Faculty of Computer Science © 2006 CMPUT 229 Representing Information Numbers, Numbers, and Numbers

© 2006 Department of Computing Science CMPUT 229 Slide’s source Yale N. Patt and Sanjay J. Patel, Introduction to Computing Systems: From bits & gates to C & Beyond, McGrawHill Press, 2001, Chapter2.

© 2006 Department of Computing Science CMPUT 229 Positional Number System

© 2006 Department of Computing Science CMPUT 229 Positional Number System          19   

© 2006 Department of Computing Science CMPUT 229 Positional Number System             The same positional system works with different basis:

© 2006 Department of Computing Science CMPUT 229 Binary System             In computers we are mostly interested on bases 2, 8, and

© 2006 Department of Computing Science CMPUT 229 Signed Integers –Problem: given 2 k distinct patterns of bits, each pattern with k bits, assign integers to the patterns in such a way that: The numbers are spread in an interval around zero without gaps. Roughly half of the patterns represent positive numbers, and half represent negative numbers. When using standard binary addition, given an integer n, the following property should hold: pattern(n+1) = pattern(n) + pattern(1)

© 2006 Department of Computing Science CMPUT 229 Sign-Magnitude Representation In a sign-magnitude representation we use the first bit of the pattern to indicate if it is a positive or a negative number.

© 2006 Department of Computing Science CMPUT 229 Sign-Magnitude Represetation What do we do with the pattern 1000?

© 2006 Department of Computing Science CMPUT 229 Sign-Magnitude Representation Having two patterns to represent 0 is wasteful. The sign-magnitude representation has the advantage that it is easy to read the value from the pattern. But does it have the binary arithmetic property? For instance, what is the result of pattern(-1) + pattern(1)? 1001 pattern(-1) pattern(1) ?? PattPatel, pp. 20

© 2006 Department of Computing Science CMPUT 229 Sign-Magnitude Representation Having two patterns to represent 0 is wasteful. The sign-magnitude representation has the advantage that it is easy to read the value from the pattern. But does it have the arithmetic property? For instance, what is the result of pattern(-1) + pattern(1)? 1001 pattern(-1) pattern(1) 1010 = ?? PattPatel, pp. 20

© 2006 Department of Computing Science CMPUT 229 Sign-Magnitude Representation Having two patterns to represent 0 is wasteful. The sign-magnitude representation has the advantage that it is easy to read the value from the pattern. But does it have the arithmetic property? For instance, what is the result of pattern(-1) + pattern(1)? 1001 pattern(-1) pattern(1) 1010 = pattern(-2) PattPatel, pp. 20

© 2006 Department of Computing Science CMPUT 229 1’s-Complement Representation A negative number is represented by “flipping” all the bits of a positive number. We still have two patterns for 0. It is still easy to read a value from a given pattern. How about the arithmetic property? Suggestion: try the following = ?? = ?? = ?? PattPatel, pp. 20

© 2006 Department of Computing Science CMPUT 229 2’s-Complement Representation A single pattern for pattern(-1) pattern(1) 0000 = pattern(0) It holds the arithmetic property. But the reading of a negative pattern is not trivial. PattPatel, pp. 20

© 2006 Department of Computing Science CMPUT 229 Binary to Decimal Conversion Problem: Given an 8-bit 2’s complement binary number: a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 find its corresponding decimal value. Because the binary representation has 8 bits, the decimal value must be in the [-2 7 ; +(2 7 -1)] =[-128;+127] interval. PattPatel, pp. 23

© 2006 Department of Computing Science CMPUT 229 if (negative = true) then Binary to Decimal Conversion a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 Solution: negative  false if (a 7 = 1) then negative  true flip all bits; compute magnitude using: PattPatel, pp. 24

© 2006 Department of Computing Science CMPUT 229 Binary to Decimal Conversion (Examples) Convert the 2’s complement integer to its decimal integer value. 1. a 7 is 1, thus we make a note that this is a negative number and invert all the bits, obtaining: We compute the magnitude: 3. Now we remember that it was a negative number, thus: PattPatel, pp. 24

© 2006 Department of Computing Science CMPUT 229 Decimal to Binary Convertion We will start with an example. What is the binary representation of ? Our problem is to find the values of each a i Because 105 is odd, we know that a 0 = 1 Thus we can subtract 1 from both sides to obtain: PattPatel, pp. 24

© 2006 Department of Computing Science CMPUT 229 Decimal to Binary Convertion (cont.) Now we can divide both sides by 2 Because 52 is even, we know that a 1 = 0 a 2 = 0 a 3 = 1 PattPatel, pp. 24

© 2006 Department of Computing Science CMPUT 229 Decimal to Binary Convertion (cont.) a 4 = 0a 5 = 1a 6 = 1 Thus we got: a 1 = 0a 4 = 0a 5 = 1a 6 = 1a 2 = 0a 3 = 1a 0 = = PattPatel, pp. 25

