Activity: Number Systems & Binary Arithmetic

To convert decimal to binary, we repeatedly divide by 2 and record the remainders from bottom to top.

1. Divide the decimal number by 2.
2. Write down the remainder (0 or 1).
3. Repeat until the quotient is 0.
4. Read remainders in reverse order (MSB to LSB).

    25 / 2 = 12 rem 1
    12 / 2 = 6 rem 0
    6 / 2 = 3 rem 0
    3 / 2 = 1 rem 1
    1 / 2 = 0 rem 1
    Result: 11001₂

Each bit in a binary number has a weight of 2ⁿ. Sum the weights of all '1' bits.

Total: 8 + 0 + 2 + 1 = 11₁₀

Similar to binary conversion, but divide by 8. Use the powers of 8 table for reference.

    173 / 8 = 21 rem 5
    21 / 8 = 2 rem 5
    2 / 8 = 0 rem 2
    Result: 255₈

Divide by 16. Remember the mapping for remainders > 9:

10=A, 11=B, 12=C, 13=D, 14=E, 15=F

    450 / 16 = 28 rem 2
    28 / 16 = 1 rem 12 (C)
    1 / 16 = 0 rem 1
    Result: 1C2₁₆

Instead of going through decimal, use groupings:

• Octal: Group binary bits in 3s (e.g., 101 011₂ = 53₈)
• Hexadecimal: Group binary bits in 4s (e.g., 1010 1100₂ = AC₁₆)

    Octal: 111 | 101 = 75₈
    Hex: 0011 | 1101 = 3D₁₆

Test your ability to cross between multiple bases.

    Step 1: 72₈ to Binary = 111 010₂
    Step 2: 0011 1010₂ to Hex = 3A₁₆

    Step 1: A5₁₆ to Binary = 1010 0101₂
    Step 2: 010 100 101₂ to Octal = 245₈

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 1 = 0 (Carry 1)
• 1 + 1 + 1 = 1 (Carry 1)

When subtracting a larger bit from a smaller one (0-1), borrow 1 from the next higher column. The borrowed 1 acts as 2 in the current column (10 - 1 = 1).

   1001 (9)
-  0110 (6)
-------
   0011 (3)

Similar to decimal multiplication. Multiply by each bit and sum the partial products.

  101
x  11
-----
  101 (101 x 1)
+1010 (101 x 10)
-----
 1111 (15₁₀)

Use the long division method. It's actually easier than decimal because you only compare if the divisor goes in 0 or 1 times.

    11 goes into 11 exactly 1 time.
    Result: 100₂ (12 / 3 = 4)

Just flip every bit: 0 → 1 and 1 → 0.

Add the 1's complement of the subtrahend. If a carry is generated, add it to the LSB (End-around carry).

    7 = 0111
    3 = 0011 → 1's comp = 1100
    Sum: 0111 + 1100 = (1)0011
    End-around carry: 0011 + 1 = 0100 (4₁₀)

Add the 2's complement. Discard the final carry for positive results.

    10 = 00001010
    4 = 00000100 → 2's comp = 11111100
    Sum: 00001010 + 11111100 = (1)00000110
    Discard carry: 00000110 (6₁₀)

In signed 2's complement, the leftmost bit (MSB) is the sign bit (0 = positive, 1 = negative).

    +13 = 00001101
    1's comp = 11110010
    2's comp = 11110011 (Result)