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Number Systems and Computer Arithmetic Winter 2014 COMP 1380 Discrete Structures I Computing Science Thompson Rivers University Course Contents Speaking Mathematically Number Systems and Computer Arithmetic Logic and Truth Tables Boolean Algebra and Logic Gates Vectors and Matrices Sets and Counting Probability Theory and Distributions Statics and Random Variables TRU-COMP1380 Number Systems 2 Unit Learning Objectives Convert a decimal number to binary number. Convert a decimal number to hexadecimal number. Convert a binary number to decimal number. Convert a binary number to hexadecimal number. Convert a hexadecimal number to binary number . Add two binary numbers. Compute the 1’s complement of a binary number. Compute the 2’s complement of a binary number. Understand the 2’s complement representation for negative integers. Subtract a binary number by using the 2’s complement addition. Multiply two binary numbers. Use of left shift and right shift. Binary division TRU-COMP1380 Number Systems 3 Unit Contents The number systems: The decimal system The binary system The hexadecimal system Computer Arithmetic: Binary addition Representation of negative integers Binary multiplication Binary division Representation of fractions TRU-COMP1380 Number Systems 4 1. Number Systems TRU-COMP1380 Number Systems 5 The Decimal System Uses 10 digits 0, 1, 2, ..., 9. Decimal expansion: 83 = 8×10 + 3 4728 = 4×103 + 7×102 + 2×101 + 8×100 84037 = ??? 43.087 = ??? Do you know addition, subtraction, multiplication and division? 1234 + 435.78 1234 – 435.78 1234 × 435.78 1234 / 435.78 TRU-COMP1380 Number Systems 6 The Binary System In computer systems, the most basic memory unit is a bit that contains 0 or 1. The data unit of 8 bits is referred as a byte that is the basic memory unit used in main memories and hard disks. All data are represented by using binary numbers. Data types such as text, voice, image and video have no meaning in the data representation. 8 bits are usually used to express English alphabets. A collection of n bits has 2n possible states(, i.e., numbers). Is it true? E.g., How many different numbers can you express using 2 bits? How many different numbers can you express using 4 bits? How many different numbers can you express using 8 bits? How many different numbers can you express using 32 bits? TRU-COMP1380 Number Systems 7 How can we store integers (i.e., positive numbers only) in a computer? E.g., A decimal number 329? Is it okay to store 3 chracters ‘3’, ‘2’, and ‘9’ for 329? How do we store characters? 32910 = ???2 TRU-COMP1380 Number Systems 8 Uses two digits 0 and 1. 02 12 102 112 1002 1012 1102 1112 10002 10012 ... TRU-COMP1380 = = = = = = = = = = 0×20 1×20 1×21 1×21 1×22 1×22 1×22 1×22 1×23 1×23 = = + + + + + + + + 0×20 1×20 0×21 0×21 1×21 1×21 0×22 0×22 = = + + + + + + 0×20 1×20 0×20 1×20 0×21 0×21 = = = = + 0×20 = + 1×20 = Number Systems 010 110 210 310 410 510 610 710 810 910 9 Powers of 2 12 102 1002 10002 1 00002 10 00002 10 000002 TRU-COMP1380 = = = = = = = 1×20 1×21 1×22 1×23 24 = 25 = 26 = = + 0×20 = + 0×21 + 0×20 = + 0×22 + 0×21 + 0×20 = Number Systems 110 210 410 810 1610 3210 6410 10 1000 0000 0000 0000 0000 0000 00002 00002 00002 00002 00002 00002 = = = = = = 