© 2006 Department of Computing Science CMPUT 229 Decimal to Binary Conversion (Another Method) We can also use repeated long division: 105/2 = 52 remainder 1 52/2 = 26 remainder 0 26/2 = 13 remainder 0 13/2 = 6 remainder 1 6/2 = 3 remainder 0 3/2 = 1 remainder 1 1/2 = 0 remainder 1

© 2006 Department of Computing Science CMPUT 229 Decimal to Binary Conversion (Another Method) We can also use repeated long division: 105/2 = 52 remainder 1 52/2 = 26 remainder 0 26/2 = 13 remainder 0 13/2 = 6 remainder 1 6/2 = 3 remainder 0 3/2 = 1 remainder 1 1/2 = 0 remainder 1 rightmost digit =

© 2006 Department of Computing Science CMPUT 229 Decimal to Binary Conversion (Negative Numbers) What is the binary representation of in 8 bits? We know from the previous slide that: = To obtain the binary representation of a negative number we must flip all the bits of the positive representation and add 1: Thus: = PattPatel, pp

© 2006 Department of Computing Science CMPUT 229 Hexadecimal Numbers (base 16) If the number $FACE represents a 2’s complement binary number, what is its decimal value? In the 68K assembler, by convention, the character $ is printed in front of an hexadecimal number to indicate base 16. First we need to look up the binary representation of F, which is Therefore $FACE is a negative number, and we have to flip all the bits. PattPatel, pp

© 2006 Department of Computing Science CMPUT 229 Hexadecimal Numbers (base 16) It is best to write down the binary representation of the number first: $FACE = Now we flip all the bits and add 1: = $0532 Then we convert 0x0532 from base 16 to base 10: $0532 = 0     16 0 =    1 = = $FACE = PattPatel, pp

© 2006 Department of Computing Science CMPUT 229 Binary Arithmetic Decimal Binary Decimal Binary = = PattPatel, pp. 25

© 2006 Department of Computing Science CMPUT 229 Overflow What happens if we try to add +9 with +11 in a 5-bit 2-complement representation? Decimal Binary = -12 ? The result is too large to represent in 5 digits, i.e. it is larger than = When the result is too large for the representation we say that the result has OVERFLOWed the capacity of the representation. PattPatel, pp. 27

© 2006 Department of Computing Science CMPUT 229 Overflow Detection What happens if we try to add +9 with +11 in a 5-bit 2-complement representation? Decimal Binary = -12 ? We can easily detect the overflow by detecting that the addition of two positive numbers resulted in a negative result. PattPatel, pp. 28

© 2006 Department of Computing Science CMPUT 229 Overflow (another example) Could overflow happen when we add two negative numbers? Decimal Binary = +14 ? Again we can detect overflow by detecting that we added two negative numbers and got a positive result. Could we get overflow when adding a positive and a negative number? PattPatel, pp. 28

© 2006 Department of Computing Science CMPUT 229 Sign-extension What is the 8-bit representation of ? What is the 16-bit representation of ? What is the 8-bit representation of ? What is the 8-bit representation of ? PattPatel, pp. 27

© 2006 Department of Computing Science CMPUT 229 Sign-extension What is the 8-bit representation of ? What is the 16-bit representation of ? What is the 8-bit representation of ? What is the 8-bit representation of ? To sign-extend a number to a larger representation, all we have to do is to replicate the sign bit until we obtain the new length. PattPatel, pp. 27

© 2006 Department of Computing Science CMPUT 229 Some Useful Numbers to Remember 2 0 = 1 10 = $ = 2 10 = $ = 4 10 = $ = 8 10 = $ = = $ = = $ = = $ = = $ = = $ = = $ = = = $0400 = 1K 2 20 = 2 10 × 2 10 = × = = $ = 1M 2 30 = 2 10 × 2 10 × 2 10 = 1 G 2 40 = 2 10 × 2 10 × 2 10 × 2 10 = 1 T

© 2006 Department of Computing Science CMPUT 229 Data and Addresses “She is 104.” “I am staying at 104.” “It costs 104.” “There is 104.” “My office is 104.” “Stop in front of 104.” “My house is 104.” In each case, how do we know what 104 is? From the context in which it is used! The same is true for data and addresses. The same number is data on one instance and address on another. “Write 100 at $0780.” “Write 100 at 100.”

© 2006 Department of Computing Science CMPUT 229 Representing Fractions In base 10 we know that: = / = /10 3 Fractions are represented in a similar fashion in base 2: = / = / = and 2 5 = 32, thus = /32 10 = Clements, pp. 152

© 2006 Department of Computing Science CMPUT 229 Fractions: Binary to Decimal Another way to convert a binary fraction into a decimal value is as follows: = 1      2 -5 = 1      = =

© 2006 Department of Computing Science CMPUT 229 Fractions: Decimal to Binary The following method works to convert decimal factions to binary: = ??  2 =  2 =  2 =  2 =  2 Done! =

© 2006 Department of Computing Science CMPUT 229