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DATA REPRESENTATION

This guide explains how digital computers represent information using the binary number system. Featuring the fundamental concepts of binary, octal, and hexadecimal formats, it covers the definitions of bits, conversions between different number bases, and the significance of the most and least significant bits. It provides clear examples of converting numbers to and from decimal, octal, and hexadecimal formats, alongside binary arithmetic operations. Ideal for students, educators, and anyone interested in the core mechanisms of computer data representation.

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DATA REPRESENTATION

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  1. DATA REPRESENTATION Digital computers store information in binary Binary allows two states * On yes true 1 * Off no false 0 Each digit in a binary number is called a bit 0 1 1 0 bit 0ff 0n 0n 0ff Binary number system == how numbers can be processed in binary CAP221

  2. Binary Numbers • Digits are 1 and 0 • 1 = true • 0 = false • MSB – most significant bit • LSB – least significant bit • Bit numbering: CAP221

  3. Number system • A number system of base (radix) r is a system that uses distinct symbol for r digits. • Numbers are represented by a string of digit symbols. CAP221

  4. Number system • The base of a number is usually specified as a subscript, e.g.: • (01000011)2, • (71203)8, • (FF078ABC)16, ...etc. • Or a letter indicating the base (d for decimal, b for binary, o for octal and h for hexadecimal) is appended to the number, e.g.: • 01000011b, • 71203o, • FF078ABCh, ...etc. CAP221

  5. Number system CAP221

  6. Number system • The value of a digit depends not only on its value but also on its position within the number → gives the power of the radix by which it is multiplied. CAP221

  7. Positional values in the decimal number system CAP221

  8. Positional values in the binary number system CAP221

  9. Example 2 10 -1 -2 • 379.25= 3*10 + 7*10 + 9*10+2*10+5*10 n n-1 • (dndn-1...d0.f1f2…fm)r = dn* ( r ) + dn-1* ( r ) +….+ 0 -1 -2 -m d0* ( r ) + f1* ( r ) + f2* ( r ) +…+ fm* ( r ) Converting to decimal CAP221

  10. Examples on converting from different bases to Decimal Convert the following to Decimal: • (1001001)2 • (203)8 • (FA07)16 Solution: • (1001001)2 = 1 + 0*21 + 0*22 + 1*23 + 0*24 + 0*25 + 1*26 = 73d • (203)8 = 3 + 0*81 + 2*82 = 131d • (FA07)16 = 7 + 0*161 + 10*162 + 15*163 = 64007d CAP221

  11. Conversion from decimal • The conversion of a decimal integer into a base r is done by: • Whole numbers: divisions by r • Fractions: repeated multiplication by r. CAP221

  12. Conversion from decimal To convert from decimal to any numbering system with base r : • The decimal number is divided by r, • Keeping the remainder aside, the result is further divided by r, and the new remainder is kept aside, • The new result is divided again by r, and so on till the result is less than r and this would be the last remainder, • The remainders make up the equivalent base-r number, with the last remainder being the most-significant digit and the first remainder being the least-significant digit. CAP221

  13. Conversion to binaryWhole number • Ex.: (67)10 = (?)2 2 67 2 33 rem 1 LSB 2 16 rem 1 2 8 rem 0 2 4 rem 0 67d= 10000112 2 2 rem 0 2 1 rem 0 0 rem 1 MSB CAP221

  14. Fraction Repeat multiplication by 2 until the fractional product is 0 Ex.: (0.3125)10 = (?)2 Carry 0.3125* 2 = 0.625 0 MSB 0.625* 2 = 1.25 1 (0.3125)10= (0.0101)2 0.25* 2 = 0.5 0 0.5* 2 = 1.00 1 LSB CAP221

  15. Conversion to base r (50) 10 = (?)8 8 50 8 6 2 8 0 6 (50)10 = (62)8 CAP221

  16. Octal and Hexadecimal number systems • Binary numbers are long. • On average, it takes about 3.3 times as many digits to represent a value in binary as it does to represent the same value in decimal. CAP221

  17. Octal and Hexadecimal number systems If a base R1 is an integral power of d another base R2 i.e. R1= R2 → Group of d digits in base R2 maps directly into one digit in base R1 And each digit in base R1 maps directly into d digits in base R2. CAP221

  18. Octal number systems 3 Base = 8 = 2 → d = 3 Digits: 0 → 7 Binary to Octal conversion: Group every 3 bits (starting from the right) Replace them by their corresponding octal digit Ex.: ( 0 1 0 1 1 0 1 0 1 )2 = ( 265 )8 ( 1 0 0 0 1 )2 = ( 21 )8 CAP221

  19. Octal to binary conversion • Replace each digit by its 3-bit binary equivalent. • Ex.: (476)8 = (100111110)2 CAP221

  20. Octal to binary conversion CAP221

  21. Binary – Octal conversion table BinaryOctalBinaryOctal 000 0 001000 10 001 1 010000 20 010 2 011 3 100 4 101 5 110 6 111 7 CAP221

  22. Hexadecimal number systems (Hex) 4 Base = 16 = 2 → d = 4 Digits: 0,1,2,…..,9,A,B,C,D,E,F Where A = 10, B = 11, C = 12, D = 13, E = 14, F = 15 CAP221

  23. Binary to Hex conversion Group each 4 bits and replace them by their corresponding hex digit. Ex.: (1 0 1 1 0 1 0 1 )2 = ( B5 )16 CAP221

  24. Hex to binary conversion Replace each digit by their 4 bit binary equivalent Ex.: ( 1D9C )16 = (0001110110011100 )2 CAP221

