ITPEC FE Morning October 2023 Question 4

2.75 — add the binary fractions column by column with carries, then weight the bits after the point by 2⁻¹, 2⁻², …

Adding 1.0101 and 1.0111 from the rightmost column:

• •2⁻⁴ column: 1 + 1 = 10, write 0, carry 1
• •2⁻³ column: 0 + 1 + 1 = 10, write 0, carry 1
• •2⁻² column: 1 + 1 + 1 = 11, write 1, carry 1
• •2⁻¹ column: 0 + 0 + 1 = 1
• •Units: 1 + 1 = 10

The sum is 10.11 in binary, which is 2 + 0.5 + 0.25 = 2.75. Check in decimal: 1.0101 is 1 + 0.25 + 0.0625 = 1.3125, 1.0111 is 1 + 0.25 + 0.125 + 0.0625 = 1.4375, and 1.3125 + 1.4375 = 2.75.

Answer (b)

Why not others:

• •(a) 2.5 — binary 10.1: the carry out of the 2⁻³ column is lost, so the 2⁻² column comes out 0 instead of 1
• •(c) 2.875 and (d) 2.9375 — binary 10.111 and 10.1111; the two lowest columns both produce 0 with a carry, so the sum has no 1 bits below 2⁻²

Key rule: Binary fractions add with the same carry rules as integers, and the k-th bit after the point is worth 2^−k; a quick decimal check of each operand catches slips.

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