What is the mathematical significance of the intersection point between $y=x$ and $y=a/x$ in the context of the Babylonian method?

Conditions

• $a > 0$
• Considering only the first quadrant ($x>0$)

Reasoning, step by step

1. Set the two functions equal to each other: $f(x) = g(x)$.
2. Substitute the definitions: $x = \frac{a}{x}$.
3. Multiply both sides by $x$: $x^2 = a$.
4. Take the square root: $x = \pm\sqrt{a}$.
5. Since $x_0 > 0$ and $a > 0$, restrict to the positive root $x = \sqrt{a}$.
6. Identify this x-coordinate as the limit of the sequence.

Example

A graph appears showing the red line $y=x$ and the green curve $y=a/x$ intersecting at $x=\sqrt{a}$ in the first quadrant.

Common misconceptions

• Thinking the intersection represents the maximum error of the approximation.
• Confusing the x-coordinate of the intersection with the y-coordinate (they are equal here, but distinct concepts).

Watch the explanation

Explore next

• Does the sequence ever actually reach the intersection point in finite steps?
• How does the curvature of $y=a/x$ affect the stability of the iteration near the fixed point?