GATE CS 2004 Q28 — Number Representations (Fixed and Floating Point)

In 3-digit floating-point arithmetic, numbers are represented in scientific notation with exactly 3 significant decimal digits (i.e., in the form $d.dd \times 10^e$). After every arithmetic operation, the result must be rounded to 3 significant digits.

1. Evaluating Expression 1: $(113. + -111.) + 7.51$

• Step 1: Evaluate $(113. + -111.)$ $$113. = 1.13 \times 10^2$$ $$-111. = -1.11 \times 10^2$$ Adding them: $$(1.13 - 1.11) \times 10^2 = 0.02 \times 10^2 = 2.00 \times 10^0$$ This result ($2.00$) already has 3 significant digits.
• Step 2: Add $7.51$ $$2.00 \times 10^0 + 7.51 \times 10^0 = 9.51 \times 10^0 = 9.51$$ This result ($9.51$) has exactly 3 significant digits.

Thus, the first expression evaluates to $9.51$.

2. Evaluating Expression 2: $113. + (-111. + 7.51)$

• Step 1: Evaluate $(-111. + 7.51)$ $$-111. = -1.11 \times 10^2$$ $$7.51 = 0.0751 \times 10^2$$ Adding them: $$(-1.11 + 0.0751) \times 10^2 = -1.0349 \times 10^2$$ Rounding $-1.0349 \times 10^2$ to 3 significant digits (the fourth digit is $4 < 5$, so we round down in magnitude): $$-1.03 \times 10^2 = -103.$$
• Step 2: Add $113.$ $$113. + (-103.) = 1.13 \times 10^2 - 1.03 \times 10^2 = 0.10 \times 10^2 = 1.00 \times 10^1 = 10.0$$ This result ($10.0$) has exactly 3 significant digits.

Thus, the second expression evaluates to $10.0$.

Therefore, the results are $9.51$ and $10.0$ respectively.