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THE CS NOTES

Number System

Number System and Base Conversion

Section titled “Number System and Base Conversion”

1. From any base to Decimal (Base-10)

  • Multiply each digit by its base raised to the positional power.
  • Add all results.
  • Example: (1011)₂ to decimal:
1×2³ + 0×2² + 1×2¹ + 1×2⁰
= 8 + 0 + 2 + 1
= 11₁₀

If the binary number is a float, you convert integer part and fractional part separately: ⭐

Steps:

  1. Integer part → Multiply each bit by 2^n according to its position (same as normal).
  2. Fractional part → Multiply each bit by 2^(-n) according to its position after the point.
  • Example: (11001.101)₂ to base-10
Integer part (11001):


1×2⁴ + 1×2³ + 0×2² + 0×2¹ + 1×2⁰
= 16 + 8 + 0 + 0 + 1
= 25


Fractional part (.101):


1×2⁻¹ + 0×2⁻² + 1×2⁻³
= 0.5 + 0 + 0.125
= 0.625


Final:
(11001.101)₂ = (25.625)₁₀

2. From Decimal to Any Base

  • Repeatedly divide the decimal number by the target base.
  • Write remainders in reverse order.
  • Example: (25)₁₀ to base-2:
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
→ 11001₂

If the decimal number is a float==, you convert ==integer part and fractional part separately: ⭐

Steps:

  1. Integer part → Use repeated division by base (same as normal).
  2. Fractional part → Multiply repeatedly by base, take the integer part each time as the next digit.
  • Example: (25.625)₁₀ to base-2
Integer part (25):


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
→ 11001


Fractional part (0.625):


0.625 × 2 = 1.25   → 1
0.25  × 2 = 0.5    → 0
0.5   × 2 = 1.0    → 1
→ .101


Final:
(25.625)₁₀ = (11001.101)₂

4. From One Non-Decimal Base to Another

  • Convert to decimal first, then to target base.
  • Example: (27)₈ to base-2:
1. (27)₈ = 2×8¹ + 7×8⁰
         = 16 + 7
         = 23₁₀


2. 23₁₀ = 10111₂

Base Conversion

Section titled “Base Conversion”

1. Binary to Decimal Conversion

Steps:

  1. Write Down the Binary Number:
    • Example: 1101
  2. Identify the Positional Values:
    • Each bit represents a power of 2, starting from (2^0) for the rightmost bit.
  3. Multiply Each Bit by Its Corresponding Power of 2:
    • For 1101:
      • 1 \times 2^3 = 8
      • 1 \times 2^2 = 4
      • 0 \times 2^1 = 0
      • 1 \times 2^0 = 1
  4. Sum the Results:
    • 8 + 4 + 0 + 1 = 13

Result:

  • Binary 1101 is decimal 13.

2. Decimal to Binary Conversion

Steps:

  1. Divide the Decimal Number by 2:
    • Record the remainder.
  2. Divide the Quotient by 2:
    • Continue dividing and recording remainders until the quotient is 0.
  3. Write Down the Remainders in Reverse Order:
    • This gives the binary representation.

Example: Convert 13 to Binary

  • 13 ÷ 2 = 6 with remainder 1
  • 6 ÷ 2 = 3 with remainder 0
  • 3 ÷ 2 = 1 with remainder 1
  • 1 ÷ 2 = 0 with remainder 1

Result:

  • Binary representation is 1101.

3. Hexadecimal to Decimal Conversion

Steps:

  1. Write Down the Hexadecimal Number:
    • Example: 0x2F3
  2. Identify the Positional Values:
    • Each digit represents a power of 16, starting from (16^0) for the rightmost digit.
  3. Convert Each Hexadecimal Digit to Decimal:
    • 2, F (15), and 3.
  4. Multiply Each Digit by Its Corresponding Power of 16:
    • For 0x2F3:
      • 3 \times 16^0 = 3
      • 15 \times 16^1 = 240
      • 2 \times 16^2 = 512
  5. Sum the Results:
    • 512 + 240 + 3 = 755

Result:

  • Hexadecimal 0x2F3 is decimal 755.