  25. Hex to binary conversion CAP221

  26. Binary – Hexadecimal conversion table BinaryHexBinaryHex 0000 0 1000 8 0001 1 1001 9 0010 2 1010 A 0011 3 1011 B 0100 4 1100 C 0101 5 1101 D 0110 6 1110 E 0111 7 1111 F (00010100)2 → (14)16 → ( 20 )10 (12.8)16 →(00010010.1000)2 → (18.5)10 CAP221

  27. Binary Arithmetic operations • Addition 0+0 = 0 0+1 = 1 1+0 = 1 1+1 = 10 (carry) CAP221

  28. Binary Arithmetic operations • Subtraction 0 - 0 = 0 0 - 1 = 1 (after borrowing) 1 - 0 = 1 1 - 1 = 0 CAP221

  29. Binary Arithmetic operations • Multiplication 0*0 = 0 0*1 = 0 1*0 = 0 1*1 = 1 CAP221

  30. ADDITION • Like decimal numbers, two numbers can be added by adding each pair of digits together with carry propagation. CAP221

  31. Example Carries CAP221

  32. SUBTRACTION • Two numbers can be subtracted by subtracting each pair of digits together with borrowing, where needed. CAP221

  33. SUBTRACTION CAP221

  34. Representation of signed binary numbers • Given a fixed number of bit positions (n), we n can represent 2 patterns: From (00..0)2 to (11..1)2. • Using unsigned numbers we can represent n 0 → 2 -1 n • Using signed numbers → 2 positive & negative patterns CAP221

  35. Complementary notations • The most commonly used way of representing signed numbers. It greatly simplifies all arithmetic operations. • Positive number is represented as it is (like an unsigned positive number). • Negative numbers are represented by the complement of unsigned number. CAP221

  36. Complementary notations • The value used for complementation is a power of the base or less one than a power of the base. • For binary numbers: we have 2’s complement and 1’s complement. • For decimals, we have 10’s complement and 9’s complement. • For hex. Numbers we have 16’s complement and F’s (15’s) complement. CAP221

  37. One’s complement • Is obtained by subtracting the number from (11..1)2 (n 1’s) → reversing each bit: changing every 1 to 0 and every 0 to 1. • Ex.: represent +(14d), -(14d) in 8-bit binary number • (+14)10 = ( 00001110)2 = (00001110)1s • (-14)10 = -( 00001110)2 = (11110001)1s CAP221

  38. For 8-bits number system: • Largest positive number: 0 1111111 = +(127)10 • smallest negative number: 1 0000000 = - (127)10 • Zeroes: 0 0000000 1 1111111 • Range: m-1 m-1 - (2 -1) to + 2 -1 (m : the number of bits) • -(127)10 to +(127)10 • The most significant bit represents the sign: 0 = +ve ; 1 = -ve. CAP221

  39. 2’s complement • Take the 1’s complement and add 1 → invert all the bits and add 1. • Ex.: 2’s complement of (01110100)2: 1’s complement = (10001011)2 add 1 → 2’s complement = (10001100)2 CAP221

  40. 2’s complement • 2’s complement of (00000000)2: 1’s complement = (11111111)2 add 1 → (100000000)2 • 2’s complement of n-bit number include only the rightmost n bits: 2’s complement of (00000000)2 = (00000000)2 CAP221

  41. 2’s complement - For 8-bit binary numbers 2’s complement of (5)d = (00000101)2: 1’s complement = (11111010)2 add 1→ (11111011)2 510 + (-5)10 → 000001012 + 111110112 = 1000000002 Carry 2’s complement of any N integer represents –N: N + (-N) = 0 CAP221

  42. 2’s complement • To find the 2’s complement of a binary number, proceeding from right to left, leave all bits unchanged up to and including the first `1`. Reverse all the remaining bits. • Range of signed value that can be represented in 2’s complemented notation is m-1 m-1 - 2 to 2 -1 CAP221

  43. For 8-bits number system: • Largest positive number: 0 1111111 = +(127)10 • smallest negative number: 1 0000000 = - (128)10 • Zero: 0 0000000 • Range: -(128)10 to +(127)10 • The most significant bit represents the sign: • 0 = +ve ; 1 = -ve. CAP221

  44. 2’s complement in hexadecimal • Each group of four bits corresponds to a single hexadecimal digit. • Subtract each hexadecimal digit from F and then add 1 (equivalent of taking the 16’s complement). CAP221

  45. Ex.: 2’s complement of (3A6E)16 1’s complement: FFFF - 3A6E C591 + 1 2’s complement: (C592)16 CAP221

  46. Comparing 2’s complement numbers Check the signs: If they differ, they determine the order If they are the same, the order of the numbers is the same as that of their representations. CAP221

  47. Ex.: (01001100)2 and (10100101)2 Different signs → the 1st # is +ve the 2nd # is –ve → 1st # > 2nd # CAP221

  48. (10110010)2 and (10111001)2 • Both numbers are negative: 1st # < 2nd # CAP221

  49. Signed binary arithmetic Overflow • Signed binary numbers are of a fixed range. • If the result of addition/subtraction goes beyond this range, overflow occurs. • Two conditions under which overflow can occur are: (i) positive add positive gives negative (ii) negative add negative gives positive CAP221

  50. 2s complement addition Algorithm: • Perform binary addition on the two numbers. • Ignore the carry out of the MSB. • Check for overflow: Overflow occurs if the carriers into and out of the MSB are different. CAP221

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