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ETE 204 – Digital Electronics

Number Systems and Conversion, Binary Arithmetic, and Representation of Negative Numbers [Lecture: 2] Instructor: Sajib Roy Lecturer, ETE, ULAB. ETE 204 – Digital Electronics. 52. What does this number represent? Consider the “context” in which it is used. 1011001.101.

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ETE 204 – Digital Electronics

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  1. Number Systems and Conversion, Binary Arithmetic, and Representation of Negative Numbers [Lecture: 2] Instructor: Sajib Roy Lecturer, ETE, ULAB ETE 204 – Digital Electronics

  2. ETE 204 - Digital Electronics 52 • What does this number represent? • Consider the “context” in which it is used.

  3. ETE 204 - Digital Electronics 1011001.101 • What is the decimal value of this number? • Consider the base (or radix) of this number.

  4. ETE 204 - Digital Electronics Number Systems

  5. ETE 204 - Digital Electronics Number Systems • R is the radix (or base) of the number system. • Must be a positive number • R digits in the number system: [0 .. R-1] • Important number systems for digital systems: • Base 2 (binary) [0, 1] • Base 8 (octal) [0 .. 7] • Base 16 (hexadecimal) [0 .. 9, A .. F]

  6. Positional Notation [a4a3a2a1a0.a-1a-2a-3]R ai = ith position in the number R = radix or base of the number ETE 204 - Digital Electronics Number Systems radix point

  7. Power Series Expansion D = an x R4 + an-1 x R3 + … + a0 x R0 + a-1 x R-1 + a-2 x R-2 + … a-m x R-m D = decimal value ai = ith position in the number R = radix or base of the number ETE 204 - Digital Electronics Number Systems

  8. Decimal 927.4510 = 9 x 102 + 2 x 101 + 7 x 100 + 4 x 10-1 + 5 x 10-2 ETE 204 - Digital Electronics Number Systems: Example

  9. Binary 1101.1012 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 + 1 x 2-1 + 0 x 2-2 + 1 x 2-3 ETE 204 - Digital Electronics Number Systems: Example

  10. Octal 326.478 = 3 x 82 + 2 x 81 + 6 x 80 + 4 x 8-1 + 7 x 8-2 ETE 204 - Digital Electronics Number Systems: Example

  11. Hexadecimal E5A.2B16 = 14 x 162 + 5 x 161 + 10 x 160 + 2 x 16-1 + 11 x 16-2 ETE 204 - Digital Electronics Number Systems: Example

  12. ETE 204 - Digital Electronics Conversion between Number Systems

  13. Use repeated division to convert a decimal integer to any other base. Conversion of a Decimal Integer ETE 204 - Digital Electronics

  14. Example: Convert the decimal number 57 to binary and to octal: ETE 204 - Digital Electronics Conversion of a Decimal Integer 57 / 2 = 28: rem = 1 = a0 28 / 2 = 14: rem = 0 = a1 14 / 2 = 7: rem = 0 = a2 7 / 2 = 3: rem = 1 = a3 3 / 2 = 1: rem = 1 = a4 1 / 2 = 0: rem = 1 = a5 5710 = 1110012 57 / 8 = 7: rem = 1 = a0 7 / 8 = 0: rem = 7 = a1 5710 = 718

  15. Use repeated multiplication to convert a decimal fraction to any other base. Conversion of a Decimal Fraction ETE 204 - Digital Electronics

  16. Example: Convert the decimal number 0.625 to binary and to octal. ETE 204 - Digital Electronics Conversion of a Decimal Fraction 0.625 * 2 = 1.250: a-1 = 1 0.250 * 2 = 0.500: a-2 = 0 0.500 * 2 = 1.000: a-3 = 1 0.62510 = 0.1012 0.625 * 8 = 5.000: a0 = 5 0.62510 = 0.58

  17. Example: Convert the decimal number 0.7 to binary. ETE 204 - Digital Electronics Conversion of a Decimal Fraction 0.7 * 2 = 1.4: a-1 = 1 0.4 * 2 = 0.8: a-2 = 0 0.8 * 2 = 1.6: a-3 = 1 0.6 * 2 = 1.2: a-4 = 1 0.2 * 2 = 0.4: a-5 = 0 0.4 * 2 = 0.8: a-6 = 0 0.710 = 0.1 0110 0110 0110 ...2 In some cases, conversion results in a repeating fraction. process begins repeating here!

  18. ETE 204 - Digital Electronics Conversion of a Mixed Decimal Number • Convert the integer part of the decimal number using repeated division. • Convert the fractional part of the decimal number using repeated multiplication. • Combine the integer and fractional parts in the new base.

