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Number Systems Exercise Solutions | Class 11 Computer Applications

Complete solutions for Chapter 2 Number Systems — binary, octal, hexadecimal conversions, binary arithmetic, 1's and 2's complement, BCD. TN State Board Class 11 Computer Applications.

Updated 16 June 202617 min read

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Number Systems: Exercise Solutions

How to Use This Page

This page provides complete, step-by-step solutions for all exercise questions from Chapter 2: Number Systems of the TN State Board Class 11 Computer Applications textbook. Work through each problem on your own first, then check your answers here. Pay special attention to the explanation after each solution — understanding why an answer is correct matters more than memorizing the result.

Key Concepts (Quick Recap)

ConceptDetails
Decimal (Base 10)Digits: 0–9; everyday number system
Binary (Base 2)Digits: 0, 1; fundamental language of computers
Octal (Base 8)Digits: 0–7; 3 binary digits = 1 octal digit
Hexadecimal (Base 16)Digits: 0–9, A–F; 4 binary digits = 1 hex digit
Decimal to BinaryDivide by 2, collect remainders bottom-up
Decimal to OctalDivide by 8, collect remainders bottom-up
Decimal to HexadecimalDivide by 16, collect remainders bottom-up
Binary to DecimalMultiply each bit by 2^position, sum results
1's ComplementFlip all bits (0 to 1, 1 to 0)
2's Complement1's complement + 1; used for negative number representation
BCDEach decimal digit is a 4-bit binary group (0000–1001)
Binary Addition0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 carry 1)

I. Choose the Correct Answer (MCQ)

1. The base of the hexadecimal number system is:
Answer
  • (a) 2
  • (b) 8
  • (c) 10
  • (d) 16 ✓
ExplanationThe hexadecimal system uses 16 distinct symbols (0–9 and A–F), so its base (radix) is 16.
2. The binary equivalent of decimal number 10 is:
Answer
  • (a) 1010 ✓
  • (b) 1100
  • (c) 1001
  • (d) 1110
Explanation10 / 2 = 5 remainder 0; 5 / 2 = 2 remainder 1; 2 / 2 = 1 remainder 0; 1 / 2 = 0 remainder 1. Reading remainders bottom-up: 1010.
3. The decimal equivalent of binary 1101 is:
Answer
  • (a) 11
  • (b) 13 ✓
  • (c) 15
  • (d) 12
Explanation1 x 2^3 + 1 x 2^2 + 0 x 2^1 + 1 x 2^0 = 8 + 4 + 0 + 1 = 13.
4. In BCD, each decimal digit is represented by:
Answer
  • (a) 2 bits
  • (b) 4 bits ✓
  • (c) 8 bits
  • (d) 16 bits
ExplanationBCD uses a 4-bit binary group to represent each decimal digit from 0 to 9. For example, decimal 5 is 0101 in BCD.
5. The 1's complement of binary 101100 is:
Answer
  • (a) 101100
  • (b) 010011 ✓
  • (c) 010100
  • (d) 101101
ExplanationTo find the 1's complement, flip every bit: 1 becomes 0 and 0 becomes 1. So 101100 becomes 010011.
6. The 2's complement of binary 1010 is:
Answer
  • (a) 0101
  • (b) 0110 ✓
  • (c) 1011
  • (d) 0100
ExplanationStep 1 — 1's complement of 1010 = 0101. Step 2 — Add 1: 0101 + 1 = 0110.
7. Which of the following is not a valid octal number?
Answer
  • (a) 457
  • (b) 238 ✓
  • (c) 670
  • (d) 123
ExplanationOctal numbers use only digits 0–7. The digit 8 in option (b) is invalid in the octal system.
8. The hexadecimal equivalent of decimal 255 is:
Answer
  • (a) EE
  • (b) AB
  • (c) FF ✓
  • (d) 99
Explanation255 / 16 = 15 remainder 15; 15 / 16 = 0 remainder 15. In hexadecimal, 15 is represented as F. So 255 = FF in hex.
9. Binary addition of 1100 + 1011 results in:
Answer
  • (a) 10111 ✓
  • (b) 11011
  • (c) 10101
  • (d) 11110
Explanation`` 1100 + 1011 ------ 10111 `` Column-wise from right: 0+1=1, 0+1=1, 1+0=1, 1+1=10 (0 with carry 1). Result: 10111.
10. The octal equivalent of binary 110101 is:
Answer
  • (a) 65 ✓
  • (b) 53
  • (c) 71
  • (d) 46
ExplanationGroup binary digits in sets of 3 from right: 110 101. 110 = 6, 101 = 5. So the octal equivalent is 65.

