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CMPUT229 - Fall 2002 Topic1: Numbers, Numbers, and Numbers José Nelson Amaral CMPUT 229 - Computer Organization and Architecture I 1 Reading Assignment Patt and Patel textbook: Chapter2 CMPUT 229 - Computer Organization and Architecture I 2 Positional Number System 329 3 2 9 923 9 2 3 9 9 2 2 3 3 923 329 CMPUT 229 - Computer Organization and Architecture I 3 Positional Number System 329 923 3102 + 2101 + 9100 9102 + 2101 + 3100 3100 + 210 + 91 300 + 20 + 9 9100 + 210 + 31 900 + 20 + 3 329 923 CMPUT 229 - Computer Organization and Architecture I 4 Positional Number System The same positional system works with different basis: 32913 92313 3132 + 2131 + 9130 9132 + 2131 + 3130 3169 + 213 + 91 50710 + 2610 + 910 9169 + 213 + 31 152110 + 2610 + 310 54210 155010 CMPUT 229 - Computer Organization and Architecture I 5 Binary System In computers we are mostly interested on bases 2, 8, and 16. 1102 92316 122 + 121 + 020 9162 + 2161 + 3160 14 + 12 + 01 410 + 210 + 010 9256 + 216 + 31 230410 + 3210 + 310 610 233910 CMPUT 229 - Computer Organization and Architecture I 6 Signed Integers Problem: given 2k 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) CMPUT 229 - Computer Organization and Architecture I 7 Signed Magnitude Pattern Value Represented Sign Magnitude 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 1001 1010 1011 1100 1101 1110 1111 In a signed magnitude representation we use the first bit of the pattern to indicate if it is a positive or a negative number. Signed Magnitude Pattern Value Represented Sign Magnitude 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 1001 -1 1010 -2 1011 -3 1100 -4 1101 -5 1110 -6 1111 -7 What do we do with the pattern 1000? Signed Magnitude Pattern Value Represented Sign Magnitude 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 -0 1001 -1 1010 -2 1011 -3 1100 -4 1101 -5 1110 -6 1111 -7 Having two patterns to represent 0 is wasteful. The signed 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 + 0001 ?? pattern(-1) pattern(1) PattPatel pp. 20 Signed Magnitude Pattern Value Represented Sign Magnitude 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 -0 1001 -1 1010 -2 1011 -3 1100 -4 1101 -5 1110 -6 1111 -7 Having two patterns to represent 0 is wasteful. The signed 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) + 0001 pattern(1) 1010 = ?? PattPatel pp. 20 Signed Magnitude Pattern Value Represented Sign Magnitude 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 -0 1001 -1 1010 -2 1011 -3 1100 -4 1101 -5 1110 -6 1111 -7 Having two patterns to represent 0 is wasteful. The signed 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) + 0001 pattern(1) 1010 = pattern(-2) PattPatel pp. 20 1’s Complement Pattern 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Value Represented Sign Magnitude 1’s complement 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 -0 -7 -1 -6 -2 -5 -3 -4 -4 -3 -5 -2 -6 -1 -7 -0 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 folllowing -1 + 1 = ?? -0 + 1 = ?? 0 + 1 = ?? PattPatel pp. 20 2’s Complement Pattern 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Value Represented Sign Magnitude 1’s complement 2’s complement 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 -0 -7 -8 -1 -6 -7 -2 -5 -6 -3 -4 -5 -4 -3 -4 -5 -2 -3 -6 -1 -2 -7 -0 -1 A single pattern for 0. 