Bisection method calculator

Enter f(x) and an interval where the function changes sign. The interval is halved until its half-width is below your tolerance, and each step is listed.

Function f(x)Use x, + − * / ^, pi, e and sin, cos, tan, exp, ln, log, sqrt, abs. Example: x^3 - x - 2.

Left end af(a) and f(b) must have opposite signs.

Right end b

ToleranceStop when the half-width of the interval is at most this. Between 0 and 1, such as 1e-6.

Maximum steps

Root (last midpoint)1.4142141

f at the root: 0.000001617
Steps taken: 20
Error bound (half-width): 9.537e-7
Step 1: a = 1, b = 2, c = 1.5, f(c) = 0.25
Step 2: a = 1, b = 1.5, c = 1.25, f(c) = -0.4375
Step 3: a = 1.25, b = 1.5, c = 1.375, f(c) = -0.1094
Step 4: a = 1.375, b = 1.5, c = 1.4375, f(c) = 0.06641
Step 5: a = 1.375, b = 1.4375, c = 1.40625, f(c) = -0.02246
Step 6: a = 1.40625, b = 1.4375, c = 1.421875, f(c) = 0.02173
Step 7: a = 1.40625, b = 1.421875, c = 1.4140625, f(c) = -0.0004272
Step 8: a = 1.4140625, b = 1.421875, c = 1.4179688, f(c) = 0.01064
Step 9: a = 1.4140625, b = 1.4179688, c = 1.4160156, f(c) = 0.0051
Step 10: a = 1.4140625, b = 1.4160156, c = 1.4150391, f(c) = 0.002336
Step 11: a = 1.4140625, b = 1.4150391, c = 1.4145508, f(c) = 0.0009539
Step 12: a = 1.4140625, b = 1.4145508, c = 1.4143066, f(c) = 0.0002633
Step 13: a = 1.4140625, b = 1.4143066, c = 1.4141846, f(c) = -0.000082
Step 14: a = 1.4141846, b = 1.4143066, c = 1.4142456, f(c) = 0.00009063
Step 15: a = 1.4141846, b = 1.4142456, c = 1.4142151, f(c) = 0.000004315
Step 16: a = 1.4141846, b = 1.4142151, c = 1.4141998, f(c) = -0.00003884
Step 17: a = 1.4141998, b = 1.4142151, c = 1.4142075, f(c) = -0.00001726
Step 18: a = 1.4142075, b = 1.4142151, c = 1.4142113, f(c) = -0.000006475
Step 19: a = 1.4142113, b = 1.4142151, c = 1.4142132, f(c) = -0.00000108
Step 20: a = 1.4142132, b = 1.4142151, c = 1.4142141, f(c) = 0.000001617

Runs in your browser: nothing you enter is sent anywhere.

How to use it

Type f(x), for example x^2 - 2.
Enter a and b with a sign change between them.
Set a tolerance, and read the root and the step table.

How it works

At each step the midpoint c = (a + b) / 2 is tested. If f(c) is 0 the search stops. Otherwise the half with the sign change is kept. The search stops when (b − a) / 2 is at most the tolerance, so the true root lies within that distance of the last midpoint.

Worked example

For f(x) = x² − 2 on [1, 2], the first midpoint is 1.5 and f(1.5) = 0.25 is positive, so the root is in [1, 1.5]. Next c = 1.25 and f is negative, so the root is in [1.25, 1.5]. After 20 steps the half-width is below 1e-6 and the result is about 1.414214, close to √2.

Tips

If the function touches zero without crossing (like x²), there is no sign change and the method cannot start.
If the interval holds several roots, the method finds one of them.

Frequently asked questions

What is the bisection method?

It finds a root of a continuous function by repeatedly halving an interval where the function changes sign. Each step keeps the half that still contains the sign change.

Why must f(a) and f(b) have opposite signs?

For a continuous function, a sign change means it crosses zero in between (the intermediate value theorem). Without it, the method has no guarantee.

How many steps will it need?

The interval halves each time, so about log₂((b − a) / tolerance) steps. From [1, 2] to a tolerance of 1e-6 takes 20.

Is it faster than Newton's method?

No. It gains about one binary digit per step, while Newton's method is faster near a root. Bisection always works once the sign change is bracketed, and needs no derivative.

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All tools in this category: Math ·Updated November 3, 2026