# Lecture 2: Number Systems

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1 Lecture 2: Number Systems Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University

2 Contents Number systems Binary arithmetic Number-base conversion Complement of numbers Binary Coded Decimal & Gray Code Logic operations Chapter 1 ECE 2610 Digital Logic 1 2

3 Number System Decimal System It has 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 It's also known as Base 10 or Radix 10 system Integer (273) 10 = = 2* * *10 0 Fraction (0.526) 10 = = 5* * *10-3 Chapter 1

4 Decimal System (543.67) 10 = 5* * * * *10-2 ( ) 10 = 2* * * * * * * *10-4 Why do we use 10 digits, anyway? Chapter 1

5 Binary System It has 2 digits: 0 and 1 It's also known as Base 2 or Radix 2 system A computer does all of its calcualtions using Binary System A 0 is analogous to a Switch is OFF A 1 is analogous to a Switch is ON A Binary Digits is call a BIT (Binary digit) A group of 8 Bits is called a Byte Examples: , , , etc. A group of 4 Bits is called a Nibble Examples: 1101, 0010, 1011, 0001, 1000, 1101, etc. Chapter 1 5

6 Why Use Binary Numbers? Easy to represent 0 and 1 using electrical values. Possible to tolerate noise. Easy to transmit data Easy to build binary circuits. AND Gate 1 0 0

7 Binary System Conversion from Binary to Decimal (101.11) 2 = 1* *2 1 +1* *2-1 +1*2-2 = /2 + 1/4 = = (5.75) 10 Chapter 1 7

8 Octal System It has 8 digits: 0, 1, 2, 3, 4, 5, 6, 7 It's also known as Base 8 or Radix 8 system Conversion from Octal to Decimal (431.65) 8 = 4* *8 1 +1* *8-1 +5*8-2 = 4*64 + 3* /8 + 5/64 = ( ) 10 Chapter 1 8

9 Hexadecimal System It has 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F It's also known as Base 16 or Radix 16 system Conversion from Hexadecimal to Decimal (A2B.DB) 16 = 10* * * * *16-2 = 10* * / /256 = ( ) 10 Chapter 1 9

10 Example: Any radix to Decimal Conversion Chapter 1 ECE 2610 Digital Logic 1 10

11 Conversion Between Number Bases Octal(base 8) Decimal(base 10) Binary(base 2) Learn to convert between bases. Already demonstrated how to convert from binary to decimal. Hexadecimal (base16)

12 Convert an Integer from Decimal to Another Base For each digit position: 1. Divide decimal number by the base (e.g. 2) 2. The remainder is the lowest-order digit 3. Repeat first two steps until no divisor remains. Example for (13) 10: Integer Quotient Remainder Coefficient 13/2 = a 0 = 1 6/2 = a 1 = 0 3/2 = a 2 = 1 1/2 = a 3 = 1 Answer (13) 10 = (a 3 a 2 a 1 a 0 ) 2 = (1101) 2

13 Convert a Fraction from Decimal to Another Base For each digit position: 1. Multiply decimal number by the base (e.g. 2) 2. The integer is the highest-order digit 3. Repeat first two steps until fraction becomes zero. Example for (0.625) 10: Integer Fraction Coefficient x 2 = a -1 = x 2 = a -2 = x 2 = a -3 = 1 Answer (0.625) 10 = (0.a -1 a -2 a -3 ) 2 = (0.101) 2

14 Powers of Two Chapter 1 ECE 2610 Digital Logic 1 14

15 Numbers with Different Bases Chapter 1 ECE 2610 Digital Logic 1

16 Binary Addition Binary addition is very simple. This is best shown in an example of adding two binary numbers carries

17 Binary Subtraction We can also perform subtraction (with borrows in place of carries). Let s subtract (10111) 2 from ( ) borrows

18 Binary Multiplication Binary multiplication is much the same as decimal multiplication, except that the multiplication operations are much simpler X

19 Binary Division 25/ Chapter 1 ECE 2610 Digital Logic 1 19

20 Convert an Integer from Decimal to Octal For each digit position: 1. Divide decimal number by the base (8) 2. The remainder is the lowest-order digit 3. Repeat first two steps until no divisor remains. Example for (175) 10: Integer Quotient Remainder Coefficient 175/8 = a 0 = 7 21/8 = a 1 = 5 2/8 = a 2 = 2 Answer (175) 10 = (a 2 a 1 a 0 ) 2 = (257) 8