1111 pattern(-1) + 0001 pattern(1) 0000 = pattern(0) It holds the arithmetic property. But the reading of a negative pattern is not trivial. PattPatel pp. 20 Binary to Decimal Conversion Problem: Given an 8-bit 2’s complement binary number: a7 a6 a5 a4 a3 a2 a1 a0 find its corresponding decimal value. Because the binary representation has 8 bits, the decimal value must be in the [-27; +(27-1)] =[-128;+127] interval. CMPUT 229 - Computer Organization and Architecture I 15 23 PattPatel pp. Binary to Decimal Conversion a7 a6 a5 a4 a3 a2 a1 a0 Solution: negative false if (a7 = 1) then negative true flip all bits; compute magnitude using: V a6 26 a5 25 a4 2 4 a3 23 a2 2 2 a1 21 a0 20 if (negative = true) then V V 1 V V CMPUT 229 - Computer Organization and Architecture I 16 24 PattPatel pp. Binary to Decimal Conversion (Examples) Convert the 2’s complement integer 11000111 to its decimal integer value. 1. a7 is 1, thus we make a note that this is a negative number and invert all the bits, obtaining: 00111000 2. We compute the magnitude: V 0 26 1 25 1 2 4 1 23 0 2 2 0 21 0 20 0 64 1 32 116 1 8 0 4 0 2 0 1 0 32 16 8 0 0 0 56 3. Now we remember that it was a negative number, thus: V 56 1 57; V 57 CMPUT 229 - Computer Organization and Architecture I 17 24 PattPatel pp. Decimal to Binary Convertion We will start with an example. What is the binary representation of 10510? 105 a6 26 a5 25 a4 2 4 a3 23 a2 2 2 a1 21 a0 20 Our problem is to find the values of each ai Because 105 is odd, we know that a0 = 1 Thus we can subtract 1 from both sides to obtain: 104 a6 26 a5 25 a4 2 4 a3 23 a2 2 2 a1 21 CMPUT 229 - Computer Organization and Architecture I 18 24 PattPatel pp. Decimal to Binary Convertion (cont.) 104 a6 26 a5 25 a4 2 4 a3 23 a2 2 2 a1 21 Now we can divide both sides by 2 52 a6 25 a5 2 4 a4 23 a3 2 2 a2 21 a1 20 Because 52 is even, we know that a1 = 0 52 a6 25 a5 2 4 a4 23 a3 2 2 a2 21 26 a6 2 4 a5 23 a4 2 2 a3 21 a2 20 a2 = 0 26 a6 2 4 a5 23 a4 2 2 a3 21 13 a6 23 a5 2 2 a4 21 a3 20 a3 = 1 12 a6 23 a5 2 2 a4 21 CMPUT 229 - Computer Organization and Architecture I 19 24 PattPatel pp. Decimal to Binary Convertion (cont.) 12 a6 23 a5 2 2 a4 21 6 a6 2 2 a5 21 a4 20 a4 = 0 6 a6 2 2 a5 21 3 a6 21 a5 20 a5 = 1 2 a6 21 1 a6 2 0 a6 = 1 Thus we got: a6 = 1 a5 = 1 a4 = 0 a3 = 1 a2 = 0 a1 = 0 a0 = 1 10510 = 011010012 CMPUT 229 - Computer Organization and Architecture I 20 25 PattPatel pp. 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 CMPUT 229 - Computer Organization and Architecture I 21 Decimal to Binary Conversion (Another Method) We can also use repeated long division: 105/2 = 52 remainder 1 rightmost digit 52/2 = 26 remainder 0 26/2 = 13 remainder 0 13/2 = 6 remainder 1 10510 = 011010012 6/2 = 3 remainder 0 3/2 = 1 remainder 1 1/2 = 0 remainder 1 CMPUT 229 - Computer Organization and Architecture I 22 Decimal to Binary Conversion (Negative Numbers) What is the binary representation of -10510 in 8 bits? We know from the previous slide that: +10510 =011010012 To obtain the binary representation of a negative number we must flip all the bits of the positive representation and add 1: 10010110 + 00000001 10010111 Thus: -10510 =100101112 CMPUT 229 - Computer Organization and Architecture I PattPatel pp.2322-23 Hexadecimal Numbers (base 16) Binary Decimal Hexadecimal 0000 0 0 0001 1 1 0010 2 2 0011 3 3 0100 4 4 0101 5 5 0110 6 6 0111 7 7 1000 8 8 1001 9 9 1010 10 A 1011 11 B 1100 12 C 1101 13 D 1110 14 E 1111 15 F By convention, the characters 0x are printed in front of an hexadecimal number to indicate base 16. If the number 0xFACE represents a 2’s complement binary number, what is its decimal value? First we need to look up the binary representation of F, which