Binary Number System

• The Binary Number System is a base-2 numeral system that uses only two digits: 0 and 1 to represent all numerical values.
• It is the fundamental number system used in digital electronics and computing, where each digit (called a bit) represents an on (1) or off (0) state of a switch or transistor.

Why Binary Number Systems is important?

• Computers use transistors (tiny switches) that can only be ON (1) or OFF (0).
• Binary number systems simplify hardware design:
  1. Reliability: Easier to distinguish between two states (voltage levels).
  2. Error resistance: Less prone to misinterpretation than multiple voltage levels.

Binary Number Structure (Place Values in Binary)

• Each digit in a binary number represents a power of 2, starting from the right (least significant bit, LSB) to the left (most significant bit, MSB).
• Rightmost digit is at position
• Each position represents a power of 2

Example 1: Binary 1011 is equivalent to (?) decimal number

Solution:

= (1×2³) + (0×2²) + (1×2¹) + (1×2⁰)

= (1×8) + (0×4) + (1×2) + (1×1)

= 8 + 0 + 2 + 1

= 11 (Decimal)

Binary number 1011 is equivalent to decimal number 11.

Binary Number to Decimal number Conversion

Steps of Conversion:

• Write down the binary number.
• Assign powers of 2 from right to left.
• Multiply each binary digit by its power of 2.
• Add up the results.

Example 2: Convert 1101₂ to Decimal

Solution:

= (1×2³) + (1×2²) + (0×2¹) + (1×2⁰)

= (1×8) + (1×4) + (0×2) + (1×1)

= 8 + 4 + 0 + 1

= 13 (Decimal)

Example 3: Convert 1101₂ to Decimal

Solution:

To convert the binary number 11011.1010 to decimal, we’ll break it into two parts:

1. Integer part (11011) → Convert using powers of 2 (left of the point).
2. Fractional part (0.1010) → Convert using negative powers of 2 (right of the point).

Step 1: Convert the Integer Part (11011) to Decimal

Each digit represents a power of 2, starting from the rightmost bit (LSB = 2⁰).

Calculation:

= (1×16) + (1×8) + (0×4) + (1×2) + (1×1)

= 16 + 8 + 0 + 2 + 1

= 27

∴ 11011 (Binary) = 27 (Decimal)

Step 2: Convert the Fractional Part (0.1010) to Decimal

Each digit represents a negative power of 2, starting from the leftmost bit (2⁻¹).

Calculation:

= (1×0.5) + (0×0.25) + (1×0.125) + (0×0.0625)

= 0.5 + 0 + 0.125 + 0

= 0.625

∴ 0.1010 (Binary) = 0.625 (Decimal)

Step 3: Combine Both Parts

Integer part: 27

Fractional part: 0.625

Answer: Final Decimal Value is

11011.1010 (Binary)=27.625 (Decimal)

Decimal to Binary Conversion

Step-by-Step (Division-by-2 Method):

1. Divide the decimal number by 2.
2. Write down the remainder (0 or 1).
3. Repeat the division with the quotient until it becomes 0.
4. The binary number is the remainders read in reverse order.

Example 4: Convert 19₁₀ to Binary

Solution:

Binary: Read remainders in reverse = 10011₂

Answer: 19₁₀ = 10011₂

Example 5: Convert 25₁₀ to Binary

Solution:

Binary: Read remainders in reverse = 11001₂

Answer: 25₁₀ = 11001₂

Example 6: Convert 107.25₁₀ to Binary

Solution:

To convert the decimal number 107.25 into binary, we’ll break it into two parts:

• Integer part (107) → Convert using division by 2
• Fractional part (0.25) → Convert using multiplication by 2

Step 1: Convert Integer Part (107) to Binary

            Method: Repeated division by 2, recording remainders.

Read remainders from bottom to top:

107 (Decimal) = 1101011 (Binary)

Step 2: Convert Fractional Part (0.25) to Binary

Method: Repeated multiplication by 2, recording integer parts.

Read integer parts from top to bottom:

0.25 (Decimal) = 0.01 (Binary)

Step 3: Combine Both Parts

Integer part: 107 (Decimal) = 1101011 (Binary)

          Fractional part: 0.25 (Decimal) = 0.01 (Binary)

Answer: Final Binary Representation is

107.25 (Decimal) = 1101011.01 (Binary)