Rounding
Rounding is replacing a number with a nearby value that is simpler or less precise, based on a specified place value.
Definition
Rounding means changing a number to a nearby, simpler value: you look at the digit just to the right of the place you are rounding to, and round up if it is $5$ or more, or keep the digit the same if it is less than $5$. More precisely, rounding to a specified place value replaces a number with the nearest multiple of the relevant power of $10$ (or other base); at the exact midpoint, standard rounding always rounds up, while banker's rounding (round half to even) rounds to the nearest even digit instead. Geometrically, rounding corresponds to a projection from $\mathbb{R}$ to a discrete subset such as $10^{-k} \cdot \mathbb{Z}$, and rounding error analysis is central to numerical analysis: the IEEE 754 floating-point standard specifies that each operation is rounded to the nearest representable value, with a relative error bounded by the machine epsilon $\varepsilon = 2^{-52}$ for double precision.
Example
Round $347$ to the nearest ten: the ones digit is $7 \ge 5$, so round up to $350$; round $342$ to the nearest ten and the ones digit $2$ rounds down to $340$. Round $3.14159$ to $3$ decimal places: the $4$th decimal is $5$, so round up to $3.142$; using banker's rounding on $2.345$, the tie is broken toward the even digit, keeping it at $2.344$. Accumulated rounding errors can cause catastrophic cancellation: computing $(1 + 10^{-16}) - 1$ in double precision returns $0$ because $1 + 10^{-16}$ rounds to $1.0000000000000002$, and the subtraction loses all significant digits.
Key Insight
Rounding makes numbers easier to work with when you do not need exact values; prices, distances, and populations are often reported as rounded numbers. Banker's rounding is standard in financial and scientific computing because it minimizes systematic bias when many values are rounded, whereas standard rounding introduces a slight upward bias. The Kahan summation algorithm compensates for rounding error in large sums by tracking a running compensation term, improving accuracy from $O(n \cdot \varepsilon)$ to $O(\varepsilon)$ for summing $n$ floating-point numbers.