is 1111. Therefore 0xFACE is a negative number, and we have to flip all the bits. CMPUT 229 - Computer Organization and Architecture I PattPatel pp.2426-27 Hexadecimal Numbers (base 16) It is best to write down the binary representation of the number first: Binary Decimal Hexadecimal 0000 0 0 0001 1 1 0010 2 2 0011 3 3 0100 4 4 0101 5 5 0110 6 6 0111 7 7 1000 8 8 1001 9 9 1010 10 A 1011 11 B 1100 12 C 1101 13 D 1110 14 E 1111 15 F 0xFACE = 1111 1010 1100 1110 Now we flip all the bits and add 1: 0000 0101 0011 0001 + 0000 0000 0000 0001 0000 0101 0011 0010 = 0x0532 Then we convert 0x0532 from base 16 to base 10: 0x0532 = 0163 + 5162 + 3161 + 2160 = 0 + 5256 + 316 + 21 = 1280 + 48 + 2 = 133010 0xFACE = -133010 CMPUT 229 - Computer Organization and Architecture I PattPatel pp.2526-27 Binary Arithmetic Decimal Binary 19 + 3 22 010011 + 000011 010110 Binary Decimal 14 - 9 5 910 = 0010012 -910 = 1101112 CMPUT 229 - Computer Organization and Architecture I 001110 + 110111 000101 26 25 PattPatel pp. Overflow What happens if we try to add +9 with +11 in a 5-bit 2-complement representation? Decimal 9 + 11 20 Binary 01001 + 01011 10100 = -12 ? The result is too large to represent in 5 digits, i.e. it is larger than 01111 = +1510. When the result is too large for the representation we say that the result has OVERFLOWed the capacity of the representation? CMPUT 229 - Computer Organization and Architecture I 27 27 PattPatel pp. Overflow Detection What happens if we try to add +9 with +11 in a 5-bit 2-complement representation? Decimal 9 + 11 20 Binary 01001 + 01011 10100 = -12 ? We can easily detect the overflow by detecting that the addition of two positive numbers resulted in a negative result. CMPUT 229 - Computer Organization and Architecture I 28 28 PattPatel pp. Overflow (another example) Could overflow happen when we add two negative numbers? Decimal - 12 + -6 -18 Binary 10100 + 11010 01110 = +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? CMPUT 229 - Computer Organization and Architecture I 29 28 PattPatel pp. Sign-extension What is the 8-bit representation of +510? 0000 0101 What is the 16-bit representation of +510? 0000 0000 0000 0101 What is the 8-bit representation of -510? 1111 1011 What is the 8-bit representation of -510? 1111 1111 1111 1011 CMPUT 229 - Computer Organization and Architecture I 30 27 PattPatel pp. Sign-extension What is the 8-bit representation of +510? 0000 0101 What is the 16-bit representation of +510? 0000 0000 0000 0101 What is the 8-bit representation of -510? 1111 1011 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. What is the 8-bit representation of -510? 1111 1111 1111 1011 CMPUT 229 - Computer Organization and Architecture I 31 27 PattPatel pp. Some Useful Numbers to Remember 20 = 110 = 0x0001 21 = 210 = 0x0002 210 = 102410 = 0100 0000 0000 = 0x0400 = 1K 22 = 410 = 0x0004 23 = 810 = 0x0008 24 = 1610 = 0x0010 220 = 210 × 210 = 102410 × 102410 = 0001 0000 0000 0000 0000 0000 = 0x0010 0000 = 1M 25 = 3210 = 0x0020 26 = 6410 = 0x0040 230 = 210 × 210 × 210 = 1 G 27 = 12810 = 0x0080 28 = 25610 = 0x0100 240 = 210 × 210 × 210 × 210 = 1 T 29 = 51210 = 0x0200 CMPUT 229 - Computer Organization and Architecture I 32 Data and Addresses “She is 104.” In each case, how do we know what 104 is? “I am staying at 104.” 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. “It costs 104.” “There is 104.” “My office is 104.” “Write 100 at 0x0780.” “Stop in front of 104.” “My house is 104.” “Write 100 at 100.” CMPUT 229 - Computer Organization and Architecture I 33