2. Fall’ 2014 Number System CSE 101

3. 2 Number System 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. 3 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. 4 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. 1 0 1 1 0 0 1 01 0 0 1 0 0 1 01 0 0 1 0 0 1 11 1 1 1

6. 5 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 0000 000 0001111 0010222 0011333 0100444 0101555 0110666 0111777 1000 1088 10011199 10101210A 10111311B 11001412C 11011513D 11101614E 11111715F

7. 6 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 10 -1 ) + (5 x 10 -2 )  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 10 -1 ) + (f 2 x 10 -2 ) + … + (f m x 10 -m )

8. 7 Number System – Base–R to Decimal Conversion  ( 1101.101) 2 = 12 3 + 12 2 + 12 0 + 12 -1 + 02 -2 12 -3 = 8 + 4 + 1 + 0.5 + 0.125 = (13.625) 10  (572.6) 8 = 58 2 + 78 1 + 28 0 + 68 -1 = 320 + 56 + 2 + 0.75 = (378.75) 10  (2A.8) 16 = 216 1 + 1016 0 + 816 -1 = 32 + 10 + 0.5 = (42.5) 10  (341.24) 5 = 35 2 + 45 1 + 15 0 + 25 -1 + 45 -2 = 75 + 20 + 1 + 0.4 + 0.16 = (96.56) 10

9. 8 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 = 8 + 1 = 2 3 + 2 0 = (1001) 2 (18) 10 = 16 + 2 = 2 4 + 2 1 = (10010) 2 (58) 10 = 32 + 16 + 8 + 2 = 2 5 + 2 4 + 2 3 + 2 1 = (111010) 2 (0.625) 10 = 0.5 + 0.125 = 2 -1 + 2 -3 = (0.101) 2

10. 9 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

11. 10 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

12. 11 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 hxadecimal 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 (10 111 011 001. 101 110) 2 = (2731.56) 8  Octal  Binary: reverse (2731.56) 8 = (10 111 011 001. 101 110) 2  Binary  Hexadecimal: Partition in groups of 4 (101 1101 1001. 1011 1000) 2 = (5D9.B8) 16  Hexadecimal  Binary: reverse (5D9.B8) 16 = (101 1101 1001. 1011 1000) 2 Binary-Octal/Hexadecimal Conversion

13. 12 Any Question Fall’12