II. Very Short Answer Questions (2 Marks)

1. Define number system.
Answer

A number system is a systematic way of representing numbers using a set of symbols (digits) and a base (radix). Each number system has a fixed base that determines the total number of distinct symbols available. For example, the decimal system has base 10 with digits 0–9, while the binary system has base 2 with digits 0 and 1.

2. What is the base value of binary and octal number systems?
Answer

The binary number system has base 2, using only the digits 0 and 1. The octal number system has base 8, using digits 0 through 7. Each digit in an octal number represents exactly 3 binary digits (bits), making conversion between binary and octal straightforward.

3. Convert binary 10110 to decimal.
10110 = 1 x 2^4 + 0 x 2^3 + 1 x 2^2 + 1 x 2^1 + 0 x 2^0 = 16 + 0 + 4 + 2 + 0 = 22
Answer

The decimal equivalent of binary 10110 is 22.

4. What is 1's complement? Give an example.
Answer

The 1's complement of a binary number is obtained by changing (flipping) every 0 to 1 and every 1 to 0. It is one method used to represent negative numbers in binary.

Example: 1's complement of 110010 = 001101

5. What is BCD? Give an example.
Answer

BCD (Binary Coded Decimal) is a coding scheme where each decimal digit (0–9) is represented by its 4-bit binary equivalent. It allows easy conversion between decimal and binary representations.

Example: Decimal number 59 in BCD is 0101 1001 (5 = 0101, 9 = 1001).

6. Convert decimal 25 to binary.
25 / 2 = 12 remainder 1 12 / 2 = 6 remainder 0 6 / 2 = 3 remainder 0 3 / 2 = 1 remainder 1 1 / 2 = 0 remainder 1
Answer

Reading remainders from bottom to top: 11001

So, 25 in decimal = 11001 in binary.

III. Short Answer Questions (3 Marks)

1. Convert decimal 156 to hexadecimal.
156 / 16 = 9 remainder 12 (C) 9 / 16 = 0 remainder 9
Answer

Reading remainders from bottom to top: 9C

Verification: 9 x 16^1 + C x 16^0 = 9 x 16 + 12 = 144 + 12 = 156 (verified)

Therefore, 156 in decimal = 9C in hexadecimal.

2. Convert octal 745 to decimal.
745 (base 8) = 7 x 8^2 + 4 x 8^1 + 5 x 8^0 = 7 x 64 + 4 x 8 + 5 x 1 = 448 + 32 + 5 = 485
Answer

Therefore, octal 745 = decimal 485.

3. Find the 2's complement of binary number 10110100 and explain its significance.
Answer

Step 1 — Find 1's complement: Flip all bits of 10110100 to get 01001011

Step 2 — Add 1:

01001011 + 1 -------- 01001100

2's complement of 10110100 = 01001100

Significance: The 2's complement is the standard method for representing negative numbers in computers. If 10110100 represents a positive number (+180 in 8 bits), then its 2's complement 01001100 represents -180. This system allows subtraction to be performed using addition circuits, simplifying CPU design.

4. Perform binary addition: 11011 + 10110
Carry: 1 1 1 1 1 0 1 1 + 1 0 1 1 0 ----------- 1 1 0 0 0 1
Answer

Step-by-step from rightmost column:

  • 1 + 0 = 1
  • 1 + 1 = 10, write 0, carry 1
  • 0 + 1 + 1 (carry) = 10, write 0, carry 1
  • 1 + 0 + 1 (carry) = 10, write 0, carry 1
  • 1 + 1 + 1 (carry) = 11, write 1, carry 1
  • Bring down final carry: 1
Result110001

Verification: 11011 (binary) = 27, 10110 (binary) = 22; 27 + 22 = 49; 110001 (binary) = 49 (verified)

5. Convert hexadecimal 2F to binary and then to decimal.
Answer

Hexadecimal to Binary: Replace each hex digit with its 4-bit binary equivalent:

  • 2 = 0010
  • F = 1111

So, 2F in hexadecimal = 00101111 in binary = 101111 (removing leading zeros)

Binary to Decimal:

101111 = 1 x 2^5 + 0 x 2^4 + 1 x 2^3 + 1 x 2^2 + 1 x 2^1 + 1 x 2^0 = 32 + 0 + 8 + 4 + 2 + 1 = 47

Therefore, 2F (hex) = 101111 (binary) = 47 (decimal)

IV. Long Answer Questions (5 Marks)

1. Explain the different number systems in detail with examples.
Answer

There are four main number systems used in computer science:

