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Binary, Octal, Hex Conversions, ISC Class 11

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Computer ScienceISC Class 11 · System of Numeration

Number Systems and Base Conversions (ISC Class 11 Computer Science)

Updated 17 October 2026 · Fact-checked

In a positional number system with base (radix) b, each digit has a weight that is a power of b; the value of a number is the sum of digit times weight. To convert a decimal integer to base b, divide repeatedly by b and read the remainders from bottom to top; for a fraction, multiply repeatedly by b and read the integer parts from top to bottom. Binary converts to octal by grouping three bits and to hexadecimal by grouping four bits.

Group bits from the right: 3 bits make one octal digit, 4 bits make one hex digit; pad with a leading 0 if needed.

Illustration drawn for QuestionBuddy.

Understand Number Systems: Positional Notation and Conversions among Decimal, Binary, Octal and Hexadecimal

Positional systems. Base 2 (digits 0, 1), 8 (0 to 7), 10 (0 to 9) and 16 (0 to 9, A to F, where A = 10 ... F = 15). The value of a number d3 d2 d1 d0 in base b is d3 b^3 + d2 b^2 + d1 b + d0; digits after the point have weights b^-1, b^-2, and so on.

To decimal. Multiply each digit by its weight and add: 1101011 (binary) = 64 + 32 + 8 + 2 + 1 = 107; A3F (hex) = 10 x 256 + 3 x 16 + 15 = 2,623; 745 (octal) = 7 x 64 + 4 x 8 + 5 = 485.

From decimal. *Integer part:* divide by the base repeatedly; the remainders, read from the last to the first, give the number. 156 / 2 gives remainders 0, 0, 1, 1, 1, 0, 0, 1 (bottom to top: 10011100); 156 in octal is 234; in hexadecimal 9C. *Fractional part:* multiply by the base; the integer parts, read in order, give the fraction. 0.625 x 2 = 1.25 (1), 0.25 x 2 = 0.5 (0), 0.5 x 2 = 1.0 (1): 0.101. So 10.375 (decimal) = 1010.011 (binary).

Binary, octal and hexadecimal shortcuts. Octal digit = 3 bits; hexadecimal digit = 4 bits (group from the binary point, padding with zeros).

Hex digit0123456789ABCDEF
Binary0000000100100011010001010110011110001001101010111100110111101111

2F (hex) = 0010 1111 = 101111 (binary) = 47; 745 (octal) = 111 100 101 = 1 1110 0101 = 1E5 (hex).

Counting. n bits give 2^n values (0 to 2^n - 1): 8 bits give 0 to 255.

Number Systems: Positional Notation and Conversions among Decimal, Binary, Octal and Hexadecimal: key facts

Value in base b
Value = sum of digit x b^position | Positions start at 0 on the right
Fractions use negative powers.
Decimal to base b
Integer: divide by b, read remainders upward | Fraction: multiply by b, read integer parts downward
Repeat.
Grouping
Binary to octal: 3 bits | Binary to hexadecimal: 4 bits
From the binary point.

How to answer Number Systems: Positional Notation and Conversions among Decimal, Binary, Octal and Hexadecimal questions

Choose the shortest route: direct conversion or via binary.

  1. 1Write the weights of the source base.
  2. 2To decimal: multiply and add.
  3. 3From decimal: divide or multiply by the target base.
  4. 4Between binary, octal and hexadecimal, group the bits.

Number Systems: Positional Notation and Conversions among Decimal, Binary, Octal and Hexadecimal in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Hexadecimal colours

Suppose a web colour code is given as FF in hexadecimal for the red channel.

What is its decimal value and binary form?

Try it first, then show the solutionSolution
  1. FF = 15 x 16 + 15 = 255.
  2. In binary: 1111 1111.

Answer: 255 and 11111111.

Memory hook: Two hex digits make one byte.

Tricks to remember Number Systems: Positional Notation and Conversions among Decimal, Binary, Octal and Hexadecimal

Short memory aids. Say each one aloud once, then use it on a question the same day.

  • Remainders upward

    Divide, then read from the last remainder to the first.

    Use it when: Integer conversion.

  • 3 and 4

    Octal groups of 3, hex groups of 4.

    Use it when: Shortcut.

  • A to F is 10 to 15

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

    Use it when: Hex digits.

Number Systems: Positional Notation and Conversions among Decimal, Binary, Octal and Hexadecimal: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Digit values · 1 mark · MCQBoard level

The hexadecimal digit B represents the decimal value:
(a) 10 (b) 11 (c) 12 (d) 13

Try it first, then show the solutionSolution
  1. A = 10, B = 11.

Answer: (b) 11

Example 2: To decimal · 3 marks · Short answerBoard level

Convert to decimal: (i) 1101011 (binary) (ii) 745 (octal) (iii) A3F (hexadecimal). (3 marks)

Try it first, then show the solutionSolution
  1. (i) 64 + 32 + 0 + 8 + 0 + 2 + 1 = 107.
  2. (ii) 7 x 64 + 4 x 8 + 5 = 448 + 32 + 5 = 485.
  3. (iii) 10 x 256 + 3 x 16 + 15 = 2,560 + 48 + 15 = 2,623.

Answer: (i) 107; (ii) 485; (iii) 2,623.

Example 3: From decimal · 4 marks · Short answerHarder board level

Convert 156 to binary, octal and hexadecimal by division, showing the remainders. (4 marks)

Try it first, then show the solutionSolution
  1. 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 upward 10011100.
  2. Octal: 156/8 = 19 r 4; 19/8 = 2 r 3; 2/8 = 0 r 2: 234.
  3. Hexadecimal: 156/16 = 9 r 12 (C): 9C.
  4. Check: 128 + 16 + 8 + 4 = 156.

