NUMBER SYSTEM AND COMPUTER CODES

Chapter 2

Prelude• Fingers, sticks, and other things for counting

were not enough!• Counting large numbers

• Count in groups

Evolution of the number system

Number systemsA set of values used to represent quantity

• Non-Positional Number Systems• count with their fingers, stones and pebbles • difficult to perform arithmetic operations • No zero, difficult to calculate large numbers• E.g. the Roman number system

• Positional Number Systems• Finite number of symbols to represent any

numbers• Symbol and it’s position defines a number• Decimal, binary, octal, hexadecimal

ASCII- American standard for Information Interchange

Base or radix• Number of unique digits

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Number Systems - Decimal• The decimal system is a base-10 system.• There are 10 distinct digits (0 to 9) to

represent any quantity. • For an n-digit number, the value that

each digit represents depends on its weight or position.

• The weights are based on powers of 10.

1024 = 1*103 + 0*102 + 2*101 + 4*100

= 1000 + 20 + 4

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Number Systems - Binary• The binary system is a base-2 system.• There are 2 distinct digits (0 and 1) to

represent any quantity. • For an n-digit number, the value of a

digit in each column depends on its position.

• The weights are based on powers of 2.

10112 = 1*23 + 0*22 + 1*21 + 1*20 =8+2+1 =1110

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Number Systems - Octal• Octal and hexadecimal systems provide

a shorthand way to deal with the long strings of 1’s and 0’s in binary.

• Octal is base-8 system using the digits 0 to 7.

• To convert to decimal, you can again use a column weighted system

• 75128 = 7*83 + 5*82 + 1*81 + 2*80 = 391410

• An octal number can easily be converted to binary by replacing each octal digit with the corresponding group of 3 binary digits 75128 = 1111010010102

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Number Systems - Hexadecimal• Hexadecimal is a base-16 system.• It contains the digits 0 to 9 and the

letters A to F (16 digit values). • The letters A to F represent the unit

values 10 to 15. • This system is often used in

programming as a condensed form for binary numbers (0x00FF, 00FFh)

• To convert to decimal, use a weighted system with powers of 16.

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Example- Value of 2001 in Binary, Octal and Hexadecimal

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Example- Conversion: Binary Octal Hexadecimal

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Converting decimal to binary, octal and hexadecimal• To convert from

decimal to a different number base such as Octal, Binary or Hexadecimal involves repeated division by that number base

• Keep dividing until the quotient is zero

• Use the remainders in reverse order as the digits of the converted number

Repeated Divide by 2

CPE1002 (c) Monash University 13

BaseN to Decimal Conversions Multiply each digit by increasing powers of the

base value and add the terms Example: 101102 = ??? (decimal)

04/03/10

Binary Addition

• Similar to decimal operation• Leading zeroes are frequently dropped.

4 Possible Binary Addition Combinations:(1) 0 (2) 0

+0 +100 01

(3) 1 (4) 1+0 +101 10

SumCarry

Ex 1,2,3

For Exam

Binary SubtractionJust like subtraction in any other base

Minuend 10110Subtrahend - 10010Difference 00100

And when a borrow is needed. Note that the borrow gives us 2 in the current bit position.

Ex 1,2

For Exam

And a full exampleAnd more ripple -

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Octal/Hex addition/subtractionOctal Addition 1 1 1 Carries 5 4 7 1 Augends + 3 7 5 4 Addend 11445 Sum

Octal Subtraction

6 10 4 10 Borrows 7 4 5 1 Minuend - 5 6 4 3 Subtrahend 1 6 0 6 Difference

Hexadecimal Addition

1 0 1 1 Carries 5 B A 9 Augend + D 0 5 8 Addend 1 2 C 0 1 Sum

Hexadecimal Subtraction

9 10 A 10 Borrows A 5 B 9 Minuend + 5 8 0 D Subtrahend 4 D A C Difference

BCDBinary-coded decimal, or BCD, is a method

of using binary digits to represent the decimal digits 0 through 9. A decimal digit is represented by four binary digits …

The binary combinations 1010 to 1111 are invalid and are not used.

ASCII Code"ask-key“- common code for

microcomputer Standard ASCII character set

• 128 decimal numbers ranging (0-127)• Assigned to letters, numbers, punctuation

marks, and the most common special characters.

The Extended ASCII Character Set • also consists of 128 decimal numbers (128-

255)• representing additional special,

mathematical, graphic, and foreign characters.

Groups of 32 characters

EBCDIC - Extended Binary Coded Decimal Interchange Code• It is an 8 bit character encoding

• Used on IBM mainframes and AS/400s. • It is descended from punched cards

• The first four bits are called the zone• category of the character

• Last four bits are the called the digit• identify the specific character

There are a number of different versions of EBCDIC, customized for different countries.

AssignmentsIOA, IA, GA, Case !@#$

Chapter 1 22

BinaryMultiplication

Division 1 1 0 1 0 Multiplicand x 1 0 1 0 Multiplier 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0

1 0 0 0 0 0 1 0 0 Product

1 0 0 1 1 1 1 1 0 11 0 0 1

1 1 0 01 0 0 1

1 1 1

1 1 0 QuotientDividend

Remainder

Divider