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When the value of the base is substituted… -2’s and 1’s complement for binary numbers -10’s and 9’s complement for decimal numbers
Lesson 1 basic theory of information
Lesson 1Basic Theory of Information
Number System Number It is a symbol representing a unit or quantity. Number System Defines a set of symbols used to represent quantity Radix The base or radix of number system determines how many numerical digits the number system uses.
Types of Number System Decimal System Binary Number System Octal Number System Hexadecimal Number System
Decimal Number System Ingenious method of expressing all numbers by means of tens symbols originated from India. It is widely used and is based on the ten fingers of a human being. It makes use of ten numeric symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Inherent Value and Positional Value The inherent value of a symbol is the value of that symbol standing alone. Example 6 in number 256, 165, 698 The symbol is related to the quantity six, even if it is used in different number positions The positional value of a numeric symbol is directly related to the base of a system. In the case of decimal system, each position has a value of 10 times greater that the position to its right. Example: 423, the symbol 3 represents the ones (units), the symbol 2 represents the tens position (10 x 1), and the symbol 4 represents the hundreds position (10 x 10). In other words, each symbol move to the left represents an increase in the value of the position by a factor of ten.
Inherent and Positional Value cont.2539 = 2X1000 + 5X100 + 3X10 + 9X1 = 2X103 + 5X102 + 3X101 + 9 x100This means that positional value of symbol 2 is 1000 or using the base 10 it is 103
Binary Number System Uses only two numeric symbols 1 and 0 Under the binary system, each position has a value 2 times greater than the position to the right.
Octal Number System Octalnumber system is using 8 digits to represent numbers. The highest value = 7. Each column represents a power of 8. Octal numbers are represented with the suffix 8.
Hexadecimal Number System Provides another convenient and simple method for expressing values represented by binary numerals. It uses a base, or radix, of 16 and the place values are the powers of 16.
Decimal Binary Hexadecimal Decimal Binary Hexadecimal 0 0000 0 8 1000 8 1 0001 1 9 1001 9 2 0010 2 10 1010 A 3 0011 3 11 1011 B 4 0100 4 12 1100 C 5 0101 5 13 1101 D 6 0110 6 14 1110 E 7 0111 7 15 1111 F
Radix Conversion The process of converting a base to another. To convert a decimal number to any other number system, divide the decimal number by the base of the destination number system. Repeat the process until the quotient becomes zero. And note down the remainders in the reverse order. To convert from any other number system to decimal, take the positional value, multiply by the digit and add.
Decimal to Binary Conversionof Fractions Division – Multiplication Method Steps to be followed Multiply the decimal fraction by 2 and noting the integral part of the product Continue to multiply by 2 as long as the resulting product is not equal to zero. When the obtained product is equal to zero, the binary of the number consists of the integral part listed from top to bottom in the order they were recorded.
Example 1: Convert 0.375 to its binary equivalent Multiplication Product Integral part 0.375 x 2 0.75 0 0.75 x 2 1.5 1 0.5 x 2 1.0 1 0.37510 is equivalent to 0.0112
Exercises Convertthe following decimal numbers into binary and hexadecimal numbers: 1. 128 2. 207 Convertthe following binary numbers into decimal and hexadecimal numbers: 1. 11111000 2. 1110110
Binary Multiplication0 x0=00 x 1 = 01 x 0 = 0 1 x 1 =1
Data Representation Data on digital computers is represented as a sequence of 0s and 1s. This includes numeric data, text, executable files, images, audio, and video. Data can be represented using 2n Numeric representation Fixed point Floating point Non numeric representation
Fixed Point Integers are whole numbers or fixed-point numbers with the radix point fixed after the least-significant bit. Computers use a fixed number of bits to represent an integer. The commonly-used bit-lengths for integers are 8-bit, 16-bit, 32-bit or 64-bit. Two representation
Unsigned magnitude Unsigned integers can represent zero and positive integers, but not negative integers. Example 1: Suppose that n=8 and the binary pattern is 0100 0001, the value of this unsigned integer is 1×2^0 + 1×2^6 = 65. Example 2: Suppose that n=16 and the binary pattern is 0001 0000 0000 1000, the value of this unsigned integer is 1×2^3 + 1×2^12 = 4104. An n-bit pattern can represent 2^n distinct integers. An n-bit unsigned integer can represent integers from 0 to (2^n)-1
Signed magnitude The most-significant bit (msb) is the sign bit, with value of 0 representing positive integer and 1 representing negative integer. The remaining n-1 bits represents the magnitude (absolute value) of the integer. The absolute value of the integer is interpreted as "the magnitude of the (n-1)-bit binary pattern".
