Rounding replaces a number with a nearby, simpler value to a specified precision — a chosen number of decimal places, or a chosen number of significant figures. The core rule: look at the digit immediately after the place you're rounding to. If it's 5 or greater, round the target digit up by one; if it's less than 5, leave the target digit as is, and drop everything after it.
Worked example one: round 4.867 to two decimal places. The digit after the second decimal place is 7 (in 4.86**7**), which is ≥ 5, so round the second decimal digit up: 4.87. Worked example two: round 128,439 to the nearest thousand. The digit right after the thousands place is 4 (128,**4**39), which is < 5, so round down: 128,000.
Significant figures work on the same up/down rule but count from the first nonzero digit rather than from the decimal point. Rounding 0.004528 to 2 significant figures: the first two significant digits are 4 and 5, and the next digit (2) is < 5, so it stays 0.0045. Rounding 3,847 to 3 significant figures: the first three significant digits are 3, 8, 4, and the next digit (7) is ≥ 5, so the third digit rounds up: 3,850.
The genuinely contested edge case is rounding a number that ends in exactly 5 with nothing after it, like rounding 2.5 to the nearest whole number. The rule taught above ("5 rounds up") gives 3, and that's the convention this calculator uses and the one taught in most schools — but it's not the only defensible convention. "Round half to even" (also called banker's rounding), used in some financial and statistical contexts specifically to avoid a systematic upward bias when rounding large batches of numbers, would round 2.5 down to 2 (rounding to the nearest even digit) and 3.5 up to 4. Both conventions are internally consistent; they just handle the exact tie differently, and it's worth knowing which one a specific context (a tax form, a lab report) expects.
Rounding to the nearest ten, hundred, or other place-value unit works identically to decimal-place rounding — the "look at the next digit" rule doesn't care whether that digit sits after the decimal point or in front of it. Rounding 3,782 to the nearest hundred: the digit right after the hundreds place is 8 (3,7**8**2), which is ≥5, so round the hundreds digit up: 3,800. A related but distinct operation worth not confusing with rounding is truncating — simply cutting off digits without applying the up/down rule at all (truncating 4.87 to one decimal place gives 4.8, discarding the 7 regardless of its value, whereas rounding 4.87 to one decimal place correctly gives 4.9). Some contexts, like displaying a countdown timer, intentionally truncate rather than round; most everyday numeric contexts expect proper rounding.
A common mistake is rounding in stages and letting an intermediate rounding affect the final result — rounding 4.449 to one decimal place first (giving 4.4, since the second decimal digit is 4) and then to a whole number (staying 4) gives a different, wrong answer compared to rounding 4.449 directly to a whole number in one step (also 4, in this case, but the two-step method can diverge from the direct method in other examples, like 2.49 → 2.5 → 3 instead of the correct direct rounding, 2.49 → 2). Round from the original number to the final target precision in one step whenever possible, not through successive intermediate roundings. Rounding also plays a specific, important role in reporting measured (rather than exactly known) values: a measurement of "12.4 cm" implies precision only to the nearest tenth of a centimeter, and writing it as "12.40 cm" or "12.400 cm" would misleadingly claim a level of precision the original measurement never actually had. This is part of why scientific and engineering contexts are careful about significant figures specifically, rather than rounding to an arbitrary fixed number of decimal places regardless of context.
For the full rule set with more edge cases, see How to Round Numbers Correctly.