert into hexadecimal

Number Conversion Question Binary to Hexadecimal:  Convert into hexadecimal A 8 A

Number Conversion Decimal to Binary  Use the following algorithm, begin with LSB int dec_value = some_number; int next_bit; while(dec_value > 0) { next_bit = dec_value % 2; dec_value = dec_value / 2; }

Number Conversion Decimal to Binary Convert 42 to binary: dec_value/2%2Result

Number Conversion Question Convert 39 to binary:

Number Conversion Question Convert 39 to binary: dec_value/2%2Result

Number Conversion Binary to Decimal  Use the following algorithm, begin with MSB int dec_value = 0; int bit_value; int bit_index = MSB_index; while(bit_index >= 0) { bit_value = word[bit_index]; dec_value = dec_value * 2 + bit_value; bit_index--; }

Number Conversion Binary to Decimal  Convert to decimal: bit_indexbit_valuedec_value

Number Conversion Binary to Decimal  Question, convert to decimal:

Number Conversion Binary to Decimal  Question, convert to decimal: bit_indexbit_valuedec_value

Number Conversion Decimal to Hexadecimal  Use the following algorithm, begin with LSB int dec_value = some_number; char hex_digit; while(dec_value > 0) { hex_digit = to_hex(dec_value % 16); dec_value = dec_value / 16; }

Number Conversion Decimal to Hexadecimal  Convert 1863 to hexadecimal dec_valuedec_value / 16dec_value % 16hex_digit

Number Conversion Decimal to Hexadecimal  Question, convert 1437 to hexadecimal

Number Conversion Decimal to Hexadecimal  Question, convert 1437 to hexadecimal dec_valuedec_value / 16dec_value % 16hex_digit D

Number Conversion Hexadecimal to Decimal  Use the following algorithm, begin with MSB int dec_value = 0; char digit_value; while(!word.off) { digit_value = to_dec(hex_digit); dec_value = dec_value * 16 + digit_value; }

Number Conversion Hexadecimal to Decimal  Convert F2AC to decimal Indexhex_valuedec_valueresult 3F A C

Number Conversion Hexadecimal to Decimal  Convert E59A to decimal

Number Conversion Hexadecimal to Decimal  Convert E59A to decimal Indexhex_valuedec_valueresult 3E A

Real Numbers A number n is a real number if n % 1 != 0 Take a lot of processing power/time Often math co-processors are used to perform arithmetic on these numbers

Fixed Point Real Numbers Similar to integer representation 8 bit example: Bit value-/ value-/

Fixed Point Real Numbers Maximum number is small Accuracy is limited Not used because of these reasons

Floating Point Real Numbers Floating point numbers are held in two parts  Mantissa  Exponent In   10 4 The mantissa is The exponent is 4

Floating Point Real Numbers Example (not IEEE float): mantissaexponent Mantissa = 809 Exponent = 

Character Encoding ASCII EBCDIC UNICODE ISO

Character Encoding ASCII  American Standard Code for Information Interchange  Characters are represented by integers  Table in your notes A = 65 B = 66 a = 96 z = = 64

Character Encoding ASCII  Special functions characters are also encoded  Examples: Carriage return = 13 Escape = 27 Space = 32

Recap Binary numbers can be signed of unsigned Twos complement gives simple addition Overflow - when the result of a arithmetic operation is too large to be represented Hexadecimal numbers – base 16 Real numbers use floating point Characters are represented as intergers