1. Decimal Number System (Base 10):
Answer
  • Uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
  • This is the number system we use in everyday life
  • Each position represents a power of 10
  • Example: 345 = 3 x 10^2 + 4 x 10^1 + 5 x 10^0 = 300 + 40 + 5 = 345
2. Binary Number System (Base 2):
Answer
  • Uses only 2 digits: 0 and 1
  • Fundamental language of computers — all data is stored and processed in binary
  • Each position represents a power of 2
  • Example: 1101 = 1 x 2^3 + 1 x 2^2 + 0 x 2^1 + 1 x 2^0 = 8 + 4 + 0 + 1 = 13
3. Octal Number System (Base 8):
Answer
  • Uses 8 digits: 0, 1, 2, 3, 4, 5, 6, 7
  • Each position represents a power of 8
  • Convenient shorthand for binary (each octal digit = 3 binary digits)
  • Example: 57 (octal) = 5 x 8^1 + 7 x 8^0 = 40 + 7 = 47 (decimal)
4. Hexadecimal Number System (Base 16):
Answer
  • Uses 16 symbols: 0–9 and A(10), B(11), C(12), D(13), E(14), F(15)
  • Each position represents a power of 16
  • Convenient shorthand for binary (each hex digit = 4 binary digits)
  • Example: 2A (hex) = 2 x 16^1 + 10 x 16^0 = 32 + 10 = 42 (decimal)

Comparison Table:

DecimalBinaryOctalHexadecimal
0000000
5010155
10101012A
15111117F
25511111111377FF
2. Explain the step-by-step procedure for converting: (a) Decimal to Binary, (b) Binary to Octal, (c) Hexadecimal to Decimal.
Answer

(a) Decimal to Binary — Division Method:

Procedure: Repeatedly divide the decimal number by 2. Record the remainder at each step. The binary equivalent is obtained by reading remainders from bottom to top (last remainder to first remainder).

Example: Convert 42 (decimal) to binary

42 / 2 = 21 remainder 0 21 / 2 = 10 remainder 1 10 / 2 = 5 remainder 0 5 / 2 = 2 remainder 1 2 / 2 = 1 remainder 0 1 / 2 = 0 remainder 1

Reading bottom to top: 101010 (binary)

Verification: 1 x 2^5 + 0 x 2^4 + 1 x 2^3 + 0 x 2^2 + 1 x 2^1 + 0 x 2^0 = 32 + 8 + 2 = 42 (verified)

(b) Binary to Octal:

Procedure: Group the binary digits in sets of 3 starting from the right. Convert each 3-bit group to its octal equivalent. If needed, add leading zeros to the leftmost group.

Example: Convert 110101 (binary) to octal

Binary: 110 101 Groups of 3: 6 5 Octal digits: 6 5
Result65 (octal)

Another example: Convert 10111001 (binary) to octal

Add leading zero: 010 111 001 Octal digits: 2 7 1
Result271 (octal)

(c) Hexadecimal to Decimal:

Procedure: Multiply each hexadecimal digit by 16 raised to the power of its position (starting from 0 on the right). Sum all the products.

Example: Convert 3B2 (hex) to decimal

3B2 (hex) = 3 x 16^2 + B x 16^1 + 2 x 16^0 = 3 x 256 + 11 x 16 + 2 x 1 = 768 + 176 + 2 = 946 (decimal)

Another example: Convert 1A3 (hex) to decimal

1A3 (hex) = 1 x 16^2 + A x 16^1 + 3 x 16^0 = 256 + 160 + 3 = 419 (decimal)
3. Explain 1's complement and 2's complement with examples. How are they used to perform binary subtraction?
Answer

1's Complement:

The 1's complement of a binary number is obtained by flipping every bit — changing all 0s to 1s and all 1s to 0s.

Example: 1's complement of 10110010 = 01001101

2's Complement:

The 2's complement is obtained by adding 1 to the 1's complement of the number.

2's complement = 1's complement + 1

Example: Find 2's complement of 10110010

Step 1 — 1's complement: 01001101 Step 2 — Add 1: 01001101 + 1 -------- 01001110

2's complement of 10110010 = 01001110

Binary Subtraction Using 2's Complement:

To subtract B from A (A - B), add A to the 2's complement of B. If there is a carry from the leftmost bit, discard it — the result is positive. If there is no carry, take the 2's complement of the result and add a negative sign.

