Approximation

An approximation is a value $\tilde{x}$ used in place of an exact value $x$, chosen so that the error $x-\tilde{x}$ is small in some specified sense. It matters because most real quantities — $\pi$, $\sqrt{2}$, a measured length, the solution of a hard equation — cannot be written down exactly or cheaply, yet decisions must still be made. The discipline of approximation is not about being sloppy: it is about quantifying and controlling the gap between what we use and what is true. A surprising fact is that some "approximations" can be made arbitrarily accurate — given enough work — so that the distinction between approximate and exact collapses in the limit, which is precisely the idea behind calculus and infinite series.

The Core Insight

Every useful approximation comes as a pair: a simpler value and a bound on how far it can be from the truth. A number without an error estimate is a guess; a number with one is a tool. The whole art is trading simplicity against accuracy, then proving the trade was safe.

Formal Statement

Approximation and Error

Let $x$ be an exact (target) value and $\tilde{x}$ an approximation to it.

The absolute error is

The relative error (when $x\ne 0$) is

The percentage error is simply $100\%\times E_{\text{rel}}$.

Symbols: $x$ = true value, $\tilde{x}$ = approximate value, $|\cdot |$ = absolute value. We say $\tilde{x}$ is correct to $n$ significant figures if $E_{\text{rel}}<\frac{1}{2}\times 10^{-n+1}$, and correct to $n$ decimal places if $E_{\text{abs}}<\frac{1}{2}\times 10^{-n}$.

Order of Approximation

A family of approximations $\tilde{x}(h)$ depending on a small parameter $h$ has error of order $p$, written $O(h^p)$, if there is a constant $C$ with

Larger $p$ means the error shrinks faster as $h\to 0$: halving $h$ divides an $O(h^2)$ error by $4$.

Taylor Approximation with Remainder

If $f$ is $(n+1)$ times differentiable near $a$, then for $x$ near $a$,

with the Lagrange remainder

The polynomial part is a systematic approximation whose error $R_n$ can be bounded explicitly.

Why It Holds / Derivation

Linear (Tangent-Line) Approximation

We derive the simplest non-trivial approximation, then bound its error.

1. Setup. Let $f$ be differentiable at $a$. Near $a$, replace the curve by its tangent line:

1. Why it's good. By the definition of derivative, $f(x)-f(a)=f^{\prime }(a)(x-a)+o(x-a)$, so $f(x)-L(x)$ vanishes faster than $(x-a)$ as $x\to a$.

1. Quantify the error. If $f^{\prime \prime }$ exists and $|f^{\prime \prime }|\le M$ on the interval, Taylor's theorem with $n=1$ gives

1. Conclusion. The tangent-line error is $O((x-a{)}^2)$: a second-order small quantity. Halving the distance from $a$ quarters the error. $■$

Brook Taylor | 1685–1731

The English mathematician Brook Taylor published the series now bearing his name in Methodus Incrementorum Directa et Inversa (1715). Though Gregory and Newton had used special cases earlier, Taylor gave the general formula expressing a function as a polynomial built from its derivatives at a point. The crucial remainder term — which turns a formal series into a controlled approximation — came later from Lagrange and Cauchy. Taylor's work underlies essentially every practical method for approximating functions on computers today.

Building Intuition

Think of approximation as zooming in. A circle looks curved from far away, but zoom into a tiny arc and it looks straight — that "straightness" is the tangent-line approximation. The more you zoom (smaller $|x-a|$), the better a straight line fits, and the leftover bend shrinks quadratically. Higher-order approximations (parabolas, cubics) are like keeping more of the curve's "personality" — its curvature, its twist — so the fit stays good over a wider window. The error term is the universe's receipt: it tells you exactly what you threw away.

Tangent-line approximation: the curve $f(x)$ versus its linear approximation $L(x)$ at $a$. The gap is the error $R_1$.

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Worked Examples

Approximating $\sqrt{50}$ by linearisation

1. $f(a)=\sqrt{49}=7$.
2. $f^{\prime }(x)=\frac{1}{2\sqrt{x}}$, so $f^{\prime }(49)=\frac{1}{14}$.
3. Linear approximation [6]: $L(50)=7+\frac{1}{14}(50-49)=7+\frac{1}{14}\approx 7.0714$.
4. Error bound via [7]: $f^{\prime \prime }(x)=-\frac{1}{4}{x}^{-3/2}$, and $|f^{\prime \prime }|\le \frac{1}{4}(49{)}^{-3/2}=\frac{1}{1372}$ on $[49,50]$. So error $\le \frac{1}{2}\cdot \frac{1}{1372}\cdot 1^2\approx 0.00036$.

Relative error of a rounded measurement

1. Rounding to nearest $0.1$ means the true value lies in $[2.45,2.55]\text{m}$.
2. Worst-case absolute error [1]: $E_{\text{abs}}=0.05\text{m}$.
3. Relative error [2]: $E_{\text{rel}}=\frac{0.05}{2.5}=0.02=2\%$.

Common Misconceptions

An approximation that gets *worse* with more terms

One might assume that adding more terms of a series always improves the estimate. For an asymptotic series, this fails. For example, approximating ${\int }_0^{\infty }\frac{e^{-t}}{1+xt}dt$ by its asymptotic expansion $\sum (-1{)}^{k}k!x^k$ at small $x$ gives improving accuracy up to an optimal number of terms, after which the (divergent) terms blow up and the approximation degrades. More effort can make things worse — error control, not term-counting, is what matters.

Connections

Approximation is the connective tissue of mathematics. The Derivative is the best linear approximation to a function. Taylor Series generalise this to polynomials of any order. Limits formalise the idea that an approximation can be driven to exactness. Newton's Method iterates linear approximations to solve equations. In statistics, Linear Regression is approximation of data by the closest line, and Numerical Integration approximates areas by sums. Even Significant Figures are a bookkeeping system for tracking how much approximation our numbers contain.

Real-World Applications

1. GPS navigation — satellite signals are processed with linearised equations (extended Kalman filters) approximating curved orbits and relativistic corrections in real time.
2. Computer arithmetic — every floating-point number (IEEE 754) is a rational approximation to a real number; $0.1$ is not stored exactly.
3. Engineering: small-angle approximation — $\sin \theta \approx \theta$ lets engineers design pendulum clocks, bridge oscillations, and optics with simple linear equations.
4. Computer graphics — curves and surfaces are approximated by millions of flat triangles; smooth shading approximates lighting integrals.
5. Weather and climate models — partial differential equations are approximated on grids using finite-difference and finite-element methods.
6. Medicine: drug dosing — pharmacokinetic curves are approximated by exponential-decay models to schedule doses.
7. Finance — option prices use the Black–Scholes formula, itself derived under approximating assumptions, and computed via numerical approximation.

Historical Development

• c. 1800 BCE — Babylonian tablet YBC 7289 records $\sqrt{2}\approx 1.41421296$, accurate to six digits.
• c. 250 BCE — Archimedes bounds $\pi$ between $\frac{223}{71}$ and $\frac{22}{7}$ using inscribed and circumscribed polygons — a rigorous interval approximation.
• c. 480 CE — Zu Chongzhi computes $\pi \approx 355/113$, correct to seven digits, unmatched for nearly a millennium.
• 1669 — Newton develops series methods and what becomes Newton's iteration for roots.
• 1715 — Brook Taylor publishes the general Taylor series.
• 1797 — Lagrange supplies a remainder term, turning the series into a bounded approximation.
• 1821 — Cauchy gives rigorous definitions of limit and error, founding analysis.
• 1885 — Weierstrass proves any continuous function on a closed interval can be uniformly approximated by polynomials (the Approximation Theorem).
• 1985 — IEEE 754 standardises floating-point approximation for all modern computers.

Subtleties & Limits

Approximation theory still holds deep open questions. Optimal approximation: for a given function and a fixed "budget" (e.g. a polynomial of degree $n$), finding the provably best approximation in the supremum norm (the Chebyshev / minimax problem) is solved in principle but computationally hard in many real settings. High-dimensional approximation suffers the curse of dimensionality: the work to approximate a function of $d$ variables to fixed accuracy can grow exponentially in $d$ — a barrier central to modern machine learning, where why deep neural networks escape this curse for many real-world functions is not fully understood. Inverse problems — recovering an exact cause from approximate, noisy effects (as in medical imaging) — remain mathematically delicate and ill-posed.

See Also