The decimal equivalent of the binary number (1011.011)2 is ________

Question

BlackTom AI · Accepted Answer

We start by interpreting the binary number (1011.011)2 and convert it to decimal to verify the options.

First, convert the integer part 1011 from binary to decimal:
- 1*(2^3) + 0*(2^2) + 1*(2^1) + 1*(2^0) = 8 + 0 + 2 + 1 = 11

Next, convert the fractional part .011 from binary to decimal by using negative powers of two:
- The first bit after the point is 0, which would be 0*(1/2) = 0
- The second bit after the point is 1, which is 1*(1/4) = 0.25
- The third bit after the point is 1, which is 1*(1/8) = 0.125
- Sum of fractional part = 0 + 0.25 + 0.125 = 0.375

Combine the integer and fractional parts: 11 + 0.375 = 11.375

Now evaluate each answer option:

Option a: (11.175)10
- This would correspond to an integer part of 11 and a fractional part of 0.175. The fractional component 0.175 is not equal to 0.375, so this does not match the binary-to-decimal conversion. The discrepancy could come from misplacing or miscomputing one of the fractional bits.

Option b: (11.375)10
- This matches the calculated value exactly: integer part 11 and fractional part 0.375. The binary fractional bits 0,1,1 after the point convert to 0, 0.25, 0.125 summing to 0.375, giving 11.375 in decimal.

Option c: (9.23)10
- This would imply an incorrect integer part (9 instead of 11) and an odd fractional part (0.23) that does not align with binary fractions. The binary integer 1011 is 11, so 9 is inconsistent with the given binary digits.

Option d: (10.123)10
- This option also misrepresents both the integer and fractional components. The binary integer 1011 corresponds to 11, not 10, and the fractional part 0.123 does not align with the binary fractional 0.375. In binary-to-decimal conversion, 1011.011 cannot yield 10.123.

Summary of the misconceptions:
- Option a likely stems from altering either the fractional sum (0.375) to 0.175 or miscomputing one of the fractional bits.
- Options c and d show a misunderstanding of the integer portion or how binary fractions map to decimal fractions, suggesting a misapplication of place-value concepts or misreading the bit pattern after the point.

Key takeaway: convert the integer part via powers of two, convert the fractional part via negative powers of two, and then add the results to obtain the correct decimal representation. In this case, the correct decimal value is 11.375, corresponding to the binary 1011.011.

FIT1047_S1_2025 Week 2 quiz (not assessed)

The decimal equivalent of the binary number (1011.011)2 is ________

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