21 Convert a Fraction from Decimal to Octal For each digit position: 1. Multiply decimal number by the base (e.g. 8) 2. The integer is the highest-order digit 3. Repeat first two steps until fraction becomes zero. Example for (0.3125) 10: Integer Fraction Coefficient x 8 = a -1 = x 8 = a -2 = 4 Answer (0.3125) 10 = (0.24) 8

22 Number Base Conversion - More Examples Chapter 1 ECE 2610 Digital Logic 1 22

23 Number Base Conversion - More Examples.5 x x x Chapter 1 ECE 2610 Digital Logic 1 23

24 Number Base Conversion - More Examples.513 x x x x Chapter 1 ECE 2610 Digital Logic 1 24

25 Octal and Hexadecimal Numbers ( ) 2 ( ) 8 ( ) 2 (2 C 6 B. F 0) 8 Chapter 1 ECE 2610 Digital Logic 1 25

26 Octal and Hexadecimal Numbers Octal to Binary ( ) 8 ( ) 2 Binary to Hexadecimal ( D) 16 ( ) 2 Chapter 1 ECE 2610 Digital Logic 1 26

27 How To Represent Signed Numbers Consider signed magnitude: = = Sign bit Magnitude Sign bit Magnitude

28 One s Complement Representation The one s complement of a binary number involves inverting all bits. 1 s comp of is s comp of is For an n bit number N the 1 s complement is (2 n -1) N. Called diminished radix complement by Mano since 1 s complement for base (radix 2). To find negative of 1 s complement number take the 1 s complement = = Sign bit Magnitude Sign bit Magnitude

29 Two s Complement Representation The two s complement of a binary number involves inverting all bits and adding 1. 2 s comp of is s comp of is For an n bit number N the 2 s complement is 2 n N. Called radix complement by Mano since 2 s complement for base (radix 2). To find negative of 2 s complement number take the 2 s complement = = Sign bit Magnitude Sign bit Magnitude

30 Two s Complement Shortcuts Algorithm 1 Simply complement each bit and then add 1 to the result. Finding the 2 s complement of ( ) 2 and of its 2 s complement N = [N] = Algorithm 2 Starting with the least significant bit, copy all of the bits up to and including the first 1 bit and then complementing the remaining bits. N = [N] =

31 Finite Number Representation Machines that use 2 s complement arithmetic can represent integers in the range -2 n-1 <= N <= 2 n-1-1 where n is the number of bits available for representing N. Note that 2 n-1-1 = ( ) 2 and 2 n-1 = ( ) 2 For 2 s complement more negative numbers than positive. For 1 s complement two representations for zero. For an n bit number in base (radix) z there are z n different unsigned values. (0, 1, z n-1 )

32 1 s Complement Addition Using 1 s complement numbers, adding numbers is easy. For example, suppose we wish to add +( ) 2 and +( ) 2. Let s compute (12) 10 + (1) 10. (12) 10 = +( ) 2 = in 1 s comp. (1) 10 = +( ) 2 = in 1 s comp Add Step 1: Add binary numbers Step 2: Add carry to low-order bit Add carry Final Result

33 1 s Complement Subtraction Using 1 s complement numbers, subtracting numbers is also easy. For example, suppose we wish to subtract +( ) 2 from +( ) 2. Let s compute (12) 10 - (1) 10. (12) 10 = +( ) 2 -(1) 10 = -( ) 2 = in 1 s comp. = in 1 s comp. Step 1: Take 1 s complement of 2 nd operand Step 2: Add binary numbers Step 3: Add carry to low order bit 1 s comp Add Add carry Final Result

34 2 s Complement Addition Using 2 s complement numbers, adding numbers is easy. For example, suppose we wish to add +( ) 2 and +( ) 2. Let s compute (12) 10 + (1) 10. (12) 10 = +( ) 2 = in 2 s comp. (1) 10 = +( ) 2 = in 2 s comp Add Step 1: Add binary numbers Step 2: Ignore carry bit Final Result Ignore

35 2 s Complement Subtraction Using 2 s complement numbers, follow steps for subtraction For example, suppose we wish to subtract +( ) 2 from +( ) 2. Let s compute (12) 10 - (1) 10. (12) 10 = +( ) 2 -(1) 10 = -( ) 2 = in 2 s comp. = in 2 s comp. Step 1: Take 2 s complement of 2 nd operand Step 2: Add binary numbers Step 3: Ignore carry bit Add Final Result s comp Ignore Carry