Answer: 10011100 (binary); 234 (octal); 9C (hexadecimal).

Example 4: Fractions and grouping · 7 marks · Long answerHarder board level

(a) Convert 10.375 to binary. (b) Convert 2F (hex) to binary and decimal. (c) Convert 745 (octal) to hexadecimal through binary. (d) Convert 110101 (binary) to octal and hexadecimal. (7 marks)

Try it first, then show the solutionSolution
  1. (a) 10 = 1010; 0.375 x 2 = 0.75 (0), 0.75 x 2 = 1.5 (1), 0.5 x 2 = 1.0 (1): 0.011. Result 1010.011.
  2. (b) 2 = 0010, F = 1111: 00101111 = 101111; decimal 2 x 16 + 15 = 47.
  3. (c) 7 = 111, 4 = 100, 5 = 101: 111100101; regroup in fours from the right: 1 1110 0101 = 1E5.
  4. (d) 110 101 = 65 (octal); 0011 0101 = 35 (hex); decimal 53.

Answer: (a) 1010.011; (b) 101111, 47; (c) 1E5; (d) 65 octal, 35 hex.

Example 5: Choose the base · 5 marks · Long answerHOTS / competency

A memory address is 1010 0111 1100 in binary. Write it in hexadecimal and octal and say why programmers prefer hexadecimal to binary for such addresses. Also find the number of different values a 12-bit address can take and the largest. (5 marks)

Try it first, then show the solutionSolution
  1. Groups of 4: 1010 0111 1100 = A7C (hex).
  2. Groups of 3 from the right: 101 001 111 100 = 5174 (octal).
  3. Hexadecimal is short (3 digits for 12 bits) and converts directly to groups of four bits.
  4. 12 bits give 2^12 = 4,096 values; the largest is 4,095 (FFF in hexadecimal).

Answer: A7C (hex); 5174 (octal); hexadecimal is compact and maps neatly to bits. A 12-bit address has 4,096 values, up to 4,095 (FFF).

Where students lose marks in Number Systems: Positional Notation and Conversions among Decimal, Binary, Octal and Hexadecimal

The slips examiners see most often, and what to write instead.

  • Mistake: Reading remainders top to bottom

    Order error.

    Wrong working

    00111001

    Right working

    10011100

    Do this: Read from the last remainder upward.

    Usually costs: 2 marks.

  • Mistake: Grouping from the left

    Not from the point.

    Wrong working

    Grouping 101001111100 into 3s from the left without padding

    Right working

    Group from the right

    Do this: Start at the binary point.

    Usually costs: 1-2 marks.

  • Mistake: Using letters wrongly in hex

    Value slip.

    Wrong working

    C = 13

    Right working

    C = 12

    Do this: A = 10 to F = 15.

    Usually costs: 1 mark.

Exam tips

  • Show the division table for each conversion.
  • Check by converting back.
  • Pad with zeros when grouping.

Practice questions on Number Systems: Positional Notation and Conversions among Decimal, Binary, Octal and Hexadecimal

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

  1. Q1 · 1 mark · MCQBoard level

    The binary number 1010 equals which hexadecimal digit?
    (a) 8 (b) 9 (c) A (d) B

    Hint

    10 in decimal.

    Answer

    (c) A

  2. Q2 · 3 marks · Short answerBoard level

    Convert 47 (decimal) to binary and 1011.101 (binary) to decimal. (3 marks)

    Hint

    32 + 8 + 4 + 2 + 1.

    Answer

    47 = 101111. 1011.101 = 8 + 2 + 1 + 0.5 + 0.125 = 11.625.

  3. Q3 · 4 marks · Short answerHarder board level

    Convert 7F (hex) to binary, octal and decimal, and 255 (decimal) to hexadecimal. (4 marks)

    Hint

    0111 1111.

    Answer

    7F = 01111111 = 177 (octal, groups 001 111 111) = 127. 255 = FF.

  4. Q4 · 5 marks · Long answerHarder board level

    Convert 0.8125 to binary and 36.5 to hexadecimal. Also find the decimal value of the octal number 17.4. (5 marks)

    Hint

    0.8125 x 2 = 1.625.

    Answer

    0.8125 x 2 = 1.625 (1); 0.625 x 2 = 1.25 (1); 0.25 x 2 = 0.5 (0); 0.5 x 2 = 1.0 (1): 0.1101. 36.5: 36 = 24 (hex); 0.5 = 0.8 (hex): 24.8. Octal 17.4 = 1 x 8 + 7 + 4/8 = 15.5.

  5. Q5 · 3 marks · Short answerHOTS / competency

    A student claims 1A + 1A in hexadecimal is 2A. Check by converting to decimal and give the correct hexadecimal sum. (3 marks)

    Hint

    26 + 26.

    Answer

    1A = 26; 26 + 26 = 52 = 3 x 16 + 4 = 34 (hex). So the correct sum is 34, not 2A (2A = 42).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

  1. 1.Digits of base 16?Show answer

    0 to 9 and A to F.

  2. 2.Octal groups of?Show answer

    3 bits.

  3. 3.Hex groups of?Show answer

    4 bits.

  4. 4.F in decimal?Show answer

    15.

  5. 5.8 bits give?Show answer

    256 values.

Number Systems: Positional Notation and Conversions among Decimal, Binary, Octal and Hexadecimal: frequently asked questions

Why binary in computers?

Two states are easy to build.

What is radix?

The base of a number system.

Are leading zeros significant?

They do not change the value.

What is the base of BCD?

Not covered here.