Example: Subtract 0011 from 1010 (i.e., 10 - 3)

Step 1: Find 2's complement of 0011 1's complement: 1100 Add 1: 1100 + 1 = 1101 Step 2: Add A to 2's complement of B 1010 + 1101 ------ 10111 Step 3: Discard the carry (leftmost 1) Result: 0111 = 7 (decimal)

Verification: 10 - 3 = 7 (verified)

Example where result is negative: Subtract 1010 from 0011 (i.e., 3 - 10)

Step 1: 2's complement of 1010 = 0110 Step 2: 0011 + 0110 = 1001 Step 3: No carry from leftmost bit — result is negative Take 2's complement of 1001: 0110 + 1 = 0111 Result: -7 (decimal)

Verification: 3 - 10 = -7 (verified)

4. Explain BCD (Binary Coded Decimal) with examples. What are its advantages and limitations?
Answer

BCD Encoding:

In BCD, each decimal digit (0–9) is represented by its 4-bit binary equivalent. The valid BCD codes are 0000 (0) through 1001 (9). The 4-bit codes from 1010 to 1111 are invalid in BCD.

BCD Code Table:

DecimalBCD Code
00000
10001
20010
30011
40100
50101
60110
70111
81000
91001

Example 1: Convert decimal 529 to BCD

5 = 0101 2 = 0010 9 = 1001 529 (decimal) = 0101 0010 1001 (BCD)

Example 2: Convert BCD 1001 0011 0111 to decimal

1001 = 9 0011 = 3 0111 = 7 BCD 1001 0011 0111 = 937 (decimal)

Advantages of BCD:

1. Easy decimal conversion — Direct digit-by-digit mapping makes conversion between decimal and BCD very simple and straightforward. 2. Human-readable — Each decimal digit is separately visible in its binary form, making it easy to interpret. 3. Accurate decimal arithmetic — No rounding errors in financial and accounting applications where exact decimal representation is critical. 4. Simple display — Easier to display on 7-segment displays and LED screens since each digit maps directly.

Limitations of BCD:

1. Wasteful storage — BCD uses 4 bits per digit but 4 bits can represent 0–15, so codes 1010–1111 (6 out of 16) are wasted. 2. Slower arithmetic — BCD arithmetic requires special correction steps (adding 6 when a digit exceeds 9), which adds processing overhead. 3. Larger memory footprint — A number requires more bits in BCD than in pure binary (e.g., 99 needs 8 bits in BCD but only 7 bits in binary). 4. Complex circuitry — Hardware for BCD arithmetic is more complex than for pure binary arithmetic.

Common Mistakes to Avoid

1. Forgetting to read remainders bottom-up: When converting decimal to another base, always read the remainders from the last division to the first (bottom to top), not top to bottom.

2. Incorrect hex digit values: Remember A=10, B=11, C=12, D=13, E=14, F=15. Do not confuse these with their positional values.

3. Missing leading zeros in binary-to-octal grouping: When grouping binary digits in sets of 3 from the right, add leading zeros to the leftmost group if it has fewer than 3 bits. Example: 10110 should be grouped as 010 110, not 10 110.

4. Adding 1 incorrectly in 2's complement: After finding the 1's complement, you must add 1 to the entire result, not just flip the last bit. For example, 2's complement of 1010 is 0101 + 1 = 0110, not 0100.

5. Ignoring carry in binary subtraction: When using 2's complement subtraction, if there is a carry out of the leftmost bit, discard it. The result is positive. If there is no carry, the result is negative — take the 2's complement of the answer and add a minus sign.

6. Invalid BCD codes: BCD only uses codes 0000 through 1001. Any 4-bit group from 1010 to 1111 is invalid. Never use these to represent decimal digits.

7. Confusing base values: Binary = base 2, Octal = base 8, Decimal = base 10, Hexadecimal = base 16. Always verify you are dividing or multiplying by the correct base.

8. Errors in binary addition carries: When adding multiple columns with carries, remember that 1+1+1 = 11 (write 1, carry 1). Do not forget to include the carry from the previous column.

Exam Checklist

  • [ ] I can convert decimal numbers to binary, octal, and hexadecimal using the division-remainder method
  • [ ] I can convert binary, octal, and hexadecimal numbers to decimal using the positional value method
  • [ ] I can convert between binary and octal (group/ungroup 3 bits)
  • [ ] I can convert between binary and hexadecimal (group/ungroup 4 bits)
  • [ ] I can perform binary addition with carries correctly
  • [ ] I can perform binary subtraction using the 2's complement method
  • [ ] I can find the 1's complement of any binary number
  • [ ] I can find the 2's complement (1's complement + 1)
  • [ ] I understand when to discard the carry in 2's complement subtraction
  • [ ] I can encode and decode BCD numbers
  • [ ] I know that BCD codes 1010–1111 are invalid
  • [ ] I can list the digits used in each number system (binary: 0–1; octal: 0–7; decimal: 0–9; hex: 0–9, A–F)
  • [ ] I always verify my answers by converting back to the original number system

Frequently Asked Questions

Repeatedly divide the decimal number by 2 and note the remainders. Read the remainders from bottom to top to get the binary equivalent.