Converting Decimal and Binary Values

Converting Between Decimal and Binary for IP Addresses

Introduction

  • Converting between decimal and binary is crucial for understanding IP addresses and networking concepts.
  • This skill is essential for working with IP ranges, prefixes, subnet masks, and routing.

Why Understand Decimal to Binary Conversion?

  • Human-readable IP address (Dotted Decimal Notation):
    • Example: 133.330.330.7
    • Four decimal numbers (0-255) separated by periods.
  • Computer-readable IP address (Binary):
    • A 32-bit binary number.
    • Four sets of eight bits (an octet or byte).
    • Each byte corresponds to the decimal number in the dotted decimal notation.

Converting Decimal to Binary

  • More complex than binary to decimal.
  • Tackle each decimal number individually.
Process

  1. Introduce the Binary Math Table:

    Position12345678
    Value1286432168421
    • Each position corresponds to a value in decimal.
    • A binary "1" in that position has the associated decimal value.
    • Example: A "1" in position 1 is 128 in decimal; a "1" in position 4 is 16 in decimal.

  2. Rules for Conversion:

    • Move from left to right (position 1 to position 8).
    • Rule 1: If the decimal number is smaller than the corresponding binary position value, write "0" in that position and move to the next.
    • Rule 2: If the decimal number is greater than or equal to the binary position value:
      • Subtract the binary position value from the decimal number.
      • Write "1" in that position.
      • Move to the next position.

  3. Example: Converting 133 to Binary:

    • Start at position 1 (128).
    • 133 is greater than 128, so apply Rule 2.
      • 133−128=5133 - 128 = 5
      • Write "1" in position 1.
    • Move to position 2 (64).
    • 5 is smaller than 64, so apply Rule 1.
      • Write "0" in position 2.
    • Repeat for positions 3, 4, and 5 (32, 16, and 8).
      • 5 is smaller than all, so write "0" in each.
    • Move to position 6 (4).
    • 5 is greater than 4, so apply Rule 2.
      • 5−4=15 - 4 = 1
      • Write "1" in position 6.
    • Move to position 7 (2).
    • 1 is smaller than 2, so apply Rule 1.
      • Write "0" in position 7.
    • Move to position 8 (1).
    • 1 is equal to 1, so apply Rule 2.
      • 1−1=01 - 1 = 0
      • Write "1" in position 8.
    • Result: 133 in decimal is 10000101 in binary.

  4. Example: Converting 33 to Binary:

    • Start at position 1 (128).
      • 33 is less than 128. Rule 1: Write "0", move to position 2.
    • Position 2 (64):
      • 33 is less than 64. Rule 1: Write "0", move to position 3.
    • Position 3 (32):
      • 33 is not less than 32. Rule 2: 33 - 32 = 1. Write "1", move to position 4.
    • Positions 4, 5, 6, 7 (16, 8, 4, 2):
      • 1 is less than all. Rule 1: Write "0" in each.
    • Position 8 (1):
      • 1 is not less than 1. Rule 2: 1 - 1 = 0. Write "1".
    • Result: 33 in decimal is 00100001 in binary.

  5. Example: Converting 7 to Binary:

    • Result: 7 in decimal is 00000111 in binary.

Converting Binary to Decimal

  • Much easier than decimal to binary.
Process

  1. Break the IP Address into Octets:

    • Each eight-bit binary value corresponds to a decimal number.
    • Example:
      • Binary: 10000101.00100001.00100001.00000111
      • Decimal: 133.33.33.7

  2. Use the Binary Math Table:

    Position12345678
    Value1286432168421

  3. Iterate Through Each Octet:

    • If there's a "1", add the corresponding decimal value from the table.
    • If there's a "0", add zero.

  4. Example:

    • Binary: 10000101
    • Calculation: 128+0+0+0+0+4+0+1=133128 + 0 + 0 + 0 + 0 + 4 + 0 + 1 = 133

  5. Repeat for All Octets:

    • Binary: 00100001
    • Calculation: 0+0+32+0+0+0+0+1=330 + 0 + 32 + 0 + 0 + 0 + 0 + 1 = 33

  6. Final Result:

    • Combine the decimal values to get the dotted decimal notation: 133.33.33.7

Practice

  • Pick random IP addresses and convert them back and forth to solidify your understanding.