IPA FE Subject A 2025 Question 2

Rounding off, up or down the part smaller than the least significant digit — a floating-point number holds only a fixed number of digits, so whatever lies beyond the last one has to be dropped.

The mantissa has a fixed number of bits. When a result needs more digits than that, the part smaller than the least significant digit is rounded off, rounded up or rounded down (truncated) to fit, and the difference from the exact value is the rounding error. For example, 1/3 = 0.333… cannot be held in full; kept to four decimal places it becomes 0.3333, an error of about 0.00003.

Answer (b)

Why not others:

• •(a) — a result larger than the maximum value the computer can handle is overflow
• •(c) — losing the lower part of the mantissa of the number with the smaller exponent is loss of trailing digits; it happens when the exponents are aligned to add or subtract a very large and a very small number, not in multiplication and division
• •(d) — losing the upper significant digits when two numbers of almost equal absolute value are subtracted (or added with opposite signs) is cancellation, also called loss of significance

Key rule: Tell the floating-point errors apart by cause: overflow (result too large), rounding error (digits beyond the last place dropped), loss of trailing digits (a large and a small number added), cancellation (nearly equal numbers subtracted).

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