How does the Babylonian method for computing square roots work geometrically using the arithmetic mean of two functions?

The Babylonian method iteratively computes the square root of a constant

a

by taking the arithmetic mean of the current estimate

x_n

and

a/x_n

. Geometrically, this is visualized by plotting the identity function

f(x)=x

and the inverse proportionality function

g(x)=a/x

. At any point

x_n

, vertical lines intersect these two curves. The height of the midpoint between these two intersection points represents the next term

x_{n+1}

. Projecting this midpoint back to the x-axis generates the new iterate. Repeating this zig-zag path shows the sequence converging rapidly to the intersection of the line and the hyperbola, which corresponds to

\sqrt{a}

Conditions

The constant $a > 0$ .
The initial guess $x_0 > 0$ .
The recurrence relation is $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$ .

Reasoning, step by step

Rewrite the iteration step as the average of two functions: $x_{n+1} = \frac{1}{2}(f(x_n) + g(x_n))$ , where $f(x) = x$ and $g(x) = \frac{a}{x}$ .
Plot the red line $y=x$ and the green curve $y=\frac{a}{x}$ on a Cartesian coordinate system.
Identify the intersection point of these two graphs, which occurs at $x=\sqrt{a}$ .
Start from a point $x_n$ on the x-axis and draw vertical lines up to intersect both curves.
Find the midpoint between the two intersection heights; this height represents the value of $x_{n+1}$ .
Project this midpoint horizontally back onto the x-axis to locate the new iterate $x_{n+1}$ .
Repeat the process to observe the points clustering tightly around the intersection, confirming convergence to $\sqrt{a}$ .

Example

The video demonstrates this by showing an animation where starting from a point on the x-axis, vertical lines are drawn up to intersect both curves. The midpoint between these two intersection heights represents the value of the next term in the sequence. This height is then projected horizontally back onto the x-axis to locate the new iterate.

Common misconceptions

Believing that the method requires solving a quadratic equation at each step.
Thinking that the convergence is slow or linear rather than quadratic.
Confusing the geometric midpoint with the arithmetic mean of the x-coordinates instead of the y-values.

Watch the explanation

BilibiliThe Babylonian method

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How does the Babylonian method for computing square roots work geometrically using the arithmetic mean of two functions?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

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Related questions

How does the Babylonian method for computing square roots work geometrically using the arithmetic mean of two functions?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

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