🔢 Binary Math for Networking

📌 Overview

Computers strictly speak Binary (0s and 1s). Networking concepts like IP Addresses, Subnet Masks, and CIDR all rely on binary math.

🏗️ The Magic Table (Powers of 2)

Memorize this table. It is the key to everything in networking.

1286432168421
$2^7$$2^6$$2^5$$2^4$$2^3$$2^2$$2^1$$2^0$

🔄 1. Decimal to Binary

Goal: Convert a normal number (e.g., 192) into Binary (11000000).

Method: The Subtraction Game

Take your number and try to subtract values from the Magic Table, starting from the left (128).

Example: Convert 168 to Binary

  1. 128: Can I subtract 128 from 168? Yes. (Rem: 40). Write 1.
  2. 64: Can I subtract 64 from 40? No. Write 0.
  3. 32: Can I subtract 32 from 40? Yes. (Rem: 8). Write 1.
  4. 16: Can I subtract 16 from 8? No. Write 0.
  5. 8: Can I subtract 8 from 8? Yes. (Rem: 0). Write 1.
  6. 4: Remainder is 0. Write 0.
  7. 2: Remainder is 0. Write 0.
  8. 1: Remainder is 0. Write 0.

Result: 10101000

🔄 2. Binary to Decimal

Goal: Convert Binary (11111100) to Decimal (252).

Method: The Addition Game

Look at the Magic Table. Wherever there is a 1, take that number and add it to your total.

Example: Convert 11111100

1286432168421
11111100

Math: $128 + 64 + 32 + 16 + 8 + 4 = \mathbf{252}$

☯️ 3. 2's Complement (Negative Numbers)

Computers don't have a minus sign (-). They use 2's Complement to represent negative integers.

How to Calculate (-N)

To find the negative version of a binary number:

  1. Invert all bits (Change 1 to 0, 0 to 1). (This is 1's Complement).
  2. Add 1 to the result.

Example: Represent -5 in 4-bit binary

  1. Start with +5: 0101
  2. Invert bits: 1010
  3. Add 1:
      1010
+    1
------
  1011
  4. Result: 1011 is -5 in 2's complement.

Shortcut: Start from the right. Leave all digits alone up to and including the first '1'. Then flip everything to the left.

010(1) -> First '1' is at the end. Flip the rest: 101 + 1 = 1011

⚡ Exam Tips

  1. All Ones: 11111111 = 255.
  2. First Bit: In 8-bit signed numbers, if the first bit is 1, the number is Negative. If 0, it is Positive.
  3. Subnetting: You mostly care about Binary to Decimal and vice-versa for calculating Network IDs and Broadcast IDs.