36 2 s Complement Subtraction: Example #2 Let s compute (13) 10 (5) 10. (13) 10 = +( ) 2 -(5) 10 = -( ) 2 Adding these two 8-bit codes ignore carry = in 2's comp = in 2's comp Discarding the carry bit, the sign bit is seen to be zero, indicating a correct result. Indeed, ( ) 2 = +(8)

37 2 s Complement Subtraction: Example #3 Let s compute (5) 10 (12) 10. (5) 10 -(12) 10 = +( ) 2 = in 2 s comp. = -( ) 2 = in 2 s comp. Adding these two 8-bit codes Here, there is no carry bit and the sign bit is 1. This indicates a negative result, which is what we expect. ( ) 2 = -( ) = -(7) 10.

38 Signed binary numbers Leftmost bit of the signed binary number represents the sign. 0 -> Positive 1 -> Negative Chapter 1 ECE 2610 Digital Logic 1 38

39 Binary Coded Decimal Code (BCD) Chapter 1 ECE 2610 Digital Logic 1 39

40 Gray Code The advantage of the Gray code over the straight binary number sequence is that only one bit in the code group changes in going from one number to the next Chapter 1 ECE 2610 Digital Logic 1 40

41 ASCII Code American Standard Code for Information Interchange ASCII is a 7-bit code, frequently used with an 8 th bit for error detection (more about that in a bit). Character ASCII (bin) ASCII (hex) Decimal Octal A B C Z a 1

42 ASCII Codes and Data Transmission ASCII Codes A Z (26 codes), a z (26 codes) 0-9 (10 codes), others Complete listing in Mano text Transmission susceptible to noise Typical transmission rates (1500 Kbps, 56.6 Kbps) How to keep data transmission accurate?

43 Parity Codes Parity codes are formed by concatenating a parity bit, P to each code word of C. In an odd-parity code, the parity bit is specified so that the total number of ones is odd. In an even-parity code, the parity bit is specified so that the total number of ones is even. P Information Bits Added even parity bit Added odd parity bit

44 Parity Code Example Concatenate a parity bit to the ASCII code for the characters 0, X, and = to produce both odd-parity and even-parity codes. Character ASCII Odd-Parity ASCII Even-Parity ASCII X =

45 Other Codes 2421 decimal code Excess-3 decimal code 8,4,-2,-1 decimal code American Standard Code for Information Interchange (ASCII) character code Error detecting codes odd parity, even parity Read section 1.7 Chapter 1 ECE 2610 Digital Logic 1 45

46 Binary Data Storage Binary cells store individual bits of data Multiple cells form a register. Data in registers can indicate different values Hex (decimal) BCD ASCII Binary Cell

47 Register Transfer Data can move from register to register. Digital logic used to process data We will learn to design this logic Register A Register B Digital Logic Circuits Register C

48 Transfer of Information Data input at keyboard Shifted into place Stored in memory NOTE: Data input in ASCII

49 Building a Computer We need processing We need storage We need communication You will learn to use and design these components.

50 Binary logic - NOT operation Student took 2610 Student did not take 2610 Student in ECE Chapter 1 ECE 2610 Digital Logic 1 50

51 Binary logic - AND operation Student took 2610 Student took 2610 AND 3300 Student took 3300 Student in ECE Chapter 1 ECE 2610 Digital Logic 1 51

52 Binary logic - OR operation Student took 2610 Student took 2610 OR 3300 Student took 3300 Student in ECE Chapter 1 ECE 2610 Digital Logic 1 52

53 Logic symbols and truth table Chapter 1 ECE 2610 Digital Logic 1 53

54 Input output signals for gates Chapter 1 ECE 2610 Digital Logic 1 54

55 Summary How to convert between binary, octal, decimal, and hexadecimal numbers? How to do the binary arithmetic operations? How to find the 1 s complement of a binary number? How to find the 2 s complement of a binary number? How to do the subtraction operation using 2 s complement method? What are the basic binary logic operations? Chapter 1 ECE 2610 Digital Logic 1 55

56 Homework 1 (Chapter 1) Number system 1.4, 1.8, 1.13 Add, subtract, and multiply in binary: (a) 1111 and 1010 (b) and (c) and Binary arithmetic 1.14, 1.18 Binary codes Convert the following decimal numbers to binary number, Gray code and BCD a) (25) 10 b) (58) 10 Convert the following Gray code numbers to binary numbers a) b) Chapter 1 ECE 2610 Digital Logic 1 56

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