Common Number Systems:

• Decimal (Base-10): Uses digits 0-9
• Binary (Base-2): Uses digits 0-1
• Octal (Base-8): Uses digits 0-7
• Hexadecimal (Base-16): Uses digits 0-9 and A-F

Decimal Place Values

In the number 4,329:

• 9 is in the ones place (10⁰ = 1)
• 2 is in the tens place (10¹ = 10)
• 3 is in the hundreds place (10² = 100)
• 4 is in the thousands place (10³ = 1000)

Therefore: 4,329 = 4×1000 + 3×100 + 2×10 + 9×1 = 4,329

Binary Place Values

In the binary number 1011:

Therefore: 1011₂ = 8 + 0 + 2 + 1 = 11₁₀

Why Binary is Important in Computing

Binary is used in computers because electronic components can easily represent two states:

• On/Off
• True/False
• High voltage/Low voltage

Each binary digit (bit) is the smallest unit of data in computing. Eight bits make a byte, which can represent 2⁸ = 256 different values.

Hexadecimal Place Values

In the hexadecimal number 2AF:

Therefore: 2AF₁₆ = 512 + 160 + 15 = 687₁₀

Why Hexadecimal is Useful

Hexadecimal is widely used in computing because:

• It's more compact than binary (one hex digit represents 4 binary digits)
• It's used for memory addresses, color codes, and machine code representation
• It's easier to read and write than long binary strings

Method: Division by 2

1. Divide the decimal number by 2
2. Write down the remainder (0 or 1)
3. Divide the quotient by 2
4. Repeat until the quotient becomes 0
5. Read the remainders from bottom to top

Method: Positional Value

1. Write down the powers of 2 for each position (from right to left, starting with 2⁰)
2. Multiply each binary digit by its corresponding power of 2
3. Add all the results

Method: Division by 16

1. Divide the decimal number by 16
2. Convert the remainder to a hexadecimal digit (10-15 → A-F)
3. Divide the quotient by 16
4. Repeat until the quotient becomes 0
5. Read the remainders from bottom to top

Method: Positional Value

1. Convert each hexadecimal digit to its decimal value (A-F → 10-15)
2. Multiply each decimal value by its corresponding power of 16 (from right to left, starting with 16⁰)
3. Add all the results

Method: Group by 4 Bits

1. Group binary digits into sets of 4, starting from the right
2. Add leading zeros to the leftmost group if needed
3. Convert each 4-bit group to its hexadecimal equivalent

Method: Convert Each Digit

1. Convert each hexadecimal digit to its 4-bit binary equivalent
2. Concatenate all the binary groups

Rules for Binary Addition

    1 1 1 1     (Carries)
      1 0 1 1   (First number)
    + 1 1 0 1   (Second number)
    ---------
    1 1 0 0 0   (Result)
                

Rules for Binary Subtraction

        1 0     (Borrows)
    1 1 0 1     (First number)
  - 0 1 0 1     (Second number)
    -------
    1 0 0 0     (Result)
                

For hexadecimal addition, you can:

1. Convert hex digits to decimal
2. Add them together
3. If the sum is 16 or greater, keep the remainder and carry the 1
4. Convert back to hex

      1 1       (Carries)
      2 A 7     (First number)
    + 3 9 F     (Second number)
    -------
      6 4 6     (Result)
                

Example: HTML Color Codes

HTML color codes use hexadecimal to represent RGB (Red, Green, Blue) values: