Fall’ 2014 Lesson - 1 Number System & Program Design CSE 101

2 This Lesson Includes Following section Number System

3 How Computers Represent Data Binary Numbers The Binary Number System Bits and Bytes Text Codes Binary Number Computer processing is performed by transistors, which are switches with only two possible states: on and off. All computer data is converted to a series of binary numbers– 1 and 0. For example, you see a sentence as a collection of letters, but the computer sees each letter as a collection of 1s and 0s. If a transistor is assigned a value of 1, it is on. If it has a value of 0, it is off. A computer's transistors can be switched on and off millions of times each second.

4 Number System The Binary Number System To convert data into strings of numbers, computers use the binary number system. Humans use the decimal system (“deci” stands for “ten”). Elementory storage units inside computer are electronic switches. Each switch holds one of two states: on (1) or off (0). We use a bit (binary digit), 0 or 1, to represent the state. ON OFF 0 (00) 1 (01) 2 (10) 3 (11) The binary number system works the same way as the decimal system, but has only two available symbols (0 and 1) rather than ten (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9).

5 Number System Bits and Bytes A single unit of data is called a bit, having a value of 1 or 0. Computers work with collections of bits, grouping them to represent larger pieces of data, such as letters of the alphabet. Eight bits make up one byte. A byte is the amount of memory needed to store one alphanumeric character. With one byte, the computer can represent one of 256 different symbols or characters

6 Number System Text Codes A text code is a system that uses binary numbers (1s and 0s) to represent characters understood by humans (letters and numerals). An early text code system, called EBCDIC (Extended Binary Coded Decimal Interchange Code), uses eight-bit codes, but is used primarily in older mainframe systems. In the most common text-code set, ASCII (American Standard Code for Information Interchange), each character consists of eight bits (one byte) of data. ASCII is used in nearly all personal computers. In the Unicode text-code set, each character consists of 16 bits (two bytes) of data CodeCharacter A B C D E

7 Number System  In general, N bits can represent 2 N different values.  For M values, bits are needed. 1 bit  represents up to 2 values (0 or 1) 2 bits  rep. up to 4 values (00, 01, 10 or 11) 3 bits  rep. up to 8 values (000, 001, 010. …, 110, 111) 4 bits  rep. up to 16 values (0000, 0001, 0010, …, 1111) 32 values  requires 5 bits 64 values  requires 6 bits 1024 values  requires 10 bits 40 values  requires 6 bits 100 values  requires 7 bits  Decimal number system, symbols = { 0, 1, 2, 3, …, 9 }  Position is important  Example:(7594) 10 = (7x10 3 ) + (5x10 2 ) + (9x10 1 ) + (4x10 0 )  In general, (a n a n-1 … a 0 ) 10 = (a n x 10 n ) + (a n-1 x 10 n-1 ) + … + (a 0 x 10 0 )  (2.75) 10 = (2 x 10 0 ) + (7 x ) + (5 x )  In general, (a n a n-1 … a 0. f 1 f 2 … f m ) 10 = (a n x 10 n ) + (a n-1 x10 n-1 ) + … + (a 0 x 10 0 ) + (f 1 x ) + (f 2 x ) + … + (f m x 10 -m )

8 Other Number System  Binary (base 2): weights in powers-of-2. Binary digits (bits): 0,1.  Octal (base 8): weights in powers-of-8. Octal digits: 0,1,2,3,4,5,6,7  Hexadecimal (base 16): weights in powers-of-16. Hexadecimal digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F BinaryOctalDecimalHexadecimal A B C D E F

9 Number System – Base–R to Decimal Conversion  ( ) 2 = 1    2 -2 12 -3 = = (13.625) 10  (572.6) 8 = 5   8 -1 = = (378.75) 10  (2A.8) 16 = 2  16 -1 = = (42.5) 10  (341.24) 5 = 3    5 -2 = = (96.56) 10

10 Number System – Decimal to Binary Conversion  Method 1: Sum-of-Weights Method  Method 2:  Repeated Division-by-2 Method (for whole numbers)  Repeated Multiplication-by-2 Method (for fractions) Sum-of-Weights Method  Determine the set of binary weights whose sum is equal to the decimal number. (9) 10 = = = (1001) 2 (18) 10 = = = (10010) 2 (58) 10 = = = (111010) 2 (0.625) 10 = = = (0.101) 2

11 Number System – Decimal to Binary Conversion To convert a whole number to binary, use successive division by 2 until the quotient is 0. The remainders form the answer, with the first remainder as the least significant bit (LSB) and the last as the most significant bit (MSB). (43) 10 = (101011) 2 Repeated Multiplication-by-2 Method (for fractions) Repeated Division-by-2 Method (for whole number) To convert decimal fractions to binary, repeated multiplication by 2 is used, until the fractional product is 0 (or until the desired number of decimal places). The carried digits, or carries, produce the answer, with the first carry as the MSB, and the last as the LSB. (0.3125) 10 = (.0101) 2

12 Number System - Conversion between Decimal to other Base  Decimal to base-R  whole numbers: repeated division-by-R  fractions: repeated multiplication-by-R In general, conversion between bases can be done via decimal: Base-2Base-3 Base-4DecimalBase-4 … ….Base-R

13 Number System - Conversion between Decimal to other Base Octal and Hexadecimal Numbers The conversion of binary, octal and hexadecimal plays an important part in digital computers. Each octal digit corresponds to three digits and each hexadecimal digit corresponds to four binary digits. The conversion from binary to octal is easily accomplished by partitioning the binary number into groups of three each, starting from the binary point and proceeding to the left and to the right. Conversion from binary to hexadecimal is similar. Conversion from the octal or hexadecimal to binary is done by procedure reverse to the above.  Binary  Octal: Partition in groups of 3 ( ) 2 = ( ) 8  Octal  Binary: reverse ( ) 8 = ( ) 2  Binary  Hexadecimal: Partition in groups of 4 ( ) 2 = (5D9.B8) 16  Hexadecimal  Binary: reverse (5D9.B8) 16 = ( ) 2 Binary-Octal/Hexadecimal Conversion

14 Any Question Fall’12