Convert the following binary into decimal A. 101011111 B. 1111111 2 C. 100001
The correct answer and explanation is :
Converting binary numbers to their decimal equivalents involves understanding the positional value of each bit in the binary system. In binary, each digit (bit) represents an increasing power of 2, starting from the rightmost bit, which represents 2⁰. The value of a binary number is the sum of these powers of 2 for each bit that is set to 1.
A. 101011111₂
To convert the binary number 101011111₂ to decimal:
- Identify the positional values: Bit Position (from right) Bit Value Positional Value (2^position) Contribution to Decimal Value 0 1 2⁰ = 1 1 × 1 = 1 1 1 2¹ = 2 1 × 2 = 2 2 1 2² = 4 1 × 4 = 4 3 1 2³ = 8 1 × 8 = 8 4 1 2⁴ = 16 1 × 16 = 16 5 0 2⁵ = 32 0 × 32 = 0 6 1 2⁶ = 64 1 × 64 = 64 7 0 2⁷ = 128 0 × 128 = 0 8 1 2⁸ = 256 1 × 256 = 256
- Calculate the decimal value: Sum of contributions = 256 + 0 + 64 + 0 + 16 + 8 + 4 + 2 + 1 = 351. Therefore, 101011111₂ = 351₁₀.
B. 1111111₂
To convert the binary number 1111111₂ to decimal:
- Identify the positional values: Bit Position (from right) Bit Value Positional Value (2^position) Contribution to Decimal Value 0 1 2⁰ = 1 1 × 1 = 1 1 1 2¹ = 2 1 × 2 = 2 2 1 2² = 4 1 × 4 = 4 3 1 2³ = 8 1 × 8 = 8 4 1 2⁴ = 16 1 × 16 = 16 5 1 2⁵ = 32 1 × 32 = 32 6 1 2⁶ = 64 1 × 64 = 64
- Calculate the decimal value: Sum of contributions = 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127. Therefore, 1111111₂ = 127₁₀.
C. 100001₂
To convert the binary number 100001₂ to decimal:
- Identify the positional values: Bit Position (from right) Bit Value Positional Value (2^position) Contribution to Decimal Value 0 1 2⁰ = 1 1 × 1 = 1 1 0 2¹ = 2 0 × 2 = 0 2 0 2² = 4 0 × 4 = 0 3 0 2³ = 8 0 × 8 = 0 4 0 2⁴ = 16 0 × 16 = 0 5 1 2⁵ = 32 1 × 32 = 32
- Calculate the decimal value: Sum of contributions = 32 + 0 + 0 + 0 + 0 + 1 = 33. Therefore, 100001₂ = 33₁₀.
Explanation of Binary to Decimal Conversion:
The binary numeral system is base-2, utilizing only two digits: 0 and 1. Each position in a binary number represents a power of 2, increasing from right to left. To convert a binary number to its decimal equivalent, follow these steps:
- Determine the positional value of each bit: Starting from the rightmost bit (least significant bit), assign it the value of 2⁰. Move leftward, incrementing the exponent by 1 for each subsequent bit.
- Multiply each bit by its positional value: If the bit is 1, multiply it by its corresponding power of 2. If it’s 0, the contribution to the decimal value is 0.
- Sum the contributions: Add all the products from the previous step to obtain the decimal equivalent of the binary number.
This method ensures an accurate conversion from binary to decimal by leveraging the inherent positional value of each bit in the binary system.