chapter2.ppt - Chapter 2 Number system and Data representation 1 Learning outcomes By the end of this Chapter you will be able to \u2022 Explain how

chapter2.ppt - Chapter 2 Number system and Data...

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1 Chapter 2 Number system and Data representation
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2 Learning outcomes By the end of this Chapter you will be able to: Explain how integers are represented in computers using: Unsigned, signed magnitude, one’s complement and two’s complement notations Explain how fractional numbers are represented in computers Fixed point and Floating point notation Explain how characters are represented in computers E.g. using ASCII and Unicode Explain how colours, images, sound and movies are represented
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3 2.1 introduction Binary number is simply a number comprised of only 0's and 1's. Computers use binary numbers because it's easy for them to communicate using electrical current -- 0 is off, 1 is on. A bit is the most basic unit of information in a computer. It is a state of “on” or “off” in a digital circuit. Sometimes they represent high or low voltage A byte is a group of eight bits.. It is the smallest possible addressable unit of computer storage A word is the number of bits (word length) whichcan be processed by a computer in a single step (e.g., 16,32 or 64) The word size in any given computer is fixed. Example : 16-bit word every word (memory location) can hold a 16-bit pattern, with each bit either 0 or 1.
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2.2 Number systems Base or Radix r system : uses distinct symbols for r digits Most common number system :Decimal, Binary, Octal, Hexadecimal Decimal System/Base-10 System Composed of 10 symbols or numerals(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0) Binary System/Base-2 System Composed of 10 symbols or numerals(0, 1) Bit = Binary digit Hexadecimal System/Base-16 System : Composed of 16 symbols or numerals(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F) Positional-value(weight) System : r 2 r 1 r 0 .r -1 r -2 r -3 Multiply each digit by an integer power of r and then form the sum of all weighted digits 4
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2.3 Conversion of number system Binary-to-Decimal Conversions 1011.101 2 = (1 x 2 3 ) + (0 x 2 2 )+ (1 x 2 1 ) + (1 x 2 o ) + (1 x 2 -1 ) + (0 x 2 -2 ) + (1 x 2 -3 ) = 8 10 + 0 + 2 10 + 1 10 + 0.5 10 + 0 + 0.125 10 = 11.625 10 Decimal-to-Binary Conversions integer use Repeated division 37 / 2 = 18 remainder 1 (binary number will end with 1) : LSB 18 / 2 = 9 remainder 0 9 / 2 = 4 remainder 1 4 / 2 = 2 remainder 0 2 / 2 = 1 remainder 0 1 / 2 = 0 remainder 1 (binary number will start with 1) : MSB Read the result upward to give an answer of 37 10 = 100101 2 5 Fraction use multiplication 0.375 x 2 = 0.750 integer 0 MSB 0.750 x 2 = 1.500 integer 1 . 0.500 x 2 = 1.000 integer 1 LSB Read the result downward .375 10 = .011 2
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6 Hex-to-Decimal Conversion 2AF 16 = (2 x 16 2 ) + (10 x 16 1 ) + (15 x 16 o ) = 512 10 + 160 10 + 15 10 = 687 10 Decimal-to-Hex Conversion 423 10 / 16 = 26 remainder 7 (Hex number will end with 7) : LSB 26 10 / 16 = 1 remainder 10 1 10 / 16 = 0 remainder 1 (Hex number will start with 1) : MSB Read the result upward to give an answer of 423 10 = 1A7 16 Hex-to-Binary Conversion 9F2 16 = 9 F 2 = 1001 1111 0010 = 100111110010 2 Table 3-2 Hex Binary Decimal 0 0000 0 1 0001 1 2 0010 2 3 0011 3 4 0100 4 5 0101 5 6 0110 6 7 0111 7 8 1000 8 9 1001 9 A 1010 10 B 1011 11 C 1100 12 D 1101 13 E 1110 14 F 1111 15 Binary-to-Hex Conversion 1 1 1 0 1 0 0 1 1 0 2 = 0 0 0 0 1 1 1 0 1 0 0 1 1 0 3 A 6 = 3A6 16
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7 2.4 Number representation Representing whole numbers(integer ) Representing fractional numbers
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8 2.4.1 Integer Representations Unsigned notation Signed magnitude notion One’s complement notation Two’s complement notation.
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