27 = 28 = 29 = 210 = 211 = 212 = 1 10 100 1000 1 0000 Can you memorize the above powers of 2? Converting to decimals 11012 = ???10 1011 00102 = ???10 1011.00102 = ???10 TRU-COMP1380 Number Systems ???10 ???10 ???10 ???10 ???10 ???10 11 Converting Decimal to Binary 23 / 2 11 / 2 5/2 2/2 1/2 => 2310 = 101112 27110 = ???2 607110 = ???2 TRU-COMP1380 Quotient 11 5 2 1 0 Number Systems Remainder 1 1 1 0 1 12 Another similar idea 27110 = ???2 256 < 271 < 512 8< 15 < 16 => 271 -> 271 = 256 + 15 = 1 0000 00002 + 15 -> 15 = 8 + 7 = 10002 + 7 = 1 0000 00002 + 15 = 1 0000 00002 + 10002 + 7 = 1 0000 00002 + 10002 + 1112 = 1 0000 11112 127110 = ???2 TRU-COMP1380 Number Systems 13 Hexadecimal Number System 010 = 00002 = 016 = 0x0 110 = 00012 = 116 = 0x1 210 = 00102 = 216 = 0x2 310 = 00112 = 316 = 0x3 410 = 01002 = 416 = 0x4 510 = 01012 = 516 = 0x5 610 = 01102 = 616 = 0x6 710 = 01112 = 716 = 0x7 810 = 10002 = 816 = 0x8 910 = 10012 = 916 = 0x9 1010 = 10102 = A16 = 0xA 1110 = 10112 = B16 = 0xB 1210 = 11002 = C16 = 0xC 1310 = 11012 = D16 = 0xD 1410 = 11102 = E16 = 0xE 1510 = 11112 = F16 = 0xF 0x23c9d6ef = ???10 14816 = ???10 TRU-COMP1380 4 bits can be used for a hexadecimal number, 0, ..., F. Please memorize it! 0x23c9d6ef = ???2 14816 = ???2 Number Systems 14 Converting Decimal to Hexadecimal Quotient 20 × 16 + 1 × 16 + 0 × 16 + Remainder 8 4 1 328 / 16 = 20 / 16 = 1 / 16 = => 32810 = 14816 = ???2 14816 = (1 × 162 + 4 × 161 + 8 × 160)10 19210 = ???16 TRU-COMP1380 Number Systems 15 Converting Binary to Hexadecimal 4DA916 = ???2 1001101101010012 = ???16 = 100 1101 1010 1001 = 4DA9 10 11102 = 0x??? = ???10 0100 1110 1011 1001 01002 = 0x??? = ???10 TRU-COMP1380 Number Systems 16 2. Computer Arithmetic TRU-COMP1380 Number Systems 17 Binary Addition How to add two binary numbers? Let’s consider only unsigned integers (i.e., zero or positive numbers only) for a while. Just like the addition of two decimal numbers. carry E.g., 10010 10010 1111 + 1001 + 1011 + 1 11011 11101 ??? + TRU-COMP1380 10111 111 ??? Number Systems 18 Binary Subtraction How to subtract a binary number? Just like the subtraction of decimal numbers. E.g., 0112 02 02 1000 10 10 10010 10010 -1 -1 -11 -11 1 ?1 ?11 Try: 101010 -101 How to do? 10010 -11 1111 1 -10 34–79=? Is subtraction easier or more difficult than addition? Why? TRU-COMP1380 Number Systems 19 In the previous slide, 10010 – 11 = 1111 What if we add 00010010 + 11111100 1 00001110 + 1 00001111 Is there any relationship between 112 and 111111002? The 8-bit 1’s complement of 112 is ??? Switching 0 1 This type of addition is called 1’s complement addition. Find the 8-bit one’s complements of the followings. 11011 -> 00011011 -> 10 -> 00000010 -> 101 -> 00000101 -> TRU-COMP1380 Number Systems 20 In the previous slide, 10010 – 11 = 1111 What if we add 00010010 + 11111101 1 00001111 Is there any relationship between 11 and 11111101? The 8-bit 2’s complement of 11 is ??? 2’s complement ≡ 1’s complement + 1 -> 11111100 + 1 = 11111101 This type of addition is called 2’s complement addition. Find the 16-bit two’s complements of the followings. 11011 -> 0000000000011011 -> 10 101 TRU-COMP1380 Number Systems 21 Another example - 101010 101 ??? What if we use 1’s complement addition or 2’s complement addition instead as followings? Let’s use 8-bit representation. 