  19. Example: Convert 48.562510 to binary. Confirm the results using the Power Series Expansion. ETE 204 - Digital Electronics Conversion of a Mixed Decimal Number

  20. ETE 204 - Digital Electronics Conversion between Bases • Conversion between any two bases can be carried out directly using repeated division and repeated multiplication. • Base A → Base B • However, it is, generally, easier to convert Base A to its decimal equivalent and then convert the decimal value to Base B. • Base A → decimal value → Base B Power Series Expansion Repeated Division, Repeated Multiplication

  21. ETE 204 - Digital Electronics Conversion between Bases • Conversion between binary and octal can be carried out by inspection. • Each octal digit corresponds to 3 bits • 101110010 . 0110012 = 5 6 2 . 3 18 • 010 011 100 . 101 0012 = 2 3 4 . 5 18 • 7 4 5 . 3 28 = 111100101 . 0110102 • 3 0 6 . 0 58 = 011 000 110 . 000 1012 • Is the number 392.248 a valid octal number?

  22. ETE 204 - Digital Electronics Conversion between Bases • Conversion between binary and hexadecimal can be carried out by inspection. • Each hexadecimal digit corresponds to 4 bits • 100110100110 . 101101012 = 9 A 6 . B 516 • 1100 1011 1000 . 1110 01112 = C B 8 . E 716 • E 9 4 . D 216 = 111010010100 . 110100102 • 1 C 7 . 8 F16 = 0001 1100 0111 . 1000 11112 • Note that the hexadecimal number system requires additional characters to represent its 16 values.

  23. ECE 301 - Digital Electronics Number Systems Base: 10 2 8 16 What is the value of 12? ETE 204 - Digital Electronics

  24. ETE 204 - Digital Electronics Binary Arithmetic

  25. ETE 204 - Digital Electronics 0 0 1 1 + 0 + 1 + 0 + 1 0 1 1 10 Sum Carry Sum Binary Addition

  26. ETE 204 - Digital Electronics 01011011 + 01110010 10110101 + 01101100 00111100 + 10101010 Binary Addition: Examples

  27. ETE 204 - Digital Electronics Borrow 0 10 1 1 - 0 - 1 - 0 - 1 0 1 1 0 Difference Binary Subtraction

  28. ETE 204 - Digital Electronics 01110101 - 00110010 10110001 - 01101100 00111100 - 10101100 Binary Subtraction: Examples

  29. ETE 204 - Digital Electronics Single-bit Addition Single-bit Subtraction What logic function is this? What logic function is this? Binary Arithmetic

  30. ETE 204 - Digital Electronics 0 0 1 1 x 0 x 1 x 0 x 1 0 0 0 1 Product Binary Multiplication

  31. ETE 204 - Digital Electronics 1011 x 0110 0110 x 1010 1001 x 1101 Binary Multiplication: Examples

  32. ETE 204 - Digital Electronics Representation of Negative Numbers

  33. ETE 204 - Digital Electronics 10011010 • What is the decimal value of this number? • Is it positive or negative? • If negative, what representation are we using?

  34. ETE 204 - Digital Electronics b b b n – 1 1 0 Magnitude MSB Unsigned number b b b b n – 1 n – 2 1 0 Magnitude Sign 0 denotes + – MSB 1 denotes Signed number Unsigned and Signed Binary Numbers

  35. For an n-bit unsigned binary number, all n bits are used to represent the magnitude of the number. ** Cannot represent negative numbers. ETE 204 - Digital Electronics Unsigned Binary Numbers

  36. ETE 204 - Digital Electronics Unsigned Binary Numbers • For an n-bit binary number 0 <= D <= 2n – 1 • where D = decimal equivalent value • For an 8-bit binary number: 0 <= D <= 28 – 1 • 28 = 256 • For a 16-bit binary number: 0 <= D <= 216 – 1 • 216 = 65536

  37. For an n-bit signed binary number, n-1 bits are used to represent the magnitude of the number; the leftmost bit is, generally, used to indicate the sign of the number. 0 = positive number 1 = negative number ETE 204 - Digital Electronics Signed Binary Numbers

  38. Representations for signed binary numbers: 1. Sign and Magnitude 2. 1's Complement 3. 2's Complement ETE 204 - Digital Electronics Signed Binary Numbers

  39. ETE 204 - Digital Electronics Sign and Magnitude • For an n-bit signed binary number, • The leftmost bit is the sign bit. • The remaining n-1 bits represent the magnitude. • Includes a representation for +0 and -0 - (2n-1 – 1) <= N <= + (2n-1 – 1)

  40. What is the Sign and Magnitude representation for the following decimal values, using 8 bits? + 97 - 68 - 97 + 68 ETE 204 - Digital Electronics Sign and Magnitude: Example

  41. Can the following decimal numbers be represented using 8-bit Sign and Magnitude representation? - 212 - 127 +128 +255 ETE 204 - Digital Electronics Sign and Magnitude: Example

  42. ETE 204 - Digital Electronics Questions?

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