1 Digital Logic Design Lecture 2 More Number Systems/Complements

2 Overview °Hexadecimal numbers Related to binary and octal numbers °Conversion between hexadecimal, octal and binary °Value ranges of numbers °Representing positive and negative numbers °Creating the complement of a number Make a positive number negative (and vice versa) °Why binary?

3 Understanding Binary Numbers °Binary numbers are made of binary digits (bits): 0 and 1 °How many items does an binary number represent? (1011) 2 = 1x x x x2 0 = (11) 10 °What about fractions? (110.10) 2 = 1x x x x x2 -2 °Groups of eight bits are called a byte ( ) 2 °Groups of four bits are called a nibble. (1101) 2

4 Understanding Hexadecimal Numbers °Hexadecimal numbers are made of 16 digits: (0,1,2,3,4,5,6,7,8,9,A, B, C, D, E, F) °How many items does an hex number represent? (3A9F) 16 = 3x x x x16 0 = °What about fractions? (2D3.5) 16 = 2x x x x16 -1 = °Note that each hexadecimal digit can be represented with four bits. (1110) 2 = (E) 16 °Groups of four bits are called a nibble. (1110) 2

5 Putting It All Together °Binary, octal, and hexadecimal similar °Easy to build circuits to operate on these representations °Possible to convert between the three formats

6 Converting Between Base 16 and Base 2 °Conversion is easy!  Determine 4-bit value for each hex digit °Note that there are 2 4 = 16 different values of four bits °Easier to read and write in hexadecimal. °Representations are equivalent! 3A9F 16 = A9F

7 Converting Between Base 16 and Base 8 1.Convert from Base 16 to Base 2 2.Regroup bits into groups of three starting from right 3.Ignore leading zeros 4.Each group of three bits forms an octal digit = A9F 16 = A9F

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9 Decimal 2 Binary

10 Octal Numbers: [Base 8],[ 0,1,3,4,5,6,7] Octal to Decimal Conversion: Example:- [2374] 8 = [ ? ] 10 =4×8 0 +7×8 1 +3×8 2 +2×8 3 =[1276] 10 Number systems () Number systems (Octal Numbers) °Octal number has base 8 °Each digit is a number from 0 to 7 °Each digit represents 3 binary bits °Was used in early computing, but was replaced by hexadecimal

11 Decimal to Octal Conversion: The Division: [359] 10 = [ ? ] 8 By using the division system: Reminder Number systems () Number systems (Octal Numbers) Quotient

12 Binary to Octal Conversion: Example:- [110101] 2 = [ ? ] 8 Here we will take 3 bits and convert it from binary to decimal by using the decimal to binary truth table: BinaryDecimal = (65) 8 { 65 { Number systems () Number systems (Octal Numbers)

13 Octal to Binary Conversion: Example:- [13] 8 = [ ? ] 2 Here we will convert each decimal digit from decimal to binary (3 bits) using the decimal to binary truth table: BinaryDecimal (13) 8 = (001011) 2 Number systems () Number systems (Octal Numbers)

14 Radix Based Conversion (Example) °Convert 1234 decimal into octal Radix 8Answer Divide by radix

15 Octals Converting to decimal from octal: –Evaluate the power series Example *8 1 2* *8 0 + =

16 Hexadecimal °Hexadecimal is used to simplify dealing with large binary values: Base-16, or Hexadecimal, has 16 characters: 0-9, A-F Represent a 4-bit binary value: (0) to (F) Easier than using ones and zeros for large binary values Commonly used in computer applications °Examples: = = C = A6 C2 16 Hex values can be followed by an “H” to indicate base-16. Example: A6 C2 H

17 Hex Values in Computers

18 Decimal to Hexadecimal DecimalHex A 11B 12C 13D 14E 15F

19 Conversion Binary to Hexadecimal = = = = 6 AC16

20 Radix Based Conversion (Example) °Convert 1234 decimal into hexadecimal Radix 16Answer 4D2 16 Divide by radix D

21 Hexadecimals – Base 16 Converting to decimal from hex: –Evaluate the power series Example 2 E A *16 1 2* * =

22 Octal to Hex Conversion  To convert between the Octal and Hexadecimal numbering systems  Convert from one system to binary first  Then convert from binary to the new numbering system

