1. EENG 2710 Chapter 1 Number Systems and Codes

2. Chapter 1 Homework 1.1c, 1.2c, 1.3c, 1.4e, 1.5e, 1.6c, 1.7e, 1.8a, 1.9a, 1.10b, 1.13a

3. Number Systems

4. Binary Number System • Uses two digits, 0 and 1. • Represents any number using the positional notation.

5. Positional Notation • The value of a digit depends on its placement within a number. • In base 10, the positional values are (starting to the left of the decimal) –1 (100), 10 (101), 100 (102), 1000 (103), etc. • In base 2, the positional values are1 (20), 2 (21), 4 (22), 8 (23), etc.

6. Binary Weights

7. Fractional Binary Weights

8. Bit • Shorthand for binary digit, a logic 0 or 1. • The most significant bit (MSB) is the leftmost bit of a binary number. • The least significant bit (LSB) is the rightmost bit of a binary number.

9. Binary Inputs • Digital circuits operate by accepting logic levels (0,1) at their input(s). • The corresponding output(s) logic level will change (0,1).

10. Binary Inputs

11. 4-Input Digital Circuit

12. Base Conversions Methods • Series substitution method • Sum powers of 2 • Radix method • Repeated Division • Repeated Multiplication

13. Series Substitution Method(Binary to Decimal)

14. Sum Powers of 2(Decimal to Binary) • Step 1: • Determine the largest power of 2 less than or equal to the number to be converted. • Place a 1 in that positional location. • Convert 5710 to binary • 6410 5710  3210

15. Sum Powers of 2 • Step 2: • Subtract the number found in Step 1 from the number to be converted. • 57 – 32 = 25 • For the new number, determine if the next lowest power of 2 is less than or equal to that number. • 25 – 16 = 9

16. Sum Powers of 2 • Step 3: • If the new power of two from Step 2 is larger, place a 0 in that positional location. • If the new value is less than or equal, place a 1 in that positional location.

17. Sum Powers of 2 • Step 4: • Repeat Steps 2 and 3 until there is nothing left to subtract. • All remaining bits are set to 0. 5710 = 1110012

18. Radix Method (Repeated Division by 2) 4610 = 1011102

19. Fractional Binary Numbers • Radix point: • The generalized decimal point. The dividing line between positive and negative powers for positional multipliers. • Binary point: • The radix point for binary numbers.

20. Fractional Binary Values • The value immediately to the right of the binary point is 2–1 = 0.5. • The next value to the right is 2–2 = 0.25. • The next value to the right is 2–4 = 0.125, and so on.

21. Series Substitution Method(Binary Fraction to Decimal Fraction)

22. Radix Method for 0.210 to Binary (Repeated Multiplication by 2) • Step 1:Multiply the decimal fraction by 2. • Step 2: Integer part is 0 or 1 left of decimal point. 0.2 x 2 = 0.4 Integer part = 0 0.4 x 2 = 0.8 Integer part = 0 0.8 x 2 = 1.6 Integer part = 1 0.6 x 2 = 1.2 Integer part = 1 0.2 x 2 = 0.4 Integer part = 0 (stop 0011repeats) Read integer parts from top to bottom Therefore, 0.210 = 0.0011 0011 0011

23. Hexadecimal Numbers • Base 16 number system. • Primarily used as a shorthand form of binary numbers.

24. Counting in Hexadecimal • Values range from 0 to F with the letters A to F used to represent the values 10 to 15 respectively. • Positional multipliers are powers of 16:160 = 1, 161 = 16, 162 = 256, etc.

