Understanding Digital Number Systems in Electronics

1. NUMBERING SYSTEM

2. Analog • Analog information is made up of a continuum of values within a given range • At its most basic, digital information can assume only one of two possible values: one/zero, on/off, high/low, true/false, etc.

4. Digital • Digital Information is less susceptible to noise than analog information • Exact voltage values are not important, only their class (1 or 0) • The complexity of operations is reduced, thus it is easier to implement them with high accuracy in digital form • BUT: Most physical quantities are analog, thus a conversion is needed

6. Digital Number Systems • Many number systems are used in digital electronics: • Decimal number system (base 10) • Binary number system (base 2) • Octal number system (base 8) • Hexadecimal number system (base16)

7. Decimal system • We use decimal numbers everyday. • It is a base-10 system. • 10 symbols: 0,1, 2, 3, 4, 5, 6,7, 8, 9 • The position of each digit in a decimal number can be assigned a weight.

8. Weights 10-2 10-3 103 102 101 100 10-1 2 7 4 5 . 2 1 4 Decimal system • Most significant digit (MSD) - the digit that carries the most weight, usually the left most • Least significant digit (LSD) - the digit that carries the least weight, usually the right most • Take example: decimal number  2745.214

9. Binary system • Difficult to design a system that works with 10 different voltage levels • Solution is base-2 (binary) system • 2 digits/symbols: 0, 1 • Examples: 0, 1, 01, 111, 101010

10. 2-2 2-3 23 22 21 20 2-1 1 0 1 1 . 1 0 1 Binary System • The position of each digit (bit) in a binary number can be assigned a weight. • For example: 1011.101 • 1011.101 is a binary number • 1 is a digit, 0 is a digit, 1 is a digit… weights LSB MSB

11. Binary System (BIT) • It takes more digits in the binary system to represent the same value in the decimal system. • Examples: 710 = 1112 1010 102 • A single binary digit is referred to as a bit. • 8 bits make a byte.

12. Binary System (BIT) • With N bits we have 2N discrete values. • For example, a 4-bit system can represent 24 or 16 discrete values. • The largest value is always 2N – 1. • For the 4-bit system, 24 – 1 = 1510. • The range of values for a 4-bit number is then 0 thru 15.

13. Hexadecimal System • Base-16 system • 16 symbols: 10 numeric digits and 6 alphabetic characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F • Compact way of writing binary system. • Widely used in computer and microprocessor applications.

14. Weights 16-2 16-3 163 162 161 160 16-1 2 A F 8 . 9 8 E Hexadecimal System • Examples: 1C16 , A8516 • The position of each digit in a hexadecimal number can be assigned a weight • For example: 2AF8.98E

15. Octal System • Base-8 system. • 8 digits: 0, 1, 2, 3, 4, 5, 6, 7 • Convenient way to express binary numbers and codes.

16. Table of Number Systems

17. Conversion between Numbers 5 Octal (base 8) 6 9 10 Decimal (base 10) Binary (base 2) 1 2 7 8 Hexadecimal (base 16) 3 4

18. 1. Decimal to Binary • Method 1: Decimal number  binary number • Method: sum-of-weights • Method 2: Decimal whole number  binary number. • Method: division-by-2 • Method 3: Decimal fraction  binary number • Method: multiplication-by-2

19. Method 1: Sum of Weights • Step 1: Find the power of two that fulfills the following: • Nearest to the given decimal number; and • Its decimal number is less than or equal to the given decimal number. • Step 2: Subtract the power of two (from Step 1) from the given decimal number. • Step 3: If the result of the subtraction in Step 2 is 0, go to Step 4. Else, repeat Steps 1 and 2 for the result of the subtraction in Step 2. • Step 4: Write out the binary number based on all the powers of two from Step 1.

20. 22 24 23 21 20 Weights 1 1 0 0 1 Binary number Method 1: Sum of Weights Example 1: Convert 2510 to binary Step 1: 22 = 4? No, because it is not the nearest. 23 = 8is nearer and still less than 25. 24 = 16 is the nearest and its decimal, 16 is less than 25. 25 = 32No, because its decimal number, 32 is more than 25. Step 2: the result of subtraction 25 – 16 = 9 Step 3: Repeat Step 1: the power of two which is nearest to 9 but less than 9 is 23 = 8 Repeat Step 2: the result of subtraction 9 – 8 = 1 Repeat Step 1: the power of two which is nearest to 1 and equal to 1 is 20 = 1 Repeat Step 2: the result of subtraction 1 – 1 = 0 Step 4: Write out the binary number based on all the powers of two from Step 1.

