Number Systems for ISC Class 11: Conversions, Binary Arithmetic and 2's Complement

Number system questions are some of the quickest marks in the ISC Class 11 Computer Science paper. They follow fixed methods, and every answer can be checked. These notes cover every conversion you need, binary arithmetic, and complements, with fully worked examples.

The four number systems

• Binary: base 2, digits 0–1
• Octal: base 8, digits 0–7
• Decimal: base 10, digits 0–9
• Hexadecimal: base 16, digits 0–9 and A–F (A = 10, B = 11, C = 12, D = 13, E = 14, F = 15)

Write the base as a subscript, for example (1011)₂ or (2AF)₁₆.

Decimal to binary, octal or hex

Divide repeatedly by the base and read the remainders from the bottom up.

(156)₁₀ to binary:

156 ÷ 2 = 78  r 0
 78 ÷ 2 = 39  r 0
 39 ÷ 2 = 19  r 1
 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 upwards: (10011100)₂

Check: 128 + 16 + 8 + 4 = 156.

Using the same method, (156)₁₀ = (234)₈ (156 ÷ 8 = 19 r 4, 19 ÷ 8 = 2 r 3, then 2) and (9C)₁₆ (156 ÷ 16 = 9 r 12, and 12 is C).

Any base to decimal

Multiply each digit by its place value and add the results.

(2AF)₁₆ = 2×16² + 10×16¹ + 15×16⁰ = 512 + 160 + 15 = (687)₁₀
(234)₈  = 2×64 + 3×8 + 4         = 128 + 24 + 4   = (156)₁₀

Binary to octal and hexadecimal (and back)

Because 8 = 2³ and 16 = 2⁴, you can convert directly, without going through decimal:

• Binary to octal: group the bits in 3s from the right.
• Binary to hex: group the bits in 4s from the right.

10011100  →  010 011 100  →  (234)₈
10011100  →  1001 1100    →  (9C)₁₆

To go the other way, replace each octal digit with 3 bits, or each hex digit with 4 bits.

Fractions

Decimal fraction to binary: multiply the fractional part by 2 repeatedly, and read the integer parts from the top down.

0.625 × 2 = 1.25  → 1
0.25  × 2 = 0.5   → 0
0.5   × 2 = 1.0   → 1
(0.625)₁₀ = (0.101)₂

Binary fraction to decimal: use negative powers of 2.

(10110.11)₂ = 16 + 4 + 2 + 0.5 + 0.25 = (22.75)₁₀

Binary addition

The rules are: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 (write 0, carry 1), and 1 + 1 + 1 = 11 (write 1, carry 1).

   1011    (11)
 + 0110    (6)
 ------
  10001    (17)

1's and 2's complement

• 1's complement: flip every bit (0 becomes 1, and 1 becomes 0).
• 2's complement: the 1's complement plus 1. Computers use it to store negative numbers.

−6 in 8-bit 2's complement:

+6           = 00000110
1's comp     = 11111001
+1           = 11111010   ← −6

Subtraction using 2's complement

To find A − B, add A to the 2's complement of B and discard the final carry.

13 − 6 (5 bits):

13          = 01101
6           = 00110 → 2's complement = 11010
01101 + 11010 = 1 00111 → discard the carry → 00111 = 7

Common mistakes

• Reading the remainders top-down instead of bottom-up for whole numbers. For fractions it is the other way round: top-down.
• Grouping bits from the left instead of the right when converting to octal or hex. Pad with zeros on the left if needed.
• Forgetting to add 1 after flipping the bits for the 2's complement.

Practice questions

1. Convert (45.375)₁₀ to binary.
2. Convert (3F7)₁₆ to octal.
3. Find 9 − 12 using 2's complement in 5 bits, and interpret the result.

Answers:

1. (101101.011)₂
2. 3F7 = 0011 1111 0111 → 001 111 110 111 → (1767)₈
3. 01001 + 10100 (the 2's complement of 01100) = 11101. There is no carry, so the answer is negative: the 2's complement of 11101 is 00011, so the result is −3.

Continue with Logic Gates and Adder Circuits and Propositional Logic to complete the ISC Class 11 hardware and logic unit.