Signed magnitude Example 1: Suppose that n=8 and the binary representation is 0 100 0001. Sign bit is 0 ⇒ positive Absolute value is 100 0001 = 65 Hence, the integer is +65 Example 2: Suppose that n=8 and the binary representation is 1 000 0001. Sign bit is 1 ⇒ negative Absolute value is 000 0001 = 1 Hence, the integer is -1
Signed magnitude Example 3: Suppose that n=8 and the binary representation is 0 000 0000. Sign bit is 0 ⇒ positive Absolute value is 000 0000 = 0 Hence, the integer is +0 Example 4: Suppose that n=8 and the binary representation is 1 000 0000. Sign bit is 1 ⇒ negative Absolute value is 000 0000B = 0 Hence, the integer is -0
Drawbacks Thedrawbacks of sign-magnitude representation are: There are two representations (0000 0000B and 1000 0000B) for the number zero, which could lead to inefficiency and confusion. Positive and negative integers need to be processed separately.
One’s Complement The most significant bit (msb) is the sign bit, with value of 0 representing positive integers and 1 representing negative integers. The remaining n-1 bits represents the magnitude of the integer, as follows: for positive integers, the absolute value of the integer is equal to "the magnitude of the (n-1)-bit binary pattern". for negative integers, the absolute value of the integer is equal to "the magnitude of the complement (inverse) of the (n-1)-bit binary pattern" (hence called 1s complement).
One’s complement Example 1: Suppose that n=8 and the binary representation 0 100 0001. Sign bit is 0 ⇒ positive Absolute value is 100 0001 = 65 Hence, the integer is +65 Example 2: Suppose that n=8 and the binary representation 1 000 0001. Sign bit is 1 ⇒ negative Absolute value is the complement of 000 0001B, i.e., 111 1110B = 126 Hence, the integer is -126
Two’s complement Again, the most significant bit (msb) is the sign bit, with value of 0 representing positive integers and 1 representing negative integers. The remaining n-1 bits represents the magnitude of the integer, as follows: for positive integers, the absolute value of the integer is equal to "the magnitude of the (n-1)-bit binary pattern". for negative integers, the absolute value of the integer is equal to "the magnitude of the complement of the (n-1)-bit binary pattern plus one" (hence called 2s complement).
Two’s complement Example 1: Suppose that n=8 and the binary representation 0 000 0000. Sign bit is 0 ⇒ positive Absolute value is 000 0000 = 0 Hence, the integer is +0 Example 2: Suppose that n=8 and the binary representation 1 111 1111. Sign bit is 1 ⇒ negative Absolute value is the complement of 111 1111B plus 1, i.e., 000 0000 + 1 = 1 Hence, the integer is -1
Floating point representationA real number is represented in exponential form (a = +- m x re)1 bit 8 bits 23 bits (single precision) 0 10000100 11010000000000000000000Sign Exponent Mantissa Radix point
COMPLEMENTS Complements are used in digital computers for simplifying the subtraction operation and for logical manipulations 2 types for each base-r system 1) r’s complement (Radix complement) 2) (r-1)’s complement (Diminished radix Complement)
Radix Complement Referred to as r’s complement The r’s complement of N is obtained as (rn)-N where r = base or radix n = number of digits N = number
Example Give the 10’s complement for the following number a. 583978 b. 5498Solution: a. N = 583978 n=6 106 - 583978 1,000,000 – 583978 = 436022 b. N = 5498 n=4 104 - 5498 10, 000 – 5498 = 4502
Diminished Radix Complements Referred to as the (r-1)s complement The (r-1)s complement of N is obtained as (rn- 1)-N where r = base or radix n = number of digits N = numberTherefore 9’s complement of N is (10n-1)- N
Example Give the (r-1)’s complement for the following number if n=6a. 567894b. 012598
SolutionUsing the formula (rn-1)-N if n = 6 r = 10 then 106 = 1, 000, 000 rn-1= 1, 000, 000 = 999, 999
A.) N = 567894 999,999 – 567894 = 432105 Therefore, the 9’s complement of 567894 is 432105. B.) N = 012598 999,999 – 012598 = 987401 Therefore, the 9’s complement of 012598 is 987401.
