Round off and compute errors

Rounding to n significant figures keeps the first n digits, counting from the first non-zero digit. The discarded part decides what happens to the last kept digit:

• less than half a unit in the nth place: leave the nth digit unchanged;
• more than half a unit: increase the nth digit by 1;
• exactly half a unit: increase the nth digit by 1 if it is odd, and leave it unchanged if it is even (the round-to-even rule of standard numerical methods texts).

Error formulas (X = given value, X′ = rounded value):

• Absolute error = |X − X′|
• Relative error = |X − X′| / |X|
• Percentage error = relative error × 100

(i) 865250

First four significant figures: 8 6 5 2. The discarded part is 50, which is exactly half a unit of the fourth place (100). The fourth digit 2 is even, so it is left unchanged: 865200. (With the simpler rule "5 or more, round up", the result would be 865300. The absolute error is 50 either way, so the errors below are the same.)

• Rounded value = 865200
• Absolute error = |865250 − 865200| = 50
• Relative error = 50 / 865250 = 5.779 × 10⁻⁵
• Percentage error = 0.00578 %

(ii) 37.46235

First four significant figures: 3 7 4 6. The next digit is 2 (less than half a unit), so no change: 37.46.

• Rounded value = 37.46
• Absolute error = |37.46235 − 37.46| = 0.00235
• Relative error = 0.00235 / 37.46235 = 6.273 × 10⁻⁵
• Percentage error = 0.00627 %

Answer: 865250 → 865200 (865300 if halves are always rounded up) with Ea = 50, Er = 5.779 × 10⁻⁵, 0.00578 %; 37.46235 → 37.46 with Ea = 0.00235, Er = 6.273 × 10⁻⁵, 0.00627 %.