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Number Systems and Computer Arithmetic

Learn about number systems and computer arithmetic, including conversion between decimal, binary, and hexadecimal numbers, addition, subtraction, multiplication, division, and representation of negative integers and fractions.

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Number Systems and Computer Arithmetic

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  1. Number Systems and Computer Arithmetic Winter 2014 COMP 1380 Discrete Structures I Computing Science Thompson Rivers University

  2. 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 Number Systems

  3. 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 Number Systems

  4. 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 Number Systems

  5. 1. Number Systems Number Systems

  6. 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 Number Systems

  7. 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 nbits has 2npossible 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? Number Systems

  8. 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 Number Systems

  9. Uses two digits 0 and 1. • 02 = 0×20 = 010 • 12 = 1×20 = 110 • 102 = 1×21 + 0×20 = 210 • 112 = 1×21 + 1×20 = 310 • 1002 = 1×22 + 0×21 + 0×20 = 410 • 1012 = 1×22 + 0×21 + 1×20 = 510 • 1102 = 1×22 + 1×21 + 0×20 = 610 • 1112 = 1×22 + 1×21 + 1×20 = 710 • 10002 = 1×23 + 0×22 + 0×21 + 0×20 = 810 • 10012 = 1×23 + 0×22 + 0×21 + 1×20 = 910 • ... Number Systems

  10. Powers of 2 • 12 = 1×20 = 110 • 102 = 1×21 + 0×20 = 210 • 1002 = 1×22 + 0×21 + 0×20 = 410 • 10002 = 1×23 + 0×22 + 0×21 + 0×20 = 810 • 1 00002 = 24 = 1610 • 10 00002 = 25 = 3210 • 10 000002 = 26 = 6410 Number Systems

  11. 1000 00002 = 27 = ???10 • 1 0000 00002 = 28 = ???10 • 10 0000 00002 = 29 = ???10 • 100 0000 00002 = 210 = ???10 • 1000 0000 00002 = 211 = ???10 • 1 0000 0000 00002 = 212 = ???10 • Can you memorize the above powers of 2? • Converting to decimals • 11012 = ???10 • 1011 00102 = ???10 • 1011.00102 = ???10 Number Systems

  12. Converting Decimal to Binary Quotient Remainder • 23 / 2 11 1 11 / 2 5 1 5 / 2 2 1 2 / 2 1 0 1 / 2 0 1 => 2310 = 101112 • 27110 = ???2 • 607110 = ???2 Number Systems

  13. Another similar idea • 27110 = ???2 256 < 271 < 512 -> 271 = 256 + 15 = 1 0000 00002 + 15 8< 15 < 16 -> 15 = 8 + 7 = 10002 + 7 => 271 = 1 0000 00002 + 15 = 1 0000 00002 + 10002 + 7 = 1 0000 00002 + 10002 + 1112 = 1 0000 11112 • 127110 = ???2 Number Systems

  14. 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 0x23c9d6ef = ???2 • 14816= ???1014816= ???2 4 bitscan be used for a hexadecimal number, 0, ..., F. Please memorize it! Number Systems

  15. Converting Decimal to Hexadecimal Quotient Remainder • 328 / 16 = 20 × 16 + 8 20 / 16 = 1 × 16 + 4 1 / 16 = 0 × 16 + 1 => 32810 = 14816 = ???2 • 14816 = (1 × 162 + 4 × 161 + 8 × 160)10 • 19210 = ???16 Number Systems

  16. Converting Binary to Hexadecimal • 4DA916= ???2 • 1001101101010012 = ???16 = 100 1101 1010 1001 = 4DA9 • 10 11102 = 0x??? = ???10 • 0100 1110 1011 1001 01002 = 0x??? = ???10 Number Systems

  17. 2. Computer Arithmetic Number Systems

  18. 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. • E.g., 10010 10010 1111 + 1001+ 1011+ 1 11011 11101 ??? 10111 + 111 ??? carry Number Systems