4. Decimal to Hexadecimal Conversion

Steps:

  1. Divide the Decimal Number by 16:
    • Record the remainder (convert to hexadecimal if needed).
  2. Divide the Quotient by 16:
    • Continue dividing and recording remainders until the quotient is 0.
  3. Write Down the Remainders in Reverse Order:
    • This gives the hexadecimal representation.

Example: Convert 755 to Hexadecimal

  • 755 ÷ 16 = 47 with remainder 7
  • 47 ÷ 16 = 2 with remainder 15 (F in hexadecimal)
  • 2 ÷ 16 = 0 with remainder 2

Result:

  • Hexadecimal representation is 0x2F7.

5. Binary to Hexadecimal Conversion

Steps:

  1. Group the Binary Digits into Sets of Four (from right to left):
    • Example: 10111011 becomes 1011 1011.
  2. Convert Each Group to Hexadecimal:
    • 1011 is B
    • 1011 is B

Result:

  • Binary 10111011 is hexadecimal 0xBB.

6. Hexadecimal to Binary Conversion

Steps:

  1. Convert Each Hexadecimal Digit to Its 4-Bit Binary Equivalent:
    • Example: 0x2F3:
      • 2 is 0010
      • F is 1111
      • 3 is 0011
  2. Combine All Binary Groups:

Result:

  • Hexadecimal 0x2F3 is binary 0010 1111 0011.

2’S Complement, BCD Code, XS-3 Code, Gray Code

Section titled “2’S Complement, BCD Code, XS-3 Code, Gray Code”

n’s Complement

  • For a number system with base n, n’s complement of a number is obtained by
    subtracting the number from nᵏ, where k = number of digits.
  • Formula
    n’s complement of X = nᵏ − X
  • Types
    • Binary (n = 2) → 2’s complement
    • Decimal (n = 10) → 10’s complement
    • Octal (n = 8) → 8’s complement
    • Hexadecimal (n = 16) → 16’s complement

  • Relation with (n−1)’s complement ⭐ n’s complement = (n−1)’s complement + 1

  • Use
    • Subtraction using addition
    • Representation of negative numbers
  • Example (Decimal)
    Number = 275
    10’s complement = 1000 − 275 = 725
  • Example (Binary)
    Number = 1010
    2’s complement = 0101 + 1 = 0110

2’s Complement

Definition: Binary number representation for signed integers== where negative numbers are obtained by taking the 1’s complement and adding 1.Use: ==Simplifies subtraction in digital systems.Steps (for negative numbers):

  1. Write the number in binary.
  2. Take 1’s complement (invert bits).
  3. Add 1 to the result.

Example (–5 in 8-bit):

+5   = 00000101
1's  = 11111010
+1   = 11111011  → –5

1. Excess-3 Code

Definition: A self-complementary decimal code obtained by adding 3 to each decimal digit and converting to 4-bit binary.
Use: Error detection, decimal calculations.

Example table:

DecimalBinaryExcess-3
000000011 (3)
100010100 (4)
200100101 (5)
300110110 (6)
401000111 (7)
501011000 (8)
601101001 (9)
701111010 (10)
810001011 (11)
910011100 (12)

2. BCD (Binary Coded Decimal)

Definition: Represents each decimal digit separately in binary (4 bits per digit).
Use: Digital displays, calculators.

Example:

                5    9
Decimal 59 = (0101 1001) in BCD

3. Gray Code

Definition: A binary code where consecutive values differ by only 1 bit (minimizes errors in transitions).
Use: Position encoders, error reduction.

Example (3-bit Gray code):

DecimalBinaryGray
0000000
1001001
2010011
3011010
4100110
5101111
6110101
7111100