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Understanding Memory Terminology and Data Representation in Computer Science

This outline provides an overview of essential concepts in memory terminology and data representation, integral to computer science. It covers the basics of binary, decimal, and hexadecimal systems, including conversions between them. Key topics include the definitions of bits, bytes, kilobytes, megabytes, and their relationships. It also explores ASCII codes and the methodologies for converting numbers between different bases, such as brute force and algorithmic approaches. This foundational knowledge is crucial for students in CSCI 1060 and anyone interested in understanding digital data representation.

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Understanding Memory Terminology and Data Representation in Computer Science

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  1. Memory Terminology &Data Representation CSCI 1060 Fall 2006

  2. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  3. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  4. Memory Terminology • Memory is comprised of sequences of binary digits — bits • Smallest measure of memory, two values, 0 or 1 (off or on) • Four bits is a nibble • Eight bits is a byte — can represent a single character • ASCII code – American Standard Code for Information Interchange

  5. Memory Terminology • 1 kilobyte = 210 bytes, not 1,000 bytes • 1 megabyte = 220 bytes (1,048,576 bytes) • 1 gigabyte = 230 bytes • 1 terabyte = 240 bytes • 1 petabyte = 250 bytes • … and so on, See Figure 2 • B, KB, MB, GB represent bytes • b, kb, mb, gb represent bits

  6. Memory Terminology How many bytes are in 4 megabytes? • 1 megabyte = 220 bytes = 1,048,576 bytes • 4 megabytes = 4 * 220 bytes = 4,194,304 bytes How many bytes are in 2 gigabytes? • 1 gigabyte = 230 bytes = 1,073,741,824 bytes • 2 gigabytes = 2 * 230 bytes = 2,147,483,648 bytes

  7. Memory Terminology How many bits are there in 32 kilobytes? • 1 byte = 8 bits • 32 kilobytes = 32 * 210 * 8 bits = 262,144 bits How many nibbles in 1 kilobyte? • 1 nibble = 4 bits, 1 byte = 8 bits • 1 kilobyte = 210 bytes * 8 bits / 4 bits = 2,048 nibbles

  8. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  9. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  10. Instruction/Data Representation Decimal Base • Decimal numbers are in base 10 • Digits are 0-9 • Increment the next space to the left when each slot is “full” • Can expand a number like 536: • 5*102 + 3*101 + 6*100 = 5*100 + 3*10 + 6*1 = 536 • Other number systems work exactly the same way

  11. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  12. Instruction/Data Representation Binary to Decimal • Binary numbers are in base 2 • Digits are 0 or 1 • Can expand a binary number like 1101 0111: • 1*27 + 1*26 + 0*25 + 1*24 + 0*23 + 1*22 + 1*21 + 1*20 = • 128 + 64 + 0 + 16 + 0 + 4 + 2 + 1 = 215 • Often, people will separate binary numbers into nibbles for readability

  13. Instruction/Data Representation Convert 0111 to decimal • 0*23 + 1*22 + 1*21 + 1*20 = 4 + 2 + 1 = 7 Convert 1100 1100 to decimal • 1*27 + 1*26 + 0*25 + 0*24 + 1*23 + 1*22 + 0*21 + 0*20 = 128 + 64 + 8 + 4 = 204 Convert 1010 0101 to decimal • 1*27 + 0*26 + 1*25 + 0*24 + 0*23 + 1*22 + 0*21 + 1*20 = 128 + 32 + 4 + 1 = 165

  14. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  15. Instruction/Data Representation Decimal to Binary • Find the powers of two that add up to the decimal number • Two methods for accomplishing this: • Brute Force • Algorithmically • Algorithmically will apply to any number system

  16. Instruction/Data Representation Decimal to Binary – Brute Force • Find powers of two up to some arbitrary number, use as a chart (See Figure 3) • Identify the biggest power of two that will go into the number and then subtract it • Repeat until you get a difference of 0 • List all powers of two as placeholders and put 1s where any power of two was used

  17. Instruction/Data Representation Decimal to Binary – Brute Force • Convert 152 from decimal to binary • 152 – 128 = 24 – 16 = 8 – 8 = 0 • _ _ _ _ _ _ _ _ • 1 0 0 1 1 0 0 0