Diminished Radix Complement Inthe binary number system r=2 then r-1 = 1 so the 1’s complement of N is (2n-1)-N When a binary digit is subtracted from 1, the only possibilities are 1-0=1 or 1-1=0Therefore, 1’s complement of a binary numeral is formed by changing 1’s to 0’s and 0’s to 1’s.
Example Compute for the 1’s complement of each of the following binary numbers a. 1001011 b. 010110101Solution: a. N=1001011 The 1’s complement of 1001011 is 0110100 b. N=010110101 The 1’s complement of 010110101 is 101001010
Subtraction with Complements The subtraction of two n-digit unsigned numbers M – N in base r can be done as follows: 1. Add the minuend M to the r’s complement of the subtrahend N. If the M >= N, the sum will produce an end carry, rn, which is discarded; what is left is the result M – N If M < N, the sum does not produce an end carry ans is equal to rn – (N – M), which is the r’s complement of (N – M). To obtain the answer, take the r’s complement of the sum and place a negative sign in front.
Using 10’s complement, subtract 72532 – 3250. M = 72532 10’s complement of N = + 96750 Sum = 169282 Discard end carry 105 = - 100000 Answer = 69282
Using 10’s complement, subtract 3250 – 72532. M = 03250 10’s complement of N = + 27468 Sum = 30718 There is no end carry. Answer = -69282 Get the 10’s complement of 30718.
Exercise Giventwo binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X – Y and (b) Y – X using 2’s complements and 1’s complement.
Binary CodesDecimal Digit (BCD) Excess-3 8421 0 0000 0011 1 0001 0100 2 0010 0101 In a digital system, 3 0011 0110 it may sometimes represent a binary 4 0100 0111 number, other 5 0101 1000 times some other 6 0110 1001 discrete quantity of information 7 0111 1010 8 1000 1011 9 1001 1100
Non NumericalRepresentations Itrefers to a representation of data other than numerical values. It refers to the representation of a character, sound or image.
Standardizations of CharacterCodesCodename DescriptionEBCDIC Computer code defined by IBM for general purpose computers. 8 bits represent one character.ASCII 7 bit code established by ANSI (American National Standards Institute). Used in PC’s.ISO Code ISO646 published as a recommendation by the International Organization for Standardization (ISO), based on ASCII 7 bit code for information exchangeUnicode An industry standard allowing computers to consistently represent characters used in most of the countries. Every character is represented with 2 bytes.
Image and Sound RepresentationsStill Images GIF Format to save graphics, 256 colors displayable JPEG Compression format for color still imagesMoving Pictures Compression format for color moving pictures MPEG-1 Data stored mainly on CD ROM MPEG-2 Stored images like vide; real time images MPEG-4 Standardization for mobile terminalsSound PCM MIDI Interface to connect a musical instrument with a computer.
Operations and Accuracy Shift operations It is the operation of shifting a bit string to the right or left. Arithmetic Shift Logical Shift Left Arithmetic Left Shift Logical left shift Right Arithmetic Right Shift Logical Right Shift
Arithmetic ShiftArithmetic Shift is an operation of shifting a bit string, except for the sign bit.Example : Shift bits by 1 ALS ARS Sign bit overflow overflow 11111010 11111010Sign bit 11110100 11111101 Insert a zero in the vacated spot
Logical Shift It shifts a bit string and inserts “0” in places made empty by the shift. Perform a logical left shift. Shift by 1 bit. 01111010 Perform a logical right shift. Shift by 1 bit. 10011001
Exercise Perform arithmetic right and logical right shifts by 3 bits on the 8th binary number 11001100.