  19. 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 10010 -1-1 -11 -11 -11 1 ?1 ?11 1111 Try: 101010 How to do? 1 34–79=? -101-10 • Is subtraction easier or more difficult than addition? Why? Number Systems

  20. 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 -> Number Systems

  21. 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 Number Systems

  22. 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. 00101010 00101010 + 11111010+ 11111011 1 00100100 1 00100101 + 1 00100101 • 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? 1’s complement addition 2’s complement addition Number Systems

  23. Can we use 8-bit 1’s complement addition for 12 – 102 = –12? 1 00000001 - 10+ 11111101<- 8-bit 1’s complement of 10 11111110 <- Is this correct? (Is this 1’s complement of 1?) • Let’s use 8-bit 2’s complement addition for 12 – 102. 00000001 +11111110<- 2’s complement of 10 11111111 <- 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? Number Systems

  24. 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. • 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? • How to represent positive integer 5? • 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. Number Systems

  25. How about -9? • 10001001 is really okay? • 00001001 (9) + 10001001 (-9) = 10010010 (-18) It is wrong! • We need a different representation for negative integers. • What is the 8-bit 1’s complement of 9? • 11110110 <- 8-bit 1’s complement of 9 • 00001001 + 11110110 <- 9 + 8-bit 1’s complement of 9 = 11111111 <- Is it zero? (1’s complement of 0?) • What is the 2’s complement of 9? • 11110111 <- 8-bit 2’s complement of 9 • 00001001 + 11110111 <- 9 + 8-bit 2’s complement of 9 = 1 00000000 <- It looks more like zero. • 2’s complement representation is used for negative integers. Number Systems

  26. 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 11110011 10000001 Number Systems

  27. 8-bit representation with 2’s complement 127 01111111 126 01111110 ... ... 2 00000010 1 00000001 0 00000000 -1 11111111 (28 – 1) -2 11111110 -3 11111101 ... ... -127 10000001 -128 10000000 • The maximum number is ? byte y = 125; y += 4; ?? • The minimum number is ? byte x = -126; x -= 5; ?? • What if we add the maximum number by 1 ??? • What if we subtract the minimum number by 1 ??? overflow +1 +1 -1 overflow +1 -1 -1 Number Systems

  28. 16-bit representation with 2’s complement ... 01111111 11111111 ... 01111111 11111110 ... ... 3 00000000 00000011 2 00000000 00000010 1 00000000 00000001 0 00000000 00000000 -1 11111111 11111111 -2 11111111 11111110 -3 11111111 11111101 ... ... ... 10000000 00000001 ... 10000000 00000000 • 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 ??? overflow +1 +1 +1 short y = 32767; y++; ??? short x = -32767; x--; ??? -1 -1 -1 overflow Number Systems

  29. 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, unsigned byte 8-bit • short, unsigned short 16-bit • int, unsigned int 32-bit • long, unsigned long 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 ??? Number Systems

  30. How to convert a negative binary number to decimal? • An example of 8-bit representation, • 10011001 = ??? Number Systems

  31. 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. Number Systems

  32. Division of Binary Numbers • Binary division? 1111 <- quotient 101 1001101 -101 1001 -101 1000 -101 111 -101 10 <- remainder • Try 1101011 / 110 • Dividing negative binary numbers: division without sign, and then put the sign. Number Systems

  33. 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 ??? Complicated implementation required Number Systems

  34. 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) 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) No new operations -- same as integer arithmetic. Number Systems

  35. How to convert decimal fractions to binary? • 0.62510 = ???2 • 0.625 * 2 = 1.25 -> 1 (0.625 = x 2-1 + y 2-2 + z 2-3 + ...) • 0.25 * 2 = 0.5 -> 0 (1.25 = x + y 2-1 + z 2-2 + ... => x = 1) • 0.5 * 2 = 1.0 -> 1 (0.25 = y 2-1 + z 2-2 + ... ) • Therefore 0.101 • 0.710 = ???2 • 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

  36. 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): 1b 8 bits 23 bits S Exponent Fraction (mantissa) Number Systems

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