RESEARCHED & INVENTED BY MR. ANIKET BHARDWAJ
▶ Watch the full research presentation on YouTube — youtu.be/GuqiDf9E5Zk
How this discovery was made, and what it improves.
On 2 October 2024, in Dwarka, New Delhi, while working on faster mental-calculation techniques for students, Mr. Aniket Bhardwaj observed that the position of a number F between its two neighbouring perfect squares can itself be used as the decimal part of its square root. This led to the discovery of the ANIKET BHARDWAJ Method (Gap-Ratio formula): the square root of any non-perfect square number equals the root of the previous perfect square plus the ratio of the two gaps — ²√F = ²√P + G1/G2.
The traditional classroom shortcut divides Gap 1 by twice the previous root (²√P + G1/2√P). The new method replaces that denominator with Gap 2 — the natural distance between the two perfect squares. This single change makes the estimate self-scaling: it is exact at both perfect-square ends and never overshoots by a full unit.
The method was then verified computationally over every whole number from 1 to 225 against calculator values. The complete verification data, working slides and this interactive page are published openly below. The finding was publicly released on 2 February 2025, the researcher's 34th birthday.
Five simple values — then compare it with the traditional shortcut it improves upon.
Previous Perfect Square
Number whose square root is required
Next Perfect Square
F − P (Gap 1)
N − P (Gap 2)
Denominator = Gap 2 → self-scaling, never overshoots.
Denominator = 2×√P → can overshoot by a full unit near the next square.
The number ladder shows where F sits between the two perfect squares — the dark arrow is Gap 1, the light arrow is Gap 2. Exactly as illustrated in the research slides.
Enter any number — the ladder and the working update instantly using the discovered formula.
Works for any non-perfect square number. For perfect squares, the formula gives the exact answer.
All 225 rows from the research Excel file, recreated live — same colour groups, same conditional formatting. Narrow the range to zoom in.
← Swipe sideways to see all columns →
| Number | Normal Values | Next Normal Values | Traditional Method | Mr. Aniket