Abstract:

This video presents a follow-up analysis of a simple formula for approximating square roots, sqrt(x) ≈ (x + b) / (2 * sqrt(b)), where b is the nearest perfect square to x. Moving beyond the integer-based analysis of Part 1, this installment investigates the formula's behavior with continuous real numbers. The analysis explores three main scenarios: the approximation for numbers between 0 and 1 (where b=1 is used), the optimal choice of b when x is exactly at the midpoint between two perfect squares, and the nature of the overall error function. Key findings reveal that using the larger perfect square at the midpoint yields a more accurate result, and a graph of the error function surprisingly shows jump discontinuities at these midpoints, making the function right-continuous rather than smoothly continuous.

Uncovering the Quirks of a Simple Square Root Formula

• 0:00 The Approximation Formula: The video recaps the simple and effective formula for approximating the square root of a number x by using the closest perfect square, b.
• 1:23 Approximating for Numbers Between 0 and 1: The creator explores how the formula behaves for real numbers between 0 and 1 by consistently using b=1 as the closest perfect square.
• 2:32 Visualizing the Error: A graph compares the approximation to the exact square root for values between 0 and 1. It shows that while the approximation is poor for very small values of x (with an error limiting to 0.5), it becomes quite accurate for x > 0.5.
• 3:29 The Midpoint Dilemma: A new problem is investigated: when a number x is exactly equidistant from two perfect squares, which one should be chosen for the formula?
• 5:04 The Midpoint Solution: It is mathematically shown that using the larger (or right) perfect square in the formula always produces a smaller error and thus a more accurate approximation.
• 6:18 Graphing the Continuous Error Function: To visualize the accuracy across a range of numbers, the presenter graphs the error between the approximation and the true square root for continuous values of x from 0 to 16.
• 7:06 A Surprising Discontinuity: The resulting graph is not a smooth, continuous curve as expected. Instead, it features sudden drops or "jumps" at the midpoints between perfect squares, which occurs precisely where the algorithm switches to using the larger perfect square.
• 8:11 Right-Continuous Function: This "jumpy" behavior means the error function is not fully continuous but is classified as right-continuous. The size of these jumps gets progressively smaller as x increases.