21. Method 2: Division-by-2 • Method 2 is used to convert only whole decimal numbers (no fraction) to binary. • Repeat the division of the decimal number with 2 until the quotient is 0. • Remainder of each division determine the binary number. First remainder represent the LSB and the last remainder is the MSB.

22. 1 1 0 0 1 Method 2: Division-by-2 Example 1: Convert 2510 to binary Quotient Remainder LSB MSB 2510 =

23. Method 3: Multiplication-by-2 • Method 3 is used to convert decimal fraction only to binary. • Repeat the multiplication until the fractional part of the product are all zeros. • The binary number is determined by the first digit in the multiplication results.

24. Method 3: Multiplication-by-2 Example 2: Convert 0.3125 to binary Carry 0 0.3125 x 2 = 0.625 MSB LSB The binary fraction is: 1 0.625 x 2 = 1.25 . 0 1 0 1 0 0.25 x 2 = 0.50 1 0.50 x 2 = 1.00

25. 2-2 2-3 23 22 21 20 2-1 1 0 1 1 . 1 0 1 2. Binary to Decimal • Only one method is used. That is the sum of weight. • Example 3: Convert 1011.101 to decimal =(1x23) + (0x22) + (1x21) + (1x20) + (1x2-1) + (0x2-2) + (1x2-3) = 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 = 11.62510

26. 3. Decimal to Hexadecimal • Method used is repeated division by-16. • Repeat the division of the decimal number with 16 until the quotient is 0. • Remainder of each division determine the hex number. First remainder represent the LSB and the last remainder is the MSB.

27. 2 8 A 65010 = Decimal to Hexadecimal Example 4: Convert 65010 to hex number Quotient Remainder (decimal) Remainder (hexadecimal) LSB MSB

28. 4. Hexadecimal to Decimal • Only one method is used. That is the sum of weight. Example 5: Convert A8516 to decimal number 162 161 160 A 8 5 = (A x 162) + (8 x 161) + (5 x 160) = (10 x 256) + (8 x 16) + (5 x 1) = 2560 + 128 + 5 = 2693

29. 5. Decimal to Octal • Method used is repeated division by-8. • Repeat the division of the decimal number with 8 until the quotient is 0. • Remainder of each division determine the oct number. First remainder represent the LSB and the last remainder is the MSB.

30. 5 4 7 35910 = Decimal to Octal Example 6: Convert 35910 to octal number Quotient Remainder LSB MSB

31. 83 82 81 80 2 3 7 4 6. Octal to Decimal • Only one method is used. That is the sum of weight. Example 7: Convert 23748 to decimal number = (2 x 83) + (3 x 82) + (7 x 81) + (4 x 80) = (2 x 512) + (3 x 64) + (7 x 8) + (4 x 1) = 1024 + 192 + 56 + 4 = 1276

32. 1111 0001 0110 1001 0011 Binary Hexadecimal 3 F 1 6 9 7. Binary to Hexadecimal • Step 1: Break the binary number into 4-bit groups, starting from LSB. • Step 2: Replace each 4-bit with the equivalent hexadecimal number. Example 8: Convert 111111000101101001 to hex number

33. Hexadecimal C F 8 E 1111 1000 1110 1100 Binary 8. Hexadecimal to Binary • Step: Replace each digit of the hexadecimal number with the equivalent 4-bit binary number. Example 9: Convert CF8E16 to binary number

34. 101 111 001 Binary Octal 5 7 1 9. Binary to Octal • Step 1: Break the binary number into 3-bit groups, starting from LSD. • Step 2: Replace each 3-bit group with the equivalent octal number. Example 10: Convert 1011110012 to octal number

35. 7 5 2 6 Octal 101 010 110 111 Binary 10. Octal to Binary • Step: Replace each digit of the octal number with the equivalent 3-bit binary number. Example 11: Convert 75268 to binary number

36. Summary of Conversion Division-by-8 Octal (base 8) Sum of weight 3 bit conversion Digit-to-3 bit Division-by-2 Decimal (base 10) Binary (base 2) Sum of weight 4 bit conversion Digit-to-4 bit Hexadecimal (base 16) Division-by-16 Sum of weight