Understanding the Binary Subtraction Problem

The question asks us to perform a subtraction between two binary numbers and then express the result in decimal (base 10). The two binary numbers are a large integer, (1000000001)₂, and a fractional number, (0.0101)₂.

To solve this, we first need to convert each binary number into its equivalent decimal representation. After obtaining the decimal values, we perform the subtraction and then identify the correct option.

Converting Binary Numbers to Decimal (Base 10)

Converting a binary number to decimal involves summing the products of each digit with the corresponding power of 2. For digits to the left of the binary point, the powers of 2 are non-negative (starting from \(2^0\)), and for digits to the right, they are negative (starting from \(2^{-1}\)).

Convert (1000000001)₂ to Decimal

This number has 10 digits. We start from the rightmost digit (position 0) and move left.

\( (1000000001)_2 = 1 \times 2^9 + 0 \times 2^8 + 0 \times 2^7 + 0 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 \)

Calculating the powers of 2:

• \(2^9 = 512\)
• \(2^0 = 1\)
• Other powers of 2 multiplied by 0 are 0.

So, the decimal value is:

\( 512 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1 = 513 \)

Thus, (1000000001)₂ = 513₁₀.

Convert (0.0101)₂ to Decimal

This is a fractional binary number. We start from the first digit to the right of the binary point (position -1) and move right.

\( (0.0101)_2 = 0 \times 2^0 + 0 \times 2^{-1} + 1 \times 2^{-2} + 0 \times 2^{-3} + 1 \times 2^{-4} \)

Calculating the negative powers of 2:

• \(2^{-1} = \frac{1}{2} = 0.5\)
• \(2^{-2} = \frac{1}{4} = 0.25\)
• \(2^{-3} = \frac{1}{8} = 0.125\)
• \(2^{-4} = \frac{1}{16} = 0.0625\)

So, the decimal value is:

\( 0 \times 1 + 0 \times 0.5 + 1 \times 0.25 + 0 \times 0.125 + 1 \times 0.0625 \)

\( = 0 + 0 + 0.25 + 0 + 0.0625 = 0.3125 \)

Thus, (0.0101)₂ = 0.3125₁₀.

Performing the Subtraction in Decimal

Now that we have the decimal equivalents, we can perform the subtraction:

\( (1000000001)_2 - (0.0101)_2 = 513_{10} - 0.3125_{10} \)

Subtracting the decimal values:

\( 513.0000 - 0.3125 = 512.6875 \)

The result of the subtraction is 512.6875 in decimal.

Comparing with Options

Let's compare our calculated result with the given options:

• Option 1: (512.6775) ₁₀
• Option 2: (512.6875) ₁₀
• Option 3: (512.6975) ₁₀
• Option 4: (512.0909) ₁₀

Our calculated result, 512.6875, exactly matches Option 2.

Conclusion

By converting both binary numbers to their decimal equivalents and performing the subtraction, we found that (1000000001)₂ – (0.0101)₂ is equal to 512.6875 in base 10.

Revision Table: Binary Subtraction and Conversion

Additional Information: Number Systems

Number systems are ways of representing numbers. Common number systems include:

• Decimal System (Base 10): Uses digits 0-9. Each position represents a power of 10. This is the system we use daily.
• Binary System (Base 2): Uses digits 0 and 1. Each position represents a power of 2. This system is fundamental in computing.
• Octal System (Base 8): Uses digits 0-7. Each position represents a power of 8.
• Hexadecimal System (Base 16): Uses digits 0-9 and letters A-F (A=10, B=11, ..., F=15). Each position represents a power of 16. Used in computing and digital electronics.

Converting between these systems is a key skill in digital electronics and computer science.

Fractional parts in any base work similarly: digits to the right of the radix point are multiplied by decreasing negative powers of the base.