23 Hex to Octal Conversion Ex : Convert E8A 16 to octal First convert the hex to binary: and re-group by 3 bits (starting on the right) Then convert the binary to octal: So E8A 16 =

24 Octal to Hex Conversion Ex : Convert to hex First convert the octal to binary: re-group by 4 bits (add leading zeros) Then convert the binary to hex: 1 E A So = 1EA 16

25 Octal to Hexadecimal Hexadecimal DecimalOctal Binary

26 Octal to Hexadecimal °Technique Use binary as an intermediary

27 Example = ? E = 23E 16

28 Hexadecimal to Octal Hexadecimal DecimalOctal Binary

29 Hexadecimal to Octal °Technique Use binary as an intermediary

30 Example 1F0C 16 = ? 8 1 F 0 C F0C 16 =

31 Example: Hex → Octal Example: °Convert the hexadecimal number 5A H into its octal equivalent. 5A H = A = Solution: °First convert the hexadecimal number into its decimal equivalent, then convert the decimal number into its octal equivalent.

32 Fractions (Example)

33 Fractions (Example)

34 Radix Based Conversion (Example) °Convert decimal into binary Radix 2 Answer FractionRadixTotal (Fraction x Radix)IntegerFraction Multiply by radix 2

35 FractionRadix Total (Fraction x Radix)IntegerFraction Radix Based Conversion (Example) °Convert to base 16 (up to 4 fractional point) Radix 16 Answer Multiply by radix 16

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37 How To Represent Signed Numbers Plus and minus sign used for decimal numbers: 25 (or +25), -16, etc..For computers, desirable to represent everything as bits. Three types of signed binary number representations: signed magnitude, 1’s complement, 2’s complement. In each case: left-most bit indicates sign: positive (0) or negative (1). Consider signed magnitude: = Sign bitMagnitude = Sign bitMagnitude

38 Complement °It has already been studied that °subtracting one number from another is the same as making one number negative and just adding them. °We know how to create negative numbers in the binary number system. °How to perform 2’s complement process. °How the 2’s complement process can be use to add (and subtract) binary numbers. °Digital electronics requires frequent addition and subtraction of numbers. You know how to design an adder, but what about a subtracter? °A subtracter is not needed with the 2’s complement process. The 2’s complement process allows you to easily convert a positive number into its negative equivalent. °Since subtracting one number from another is the same as making one number negative and adding, the need for a subtracter circuit has been eliminated.

39 3-Digit Decimal A bicycle odometer with only three digits is an example of a fixed-length decimal number system. The problem is that without a negative sign, you cannot tell a +998 from a -2 (also a 998). Did you ride forward for 998 miles or backward for 2 miles? Note: Car odometers do not work this way forward (+) backward (-)

40 Negative Decimal How do we represent negative numbers in this 3-digit decimal number system without using a sign?  Cut the number system in half.  Use 001 – 499 to indicate positive numbers.  Use 500 – 999 to indicate negative numbers.  Notice that 000 is not positive or negative pos(+) neg(-)

41 3-Digit Decimal (Examples) (-5) + 2 (-3) (-3) 3 (-2) + (-3) (-5)  003 Disregard Overflow  995 Disregard Overflow It Works!

42 Complex Problem °The previous examples demonstrate that this process works, but how do we easily convert a number into its negative equivalent? °In the examples, converting the negative numbers into the 3-digit decimal number system was fairly easy. To convert the (-3), you simply counted backward from 1000 (i.e., 999, 998, 997). °This process is not as easy for large numbers (e.g., -214 is 786). How did we determine this? °To convert a large negative number, you can use the 10’s Complement Process.