1’s complement addition + 1 + 00101010 11111010 00100100 1 00100101 + 1 00101010 11111011 00100101 2’s complement addition What does this mean? A – B = A + (–B), where A and B are positive Is –B equal to the 1’s complement or the 2’s complement of B? TRU-COMP1380 Number Systems 22 Can we use 8-bit 1’s complement addition for 12 – 102 = –12 ? - 1 10 + 00000001 11111101 11111110 <- 8-bit 1’s complement of 10 <- Is this correct? (Is this 1’s complement of 1?) Let’s use 8-bit 2’s complement addition for 12 – 102. 00000001 + 11111110 11111111 <- 2’s complement of 10 <- Correct? (2’s complement of 1?) 12 – 102 = 12 + (–102) Subtraction can be converted to addition with negative numbers. Then, how to represent negative binary numbers, i.e., signed integers? TRU-COMP1380 Number Systems 23 Representation of Negative Binaries Representation of signed integers 8 or 16 or 32 bits are usually used for integers. Let’s use 8 bits for examples. How to represent positive integer 5? The left most bit (called most significant bit) is used as sign. When the MSB is 0, positive integers. When the MSB is 1, negative integers. The other 7 bits are used for integers. What signed integers can be described using the 8 bit representation? 00001001 How about -9? 10001001 is really okay? 00001001 (9) + 10001001 (-9) = 10010010 (-18) It is wrong! We need a different representation for negative integers. TRU-COMP1380 Number Systems 24 How about -9? What is the 8-bit 1’s complement of 9? 11110110 00001001 + 11110110 = 11111111 <- 8-bit 1’s complement of 9 <- 9 + 8-bit 1’s complement of 9 <- Is it zero? (1’s complement of 0?) What is the 2’s complement of 9? 10001001 is really okay? 00001001 (9) + 10001001 (-9) = 10010010 (-18) It is wrong! We need a different representation for negative integers. 11110111 00001001 + 11110111 = 1 00000000 <- 8-bit 2’s complement of 9 <- 9 + 8-bit 2’s complement of 9 <- It looks more like zero. 2’s complement representation is used for negative integers. TRU-COMP1380 Number Systems 25 12 – 102 = 12 + (–102) ??? 00000001 + 11111110 <- 2’s complement of 10, i.e., -102 11111111 <- 2’s complement of 1, i.e., -1 (= 1 – 2) 1010102 – 10011012 = 001010102 + (–010011012) ??? 100102 – 112 ??? 102 – 1112 ??? –102 – 12 ??? Is the two’s complement of the two’s complement of an integer the same integer? What is x when the 8-bit 2’s complement of x is 11111111 TRU-COMP1380 11110011 Number Systems 10000001 26 8-bit representation with 2’s complement 127 01111111 126 01111110 overflow ... ... 2 00000010 1 00000001 0 00000000 -1 11111111 (28 – 1) -2 11111110 -3 11111101 ... ... -127 10000001 -1 -128 10000000 -1 -1 The maximum number is ? byte The minimum number is ? byte What if we add the maximum number by 1 ??? What if we subtract the minimum number by 1 ??? TRU-COMP1380 Number Systems +1 +1 +1 overflow y = 125; y += 4; ?? x = -126; x -= 5; ?? 27 16-bit representation with 2’s complement ... ... ... 3 2 1 0 -1 -2 -3 ... ... ... 01111111 11111111 01111111 11111110 ... 00000000 00000011 00000000 00000010 00000000 00000001 00000000 00000000 11111111 11111111 11111111 11111110 11111111 11111101 ... 