Number Systems and Data Representation: Binary, Octal, Decimal, Hexadecimal and Logic Gates संख्या पद्धति और डेटा निरूपण: Binary, Octal, Decimal, Hexadecimal और Logic Gates

Number Systems and Data Representation

Numbers, characters, images and other forms of computer data are ultimately represented using bits. Number systems help us understand binary data, memory values and the logical operation of computers.

1. Number System and Base

The number of distinct digits available in a number system is called its base or radix. In a positional number system, a digit’s value depends on both the digit and its position.

In hexadecimal, A, B, C, D, E and F represent decimal values 10, 11, 12, 13, 14 and 15 respectively.

2. Positional Value

For an integer written in base b, positional weights from the right are b⁰, b¹, b² and so on.

Examples:

(352)₁₀ = 3 × 10² + 5 × 10¹ + 2 × 10⁰

(1011)₂ = 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰

= 8 + 0 + 2 + 1 = (11)₁₀

Positions to the right of a radix point have weights b⁻¹, b⁻² and so on.

3. Binary-to-Decimal Conversion

Multiply each binary digit by its positional weight and add the results.

Example: Convert (110101)₂ to decimal

= 1 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰
= 32 + 16 + 0 + 4 + 0 + 1
= (53)₁₀

Example: Convert (101.101)₂ to decimal

= 1 × 2² + 0 × 2¹ + 1 × 2⁰ + 1 × 2⁻¹ + 0 × 2⁻² + 1 × 2⁻³
= 4 + 0 + 1 + 0.5 + 0 + 0.125
= (5.625)₁₀

4. Decimal Integer to Binary

Repeatedly divide the decimal integer by 2 and record each remainder. Read the remainders from bottom to top.

Example: Convert (45)₁₀ to binary

Reading the remainders from bottom to top gives (45)₁₀ = (101101)₂.

5. Octal and Hexadecimal to Decimal

Octal example

(157)₈
= 1 × 8² + 5 × 8¹ + 7 × 8⁰
= 64 + 40 + 7
= (111)₁₀

Hexadecimal example

(2AF)₁₆
= 2 × 16² + 10 × 16¹ + 15 × 16⁰
= 512 + 160 + 15
= (687)₁₀

When calculating, treat hexadecimal A–F as their corresponding numerical values.

6. Direct Relationship between Binary, Octal and Hexadecimal

Since 8 = 2³, one octal digit corresponds to exactly three binary bits. Since 16 = 2⁴, one hexadecimal digit corresponds to four binary bits.

Binary to octal

Group (11010111)₂ into sets of three bits from the right:

011 | 010 | 111 = 3 | 2 | 7

Therefore, (11010111)₂ = (327)₈.

Binary to hexadecimal

Group the same number into sets of four bits from the right:

1101 | 0111 = D | 7

Therefore, (11010111)₂ = (D7)₁₆.

An incomplete leftmost group can be padded with leading zeros without changing the value.

7. Binary Addition

Example

  1011
+ 0110
————
10001

Decimal check: 11 + 6 = 17, and (10001)₂ equals 17.

8. 1’s Complement and 2’s Complement

To obtain the 1’s complement, change every 0 to 1 and every 1 to 0.

The 1’s complement of (00101101)₂ is (11010010)₂.

To obtain the 2’s complement, add 1 to the 1’s complement.

1’s complement: 11010010
+ 1
2’s complement: 11010011

Two’s-complement representation is widely used for signed binary integers.

9. Unsigned and Signed Values

With n bits, the total number of distinct patterns is 2ⁿ.

• n-bit unsigned range: 0 to 2ⁿ − 1.
• n-bit two’s-complement signed range: −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1.

Eight-bit example:

• Unsigned range: 0 to 255.
• Two’s-complement signed range: −128 to 127.

If a question asks for the maximum eight-bit value, check whether it specifies unsigned or signed representation.

