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Chapter 1 Representing Data in a Computer

Chapter 1 Representing Data in a Computer. 1.1 Binary and Hexadecimal Numbers. Decimal Numbers. Base 10 4053 = 3 x 1 + 5 x 10 + 0 x 10 2 + 4 x 10 3. 1s. 10s. 100s. 1000s. Binary Numbers. Base 2, using bits (binary digits) 0 and 1

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Chapter 1 Representing Data in a Computer

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  1. Chapter 1 Representing Data in a Computer

  2. 1.1 Binary and Hexadecimal Numbers

  3. Decimal Numbers Base 10 4053 = 3 x 1 + 5 x 10 + 0 x 102 + 4 x 103 1s 10s 100s 1000s

  4. Binary Numbers Base 2, using bits (binary digits) 0 and 1 10111 = 1 x 1 + 1 x 2 + 1 x 22 + 0 x 23 + 1 x 24 1s 101112 = 2310 2s 4s 8s 16s

  5. Hexadecimal Numbers • Base 16 • Digits • 0-9 same as decimal • A for 10 • B for 11 • C for 12 • D for 13 • E for 14 • F for 15

  6. Hexadecimal to Decimal Base 16 5CB = 11 x 1 + 12 x 16 + 5 x 162 5CB16 = 148310 1s 16s 256s

  7. Using Windows Calculator

  8. Converting decimal to hex • Use a calculator that does hex calculations or • Use this algorithm repeat divide DecimalNumber by 16, getting Quotient and Remainder;Remainder (in hex) is the next digit (right to left);DecimalNumber := Quotient;until DecimalNumber = 0;

  9. 1.2 Character Codes

  10. Character Codes • Letters, numerals, punctuation marks and other characters are represented in a computer by assigning a numeric value to each character • The system commonly used with microcomputers is the American Standard Code for Information Interchange (ASCII)

  11. Examples of ASCII Codes • Printable characters • Uppercase M codes as 4D16 = 10011012 • Lowercase m codes as 6D16 = 11011012 • Numeral 5 codes as 3516 • Space codes as 2016 • Control characters • Backspace codes as 0816 • Carriage return CR (0D16) and linefeed LF (0A16) together create a line break

  12. 1.3 Unsigned and Signed Integers

  13. Standard Number Lengths • Byte – 8 bits • Word – 16 bits • Doubleword – 32 bits • Quadword – 64 bits

  14. Unsigned Representation • Just binary in one of the standard lengths • E47A is the word-length unsigned representation for the decimal number 58490

  15. Signed Representation • 2’s complement representation used in 80x86 • One of the standard lengths • High-order (leading) bit gives sign • 0 for positive • 1 for negative • For a negative number, you must perform the 2’s complement operation to find the corresponding positive number

  16. 2’s Complement Operation • +/- button on many calculators • Manually by subtracting from 100…0 • E47A represents a negative word-length signed number since E = 1110 • 10000 – E47A = 1B86 = 704610,so E47A is the 2’s complement signed representation for -7046 minus

  17. Multiple Interpretations • One pattern of bits can have many different interpretations • The word FE89 can be interpreted as • An unsigned number whose decimal value is 65513 • A signed number whose decimal value is -375

  18. 1.4 Integer Addition and Subtraction

  19. Addition • Same for unsigned and 2’s complement signed numbers, but the results may be interpreted differently • The two numbers will be byte-size, word-size, doubleword-size or quadword-size • Add the bits and store the sum in the same length as the operands, discarding an extra bit (if any)

  20. Carry • If the sum of two numbers is one bit longer than the operand size, the extra 1 is a carry (or carry out). • A carry is discarded when storing the result, but we’ll see how the 80x86 CPU records it. • For unsigned numbers, a carry means that the result was too large to be stored – the answer is wrong.

  21. Carry In • A 1 carried into the high-order (sign, leftmost) bit position during addition is called a carry in. • Byte-length example carry in carry out

  22. Overflow • Overflow occurs when there is a carry in but no carry out, or when there is a carry out but no carry in. It recorded by the CPU. • If overflow occurs when adding two signed numbers, then the result will be incorrect. • Word-length example • Overflow, so AC99 incorrectif viewed as sum of signed numbers • No carry, so AC99 correct ifviewed as sum of unsigned numbers 483F + 645A AC99

  23. Subtraction • Take the 2’s complement of second number and add it to the first number • Overflow occurs in subtraction if it occurs in the corresponding addition • CF is set for a borrow in subtraction – when the second number is larger than the first as unsigned numbers. There is a borrow in subtraction when there is not a carry in the corresponding addition.

  24. 1.5 Other Systems For Representing Numbers

  25. 1’s complement • Similar to 2’s complement but to negate a number you flip all its bits. • The 1’s complement of 01110000 is 10001111 • 1’s complement rarely used for signed integers in modern systems

  26. Binary Coded Decimal • BCD uses four bits to encode each decimal digit • 479 could be encoded in two bytes as0000 0100 0111 1001 • The Intel architecture has only a few instructions to do arithmetic on BCD numbers

  27. Floating Point • IEEE single precision is a popular 32-bit format • Write the number in base 2 “scientific notation” • Sign bit (0 positive, 1 negative) • 8 bits for exponent (actual exponent plus a bias of 127) • 23 bits for fraction (omitting the leading 1)

  28. Floating Point Example • 78.37510 = 1001110.0112 • 1001110.0112 = 1.001110011  26 • 0 10000101 00111001100000000000000 sign + exponent,127+6 in binary fraction, with leading 1 removed and trailing 0’s added to make 23 bits

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