10000000 00000001 10000000 00000000 +1 +1 overflow +1 short y++; short x--; overflow y = 32767; ??? x = -32767; ??? -1 -1 -1 The maximum number is ? What if we add the maximum number by 1 ??? The minimum number is ? What if we subtract the minimum number by 1 ??? TRU-COMP1380 Number Systems 28 Note that computers use the 8-bit representation, the 16-bit representation, the 32-bit representation and the 64-bit representation with 2’complement for negative integers. In programming lanaguages byte, short, int, long, unsigned byte unsigned short unsigned int unsigned long 8-bit 16-bit 32-bit 64-bit When we use the 32-bit representation with 2’s complement, The maximum number is ? What if we add the maximum number by 1 ??? The minimum number is ? What if we subtract the minimum number by 1 ??? TRU-COMP1380 Number Systems 29 How to convert a negative binary number to decimal? An example of 8-bit representation, 10011001 = ??? TRU-COMP1380 Number Systems 30 Multiplication of Binary Numbers How to multiply two decimal numbers? E.g., 1001101 × 1 = ??? 1001101 × 10 = ??? 1001101 × 100 = ??? 1001101 × 101 = 1001101 × 10 × 10 + 1001101 × 0 + 1001101 × 1 What if we shift 1001101 left by one bit? 1001 -> 10010 What if we shift 1001101 left by two bits? 1001 -> 100100 Multiplication by a power of 2. TRU-COMP1380 Number Systems 31 Division of Binary Numbers Binary division? 1111 101 1001101 -101 1001 -101 1000 -101 111 -101 10 <- quotient <- remainder Try 1101011 / 110 Dividing negative binary numbers: division without sign, and then put the sign. TRU-COMP1380 Number Systems 32 Binary division by a power of 2? 1001101 / 10 = 100110 1001101 / 100 = 10011 1001101 / 1000 = 1001 What if we shift 1001101 right by 1 bit? What if we shift 1001101 right by 2 bits? What if we shift 1001101 right by 3 bits? 1001101 / 101 ??? TRU-COMP1380 Complicated implementation required Number Systems 33 Fractions: Fixed-Point How can we represent fractions? Use “binary point” to separate positive from negative powers of two -like “decimal point.” 2’s complement addition and subtraction still work. (Assuming binary points are aligned) 2-1 = 0.5 2-2 = 0.25 2-3 = 0.125 00101000.101 (40.625) + 11111110.110 (-1.25) (2’s complement) 00100111.011 (39.375) No new operations -- same as integer arithmetic. TRU-COMP1380 Number Systems 34 How to convert decimal fractions to binary? 0.62510 = ???2 0.625 * 2 = 1.25 -> 1 0.25 * 2 = 0.5 -> 0 0.5 * 2 = 1.0 -> 1 Therefore 0.101 (0.625 = x 2-1 + y 2-2 + z 2-3 + ...) (1.25 = x + y 2-1 + z 2-2 + ... => x = 1) (0.25 = y 2-1 + z 2-2 + ... ) 0.710 = ???2 TRU-COMP1380 0.7 * 2 = 1.4 -> 1 0.4 * 2 = 0.8 -> 0 0.8 * 2 = 1.6 -> 1 0.6 * 2 = 1.2 -> 1 0.2 * 2 = 0.4 -> 0 0.4 * 2 = 0.8 -> 0 0.8 * 2 = 1.6 -> 1 ... Therefore 0.1011001... How to deal with big numbers and small numbers? Number Systems 35 Very Large and Very Small: Floating-Point Large values: 6.023 × 1023 -- requires 79 bits Small values: 6.626 × 10-34 -- requires > 110 bits How to handle those big/small numbers? Use equivalent of “scientific notation”: F × 2E Need to represent F (fraction), E (exponent), and sign. 6.023 × 1023 = 0.6023 × 1024 6.626 × 10-34 = 0.6626 × 10-33 00101000.101 (40.62510) = 0.101000101 × 26. Store 6 in the exponent and 101000101 in mantissa. (Try the multiplication) IEEE 754 Floating-Point Standard (32-bits): TRU-COMP1380 1b 8 bits 23 bits S Exponent Fraction (mantissa) Number Systems 36