25. Hexadecimal vs. Decimal Numbers

26. Counting In Hexadecimal 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F 10,11,12,13,14,15,16,17,18,19,1A,1B,1C,1D,1E,1F 20,21,22,23,24,25,26,27,28,29,2A,2B,2C,2D,2E,2F 30,31,32,33,34,35,36,37,38,39,3A,3B,3C,3D,3E,3F

27. Decimal-to-Hexadecimal Conversion(Repeated division by 16) 3158110 = 7B5DH

28. Hex-Decimal-Binary Table

29. Conversion Between Hexadecimal and Binary • Each hexadecimal digit represents 4 binary bits. F D 6 9 1111 1101 0110 1001 FD69H = 11111101011010012

30. Signed/Unsigned Binary Numbers • Signed Binary Number: • A binary number of fixed length whose sign (+/–) is represented by one bit (usually MSB) and its magnitude by the remaining bits. • Unsigned Binary Number: • A binary number of fixed length whose sign is not specified by a bit. All bits are magnitude and the sign is assumed +.

31. Unsigned Binary Arithmetic • Sum: • Result of an Addition Operation of two (or more) binary numbers (operands). • Carry: • A digit (or bit) that is carried over to the next most significant bit during an n-Bit addition operation. • The carry bit is a 1 if the result was too large to be expressed in n bits.

32. Basic Rules (Unsigned)

33. Basic Subtraction • Basic Subtraction of x = a–b, with a = minuend, b = subtrahend, and x = difference or result. • Requires a Borrow Bit if a < b. • There are other forms of subtraction such as 2’s Complement Addition used by microprocessors (such as in a PC).

34. Binary Subtraction with Borrow Examples

35. Signed Binary Numbers • Sign Bit: • A bit (usually the MSB) that indicates whether a number is positive (= 0) or negative (= 1). • Magnitude Bits: • The bits of a signed binary number that tell how large it is in value.

36. Signed Binary Numbers • True-Magnitude Form: • A form of signed binary whose magnitude bits are the TRUE binary form (not complements). • 1’s Complement: • A form of signed binary in which negative numbers are created by complementing all bits. • 2’s Complement: • A form of signed binary in which the negative numbers are created by complementing all the bits and adding a 1 (1’s Complement + 1).

37. True-Magnitude Form • 5-Bit Numbers Negative Sign (S = 1) • +25 = 011001 (Note sign bit (MSB) Sign = 0) • –25 = 111001 (Same as +25 with sign = 1) • +12 = 001100 • –12 = 101100

38. 1’s Complement Form • 8-Bit 1’s Complement Negative (S = 1) • +57 = 00111001 • –57 = 11000110 (All Bits Inverted) • +72 = 01001000 • –72 = 10110111

39. 2’s Complement Form 57 = 0011 1001 -57 = 1100 0110 + 1 1100 0111

40. Signed Binary Addition (8-Bit) Signed Addition Positive (S = 0) +30 = 00011110 +75 = 01001011 105 01101001

41. Subtraction with 1’s Complement • Add the 1’s Complement and then Carry. • Uses an End around carry addition method. (80 – 65)

42. 2’s Complement Subtraction • Add 2’s Complement to Minuend. (80 – 65) Discord Carry Bit From Results

43. 2’s Complement Subtraction 20010 – 510 = 19510 (Use 16 bit word) 20010 = 00000000110010002 510 = 00000000000001012 -510 = 11111111111110102 = 1’s complement + 1 11111111111110112 00000000110010002 + 11111111111110112 100000000110000112 = 00000000110000112 = 19510

44. Negative Results • If the True-Magnitude Form is used for subtraction, the results are incorrect. • If the result is from 1’s or 2’s Complement and the result is negative (S = 1), the magnitude is found by taking the complement of the result.

45. Negative Result Example Thus, = -1510

46. More Binary Addition

47. More Binary Subtraction

48. Binary Multiplication

49. Binary Division

50. Range of Signed Numbers • Range of Positive Numbers is 0 to 2n– 1 for a number with n magnitude bits. • Range of Negative Numbers is –1 to –2n for a number with n magnitude bits. • 8-Bit Example: 8-Bit Number Range is –2n x  +2n – 1 or –128 to +127