An approximation is a simpler value, function, or solution used in place of a target when an exact answer is unavailable, unnecessary, or too costly to compute. Like a map that leaves out detail to make navigation easier, a mathematical approximation is useful only relative to a task: the right method depends on what is being approximated and which errors matter.
What does approximation mean in mathematics?
In everyday use, to approximate is to get close without being exact. In mathematics and computing, an approximation is a stand-in for a specified target. That target might be a number, a function, measured data, or the solution to an equation. Whether the stand-in is adequate depends on how it will be used and how its error is judged.
For example, UC Berkeley Math 128A notes by Per-Olof Persson (Fall 2026) compare the exact value p = 1 with the approximation p* = 1.001. The absolute error is |p − p*| = 0.001, or 10−3. Since the target has magnitude 1, the relative error is also 0.001. Applying the significant-digit rule stated in those notes, this example meets their criterion for three significant digits. It is a worked example, not a universal rule for deciding how many digits are enough. Berkeley Math 128A notes.
How do you measure the error of an approximation?
Error is the difference between the target and its approximation. Two common ways to express it answer different questions:
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- Absolute error is the size of the discrepancy, typically |target − approximation|. It uses the same units as the quantity being approximated.
- Relative error compares the discrepancy with the target’s magnitude, typically |target − approximation| / |target|. It is defined this way only when the target is nonzero.
Absolute error is useful when a fixed-unit tolerance matters, such as being within a specified distance. Relative error helps judge a discrepancy against the scale of the target: an error of 0.01 means something different when estimating 0.1 than when estimating 1,000. Neither measure is automatically the right one; the application determines what counts as acceptable.
Why does the goal determine which approximation is best?
There is no single best approximation independent of a goal. A method that exactly matches a set of sample values may behave differently between those values from one chosen to keep the largest error small across an entire interval. Virginia Tech’s lecture notes introduce approximation theory through interpolation and error optimization, and emphasize that the choice of interpolation points can strongly affect quality. Virginia Tech lecture 12: Introduction to Approximation Theory.
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| Method or objective | What it tries to achieve | Where the criterion applies |
|---|---|---|
| Interpolation | Construct a polynomial or piecewise polynomial that matches specified data values. | At the selected data points; behavior elsewhere depends on the method and point placement. |
| Minimax approximation | Minimize the maximum absolute error. | Across the domain over which the maximum error is defined, such as an interval. |
| Least squares | Minimize the integrated square of the error. | Across the specified domain and measure used in the integral. |
| Least absolute deviations | Minimize the integral of the absolute error. | Across the specified domain and measure used in the integral. |
These criteria can select different approximants because they penalize error differently. Interpolation prioritizes exact matches at chosen points; minimax focuses on the worst discrepancy; least squares gives greater influence to larger errors by squaring them; least absolute deviations accumulates error without squaring it. Ask what must be matched or controlled before calling a result “best.”
Interpolation: match selected values
Interpolation replaces data or a function with a polynomial, or a piecewise polynomial, that agrees at selected points. It is useful when those points must be reproduced exactly, but a perfect fit at the sample points alone does not guarantee small error throughout the surrounding interval. The placement of the points matters.
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Minimax: control the largest error
Minimax approximation targets the greatest absolute discrepancy over a chosen domain. It is relevant when no part of that domain should have an especially large error. Its guarantee is tied to the domain and error definition used; it does not mean the error is equally small under every other criterion.
Least squares and least absolute deviations: aggregate error
Least squares minimizes an integrated squared error, while least absolute deviations minimizes an integrated absolute error. The integrals make the domain and any weighting part of the definition of the objective. These goals are not interchangeable with exact matching or minimization of the worst-case error.
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What changes when a computer calculates an approximation?
Computers represent numbers with finite precision, so many real values cannot be stored exactly and arithmetic operations may round. This introduces arithmetic effects in addition to the error from replacing a mathematical target with a simpler stand-in. Keeping those sources distinct helps diagnose a poor result:
- Approximation or method error comes from the chosen mathematical stand-in or procedure.
- Arithmetic error comes from finite-precision representation and operations, including rounding.
One important arithmetic hazard is cancellation: subtracting nearly equal numbers can leave a small result whose accuracy is disproportionately affected by errors in the inputs or preceding calculations. Berkeley’s notes discuss finite-digit arithmetic, cancellation, stability, and error growth. Berkeley Math 128A notes.
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Stability concerns whether small changes or errors in inputs and intermediate calculations remain controlled or grow as a method runs. It is therefore useful to consider not just the error objective, but also how the computation behaves under finite precision. The notes provide this conceptual framework; they do not establish a numerical cost comparison among the approximation methods listed above.
How does approximation error change as a calculation is refined?
Convergence asks whether an approximation gets closer to its target as a relevant parameter changes—for example, as a step size shrinks, a polynomial degree increases, or an iteration continues. An error statement such as O(hp) describes how error scales with step size h for a particular method under its assumptions; p is its order of accuracy.
Berkeley’s notes use a first-order difference-quotient example in which halving h halves the error. That behavior is specific to the example and should not be assumed for every approximation. A useful comparison asks what parameter is being refined, what error measure is tracked, and whether the method’s assumptions apply.
Where does approximation theory fit in numerical analysis?
Approximation is one part of a broader computational toolkit. MIT OpenCourseWare’s archived Spring 2012 undergraduate course includes notes on interpolation, nonlinear equations, ODE methods, Fourier analysis, and spectral interpolation, differentiation, and quadrature. These are the contents of that archived offering, not a current syllabus or a required learning sequence. MIT OpenCourseWare: Introduction to Numerical Analysis, Spring 2012 lecture notes.
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For further study, SIAM describes Approximation Theory and Approximation Practice, Extended Edition as a computationally oriented treatment of classical polynomial and rational approximation, illustrated with Chebfun. The publisher presents it for advanced undergraduates and graduate students with prior numerical-analysis and complex-analysis background, so it is an optional next step for mathematically prepared readers rather than a prerequisite for understanding the ideas here. SIAM book page.
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