10. BCD — Binary-Coded Decimal

In BCD, each decimal digit is represented separately using four bits.

BCD representation of decimal 59:

5 = 0101 and 9 = 1001

Therefore, BCD for (59)₁₀ is 0101 1001.

Four-bit patterns from 1010 to 1111 are not valid individual decimal digits in ordinary BCD because a decimal digit ranges only from 0 to 9.

11. Character Representation: ASCII and Unicode

Characters are represented in computers using numerical code values. A character encoding specifies how code values relate to characters and how they are stored as bytes.

ASCII

ASCII stands for American Standard Code for Information Interchange. Original ASCII is a seven-bit code and therefore contains 128 code positions.

Several eight-bit “extended ASCII” arrangements have existed, but they are not one universal extension of the original seven-bit ASCII.

Unicode

Unicode provides a universal character-encoding standard for representing text from different languages and writing systems.

UTF-8, UTF-16 and UTF-32 are different ways of encoding Unicode characters into code units or bytes. The statement “every Unicode character always occupies 16 bits” is incorrect.

12. Basic Logic Gates

Logic gates perform Boolean operations on binary inputs. Here, 0 can be treated as False and 1 as True.

• AND: Output is 1 only when all inputs are 1.
• OR: Output is 1 when at least one input is 1.
• NOT: Inverts one input; 0 becomes 1 and 1 becomes 0.
• XOR: For two inputs, output is 1 when the inputs differ.
• NAND: Inverse of AND.
• NOR: Inverse of OR.
• XNOR: Inverse of XOR; output is 1 when two inputs are equal.

13. Quick Revision

• Binary has base 2; octal 8; decimal 10; hexadecimal 16.
• Hexadecimal A–F represent values 10–15.
• One octal digit corresponds to three bits; one hexadecimal digit corresponds to four bits.
• Binary 1 + 1 produces 10.
• Two’s complement = one’s complement + 1.
• The eight-bit unsigned range is 0–255.
• BCD converts each decimal digit separately into four bits.
• Original ASCII is a seven-bit code.
• Unicode represents text from many languages and writing systems.
• AND requires all 1s; OR requires at least one 1; two-input XOR requires different inputs.

14. Practice Questions

These are original practice questions, not verified previous-year questions. Each question has one correct answer.

Question 1. Which is a valid binary number?

A. 12001
B. 101101
C. 7821
D. 2A01

Question 2. What is the decimal value of (110101)₂?

A. 45
B. 51
C. 53
D. 55

Question 3. What is the binary representation of (45)₁₀?

A. 101101
B. 110101
C. 101011
D. 111001

Question 4. What is the decimal value of (157)₈?

A. 95
B. 105
C. 111
D. 127

Question 5. What decimal value does hexadecimal F represent?

A. 10
B. 14
C. 15
D. 16

Question 6. What is the hexadecimal representation of (10111100)₂?

A. AC
B. BC
C. B8
D. C8

Question 7. What is the 1’s complement of (00110110)₂?

A. 11001001
B. 11001010
C. 00110111
D. 10110110

Question 8. In ordinary BCD, the four-bit pattern 1010:

A. Represents decimal digit 10.
B. Represents decimal digit 9.
C. Is not a valid code for one decimal digit.
D. Represents decimal digit 1.

Question 9. What is the maximum decimal value of an eight-bit unsigned binary integer?

A. 127
B. 128
C. 254
D. 255

Question 10. Which statement about original ASCII is correct?

A. It is a seven-bit character code.
B. It only compresses images.
C. It is a base-16 number system.
D. It is used only for audio.

Question 11. What is the XOR output when A = 1 and B = 1?

A. 0
B. 1
C. 10
D. Cannot be determined

Question 12. What is the principal purpose of Unicode?

A. Adding decimal numbers only
B. Representing text from different languages and writing systems
C. Reducing image size only
D. Measuring hard-disk speed