How to Find √2 — The Bisection Method Explained Visually

What number, when you square it, gives exactly two?

You can never write it down. But you can hunt it down — one bit at a time.

This is the Bisection Method — the oldest and most reliable root-finding algorithm in numerical mathematics. And it is secretly the same thing as binary search, applied to the real number line.

THE SETUP
Define f(x) = x² − 2. When f(x) = 0, x² = 2. So the root is √2.

At x = 1: f(1) = −1 (below the axis)
At x = 2: f(2) = +2 (above the axis)

The curve is continuous. It cannot jump from below to above without crossing. This is the Intermediate Value Theorem. The root is trapped between 1 and 2.

THE ALGORITHM
Check the midpoint. Is it too big or too small?
- Too big → keep the left half
- Too small → keep the right half

That is it. One question per step. The interval halves every time.

STEP BY STEP
- m = 1.5 → f = +0.25 (too big) → keep [1, 1.5]
- m = 1.25 → f = −0.44 (too small) → keep [1.25, 1.5]
- m = 1.375 → f = −0.11 (too small) → keep [1.375, 1.5]
- And it keeps going...

THE GUARANTEE
After n steps, the maximum error is:
error ≤ (b − a) / 2ⁿ

- 10 steps → error less 0.001
- 20 steps → error less 0.000001
- 34 steps → 10 correct decimal digits

And it never fails. Unlike Newton's method which can diverge, bisection always converges. Always.

THE RESULT
After 50 steps: √2 = 1.4142135623...

An irrational number. Infinite. Non-repeating. Found one bit at a time.

THE DEEPER CONNECTION
Bisection = Binary Search on ℝ

The same algorithm every computer uses to find a word in a dictionary. But applied to the continuous number line. One question per step. Left half or right half.

Next video: Newton's Method — much faster convergence, but it comes with a catch.

Follow @dailymathvisuals for more visual math.

Timestamps:
0:00 — What number squared gives 2?
0:20 — The curve f(x) = x² − 2
0:40 — The Intermediate Value Theorem
1:00 — First cut: m = 1.5
1:20 — Second cut: m = 1.25
1:35 — Third cut: m = 1.375
1:50 — Accelerating: steps 4–10
2:10 — The error guarantee
2:35 — The digits of √2
2:50 — Bisection = Binary Search
3:05 — Outro

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Видео How to Find √2 — The Bisection Method Explained Visually канала DailyMathVisuals