43 Complement The 10’s Complement process uses base-10 (decimal) numbers. Later, when we’re working with base-2 (binary) numbers, you will see that the 2’s Complement process works in the same way. First, complement all of the digits in a number. A digit’s complement is the number you add to the digit to make it equal to the largest digit in the base (i.e., 9 for decimal). The complement of 0 is 9, 1 is 8, 2 is 7, etc. Second, add 1. Without this step, our number system would have two zeroes (+0 & -0), which no number system has.

44 10’s Complement Examples   785 Example #1 Example #2 Complement DigitsAdd 1Complement DigitsAdd 1 786

45 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 bitMagnitude = Sign bitMagnitude

46 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 -1) – N + 1. 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 bitMagnitude = Sign bitMagnitude

47 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] =

48 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 oFor 2’s complement more negative numbers than positive. oFor 1’s complement two representations for zero. oFor an n bit number in base (radix) z there are z n different unsigned values. (0, 1, …z n-1 )

49 1’s Complement Addition °Using 1’s complement numbers, adding numbers is easy. °For example, suppose we wish to add +(1100) 2 and +(0001) 2. °Let’s compute (12) 10 + (1) 10. (12) 10 = +(1100) 2 = in 1’s comp. (1) 10 = +(0001) 2 = in 1’s comp Add carry Final Result Step 1: Add binary numbers Step 2: Add carry to low-order bit Add

50 1’s Complement Subtraction °Using 1’s complement numbers, subtracting numbers is also easy. °For example, suppose we wish to subtract +(0001) 2 from +(1100) 2. °Let’s compute (12) 10 - (1) 10. (12) 10 = +(1100) 2 = in 1’s comp. (-1) 10 = -(0001) 2 = in 1’s comp Add carry Final Result 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

51 2’s Complement Addition °Using 2’s complement numbers, adding numbers is easy. °For example, suppose we wish to add +(1100) 2 and +(0001) 2. °Let’s compute (12) 10 + (1) 10. (12) 10 = +(1100) 2 = in 2’s comp. (1) 10 = +(0001) 2 = in 2’s comp Final Result Step 1: Add binary numbers Step 2: Ignore carry bit Add Ignore

52 2’s Complement Subtraction °Using 2’s complement numbers, follow steps for subtraction °For example, suppose we wish to subtract +(0001) 2 from +(1100) 2. °Let’s compute (12) 10 - (1) 10. (12) 10 = +(1100) 2 = in 2’s comp. (-1) 10 = -(0001) 2 = in 2’s comp Final Result Step 1: Take 2’s complement of 2 nd operand Step 2: Add binary numbers Step 3: Ignore carry bit 2’s comp Add Ignore Carry

53 2’s Complement Subtraction: Example #2 °Let’s compute (13) 10 – (5) 10. (13) 10 = +(1101) 2 = (01101) 2 (-5) 10 = -(0101) 2 = (11011) 2 °Adding these two 5-bit codes… °Discarding the carry bit, the sign bit is seen to be zero, indicating a correct result. Indeed, (01000) 2 = +(1000) 2 = +(8) carry

54 2’s Complement Subtraction: Example #3 °Let’s compute (5) 10 – (12) 10. (-12) 10 = -(1100) 2 = (10100) 2 (5) 10 = +(0101) 2 = (00101) 2 °Adding these two 5-bit codes… °Here, there is no carry bit and the sign bit is 1. This indicates a negative result, which is what we expect. (11001) 2 = -(7)

55 1’s Complement

56 2’s Complement 1s complement with negative numbers shifted one position clockwise Only one representation for 0 One more negative number than positive number High-order bit can act as sign bit

57 Overflow °Overflow conditions °Add two positive numbers to get a negative number °Add two negative numbers to get a positive number

58 Two’s Complement Number System  

59 Summary °Binary numbers can also be represented in octal and hexadecimal °Easy to convert between binary, octal, and hexadecimal °Signed numbers represented in signed magnitude, 1’s complement, and 2’s complement °2’s complement most important (only 1 representation for zero). °Important to understand treatment of sign bit for 1’s and 2’s complement.