Estimation
Estimation is the process of finding an approximate value that is close enough to the exact answer for a given purpose.
Definition
Estimation means finding an answer that is close to the exact answer without doing all the precise calculation, useful whenever an approximate answer is good enough. It typically works by rounding inputs to convenient values before computing: front-end estimation (rounding to the leading digit) is fast but less precise, while compatible-numbers estimation uses nearby numbers that divide or factor nicely. In analysis and numerical methods, estimation more formally means bounding the error of an approximation, in the form $|f(x) - \text{approx}| \le C$ for some constant $C$, or asymptotic estimates like $f(n) = O(g(n))$, where the quality of an estimate is judged by its tightness and the cost of computing it.
Example
You want to know if $\$12.75 + \$9.45$ fits in your $\$25$ budget: estimating $\$13 + \$9 = \$22$ tells you yes, without needing the exact answer. Estimate $392 \times 48$ by rounding to $400 \times 50 = 20{,}000$; the actual answer is $18{,}816$, about $6\%$ high, acceptable for a quick check, and estimation is also used to catch arithmetic errors: if a calculator says $392 \times 48 = 1{,}881.6$, the estimate of $20{,}000$ immediately reveals a decimal-point error. Stirling's approximation, $n! \sim \sqrt{2\pi n} \cdot (n/e)^n$, gives $3{,}598{,}696$ for $n=10$ against the true $10! = 3{,}628{,}800$, a relative error under $0.1\%$.
Key Insight
Good estimators round to numbers that are easy to compute mentally; estimation is a sign of strong number sense, not laziness, and it is a reliable way to catch arithmetic errors. Fermi estimation, the skill of estimating unknown quantities from first principles (like "how many piano tuners are in Chicago?"), is a core tool in physics, engineering, and data science, developing quantitative intuition and order-of-magnitude thinking.