  18. Instruction/Data Representation Decimal to Binary – Brute Force • Convert 201 from decimal to binary • 201 – 128 = 73 – 64 = 9 – 8 = 1 – 1 = 0 • _ _ _ _ _ _ _ _ • 1 1 0 0 1 0 0 1

  19. Instruction/Data Representation Decimal to Binary – Algorithmically • Continually divide the number by two • When you reach 1, divide one more time • Take the remainder (guaranteed to be either 1 or 0) and form a string of 1s and 0s • Reverse the string to get the binary representation

  20. Instruction/Data Representation Decimal to Binary – Algorithmically • Convert 152 from decimal to binary • 152 / 2 = 76 r 0 • 76/ 2 = 38 r 0 • 38 / 2 = 19 r 0 • 19 / 2 = 9 r 1 • 9 / 2 = 4 r 1 • 4 / 2 = 2 r 0 • 2 / 2 = 1 r 0 • 1 / 2 = 0 r 1 • Read remainders from bottom to top • (152)10 = (1001 1000)2

  21. Instruction/Data Representation Decimal to Binary – Algorithmically • Convert 201 from decimal to binary • 201 / 2 = 100 r 1 • 100/ 2 = 50 r 0 • 50 / 2 = 25 r 0 • 25 / 2 = 12 r 1 • 12 / 2 = 6 r 0 • 6 / 2 = 3 r 0 • 3 / 2 = 1 r 1 • 1 / 2 = 0 r 1 • Read remainders from bottom to top • (201)10 = (1100 1001)2

  22. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  23. Instruction/Data Representation Hexadecimal to Decimal • Hexadecimal numbers are in base 16 • Digits are 0-F (10 = A, 11 = B, etc.) • Can expand a hexadecimal number like 1128: • 1*163 + 1*162 + 2*161 + 8*160 = • 4096 + 256 + 32 + 8 = 4392 • Can expand a hexadecimal number like AF: • A*161 + F*160 = • 160 + 15 = 175

  24. Instruction/Data Representation Convert 589 to decimal • 5*162 + 8*161 + 9*160= 1280 + 128 + 8 = 1417 Convert FA8 to decimal • F*162 + A*161 + 8*160 = 3840 + 160 + 8 = 4008 Convert 1531 to decimal • 1*163 + 5*162 + 2*161 + 1*160= 4096 + 1280 + 48 + 1 = 5425

  25. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  26. Instruction/Data Representation Decimal to Hexadecimal • Continually divide the number by sixteen • When you reach a number fifteen or below, stop, that is your last digit (convert if necessary) • Reverse the string of remainders to get the hexadecimal representation

  27. Instruction/Data Representation Decimal to Hexadecimal • Convert 152 from decimal to hexadecimal • 152 / 16 = 9 r 8 • 9 / 16 = 0 r 9 • Read remainders from bottom to top • (152)10 = (98)16

  28. Instruction/Data Representation Decimal to Hexadecimal • Convert 201 from decimal to hexadecimal • 201 / 16 = 12 r 9 • 12 / 16 = 0 r 12 • Read remainders from bottom to top • (201)10 = (C9)16

  29. Instruction/Data Representation Decimal to Hexadecimal • Convert 5645 from decimal to hexadecimal • 5645 / 16 = 352 r 13 • 352 / 16 = 22 r 0 • 22 / 16 = 1 r 6 • 1 / 16 = 0 r 1 • Read remainders from bottom to top • (5645)10 = (160D)16

  30. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  31. Instruction/Data Representation Hexadecimal to Binary • Substitute the binary value of each digit • Can use intermediate step of converting each “slot” to decimal first (Do NOT calculate overall value) • A8 = 10 8 = 1010 1000 • 98 = 9 8 = 1001 1000 • C9 = 12 9 = 1100 1001

  32. Outline • Memory Terminology • Instruction/Data Representation • Decimal Base • Binary to Decimal • Decimal to Binary • Hexadecimal to Decimal • Decimal to Hexadecimal • Hexadecimal to Binary • ASCII Codes

  33. ASCII Codes • American Standard Code for Information Interchange • Used to represent data • For instance, the letter ‘A’ is a decimal 65, binary value of 0100 0001

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