The Babylonian method

This video demonstrates the geometric intuition behind the Babylonian method for computing square roots using an animation. It starts by presenting the recurrence relation $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$ and decomposes it into two functions, $f(x)=x$ and $g(x)=\frac{a}{x}$. By plotting these on a Cartesian coordinate system, the video shows how taking the arithmetic mean of their values at any point $x_n$ generates the next term $x_{n+1}$. The visual iteration process reveals that the sequence converges rapidly to the intersection of the line and the hyperbola, which corresponds to $\sqrt{a}$.

Before you watch

• Sequences and Series limits
• Monotone Convergence Theorem
• Graphs of linear and rational functions