Bisection Method: Explained with Examples, Notes and FAQs

The bisection method finds where a function crosses zero by halving an interval. Start with [a, b] where the function has opposite signs at the ends, so a root must lie inside. Check the midpoint: keep the half that still brackets the root and discard the other. Each step halves the interval, trapping the root in a smaller and smaller box.

Bisection is guaranteed to work: for a continuous function with a bracketing starting interval, it always converges. Faster methods can fail or wander off, so bisection is the safe fallback engineers trust, and the basis of more advanced bracketing methods.

Key Concepts

• Root: a value r with f(r) = 0, the target of the search.
• Bracket: an interval [a, b] with f(a) × f(b) < 0 — by the Intermediate Value Theorem, a continuous f has at least one root inside.
• Tolerance (ε): the required accuracy; stop when the interval width b − a drops below it.
• Error bound: after n steps the midpoint is within (b₀ − a₀) / 2^n of the root.

Formulas

• Midpoint: c = (a + b) / 2.
• Bracket update: if f(a) × f(c) < 0, the root is in [a, c], so set b = c; otherwise set a = c.
• Error after n iterations: E_n ≤ (b₀ − a₀) / 2^n, assuming f is continuous and the bracket holds a root.
• Iterations needed: n ≥ log_2((b₀ − a₀) / ε), rounded up, to reach tolerance ε.

Worked Example

Find the real root of f(x) = x³ − x − 2. Since f(1) = 1 − 1 − 2 = −2 and f(2) = 8 − 2 − 2 = 4, [1, 2] brackets a root. Iteration 1: c = 1.5, f(1.5) = 3.375 − 1.5 − 2 = −0.125 (same sign as f(1)) → bracket [1.5, 2]. Iteration 2: c = 1.75, f(1.75) = 5.359 − 1.75 − 2 = 1.609 → bracket [1.5, 1.75]. Iteration 3: c = 1.625, f(1.625) = 4.291 − 1.625 − 2 = 0.666 → bracket [1.5, 1.625]. Iteration 4: c = 1.5625, f(1.5625) = 3.815 − 1.5625 − 2 = 0.253 → bracket [1.5, 1.5625]. Iteration 5: c = 1.53125, f(1.53125) = 3.590 − 1.53125 − 2 = 0.059 → bracket [1.5, 1.53125]. Iteration 6: c = 1.515625, f(1.515625) = 3.481 − 1.515625 − 2 = −0.035 → bracket [1.515625, 1.53125]. The midpoint 1.5234 is within 0.0156 of the true root 1.52138, matching the bound (2 − 1)/2^6 = 1/64 ≈ 0.0156.

Common Misconceptions

• "Bisection works on any interval." It needs opposite signs at the ends — without f(a) × f(b) < 0 the method cannot start, even if a root hides inside.
• "It finds all roots." Each run finds only the bracketed root — five roots need five brackets and five runs.
• "It works across a discontinuity." A sign change can happen at a jump (like 1/x near zero) with no root at all — continuity is essential.

Real-World Applications

• Solving physics equations such as finding the equilibrium position of a spring system with nonlinear forces.
• Economics: finding the break-even point where profit equals zero from a profit function.
• Chemistry: finding the pH at which an acid-base mixture reaches a target concentration.
• Computer graphics: ray tracing uses bisection-like bracketing to find where a ray hits a surface.

Bisection traps a root by repeatedly halving a sign-changing interval, with guaranteed error (b₀ − a₀)/2^n. Slow but unfailing on continuous functions, it is the safest root-finder and the standard fallback when faster methods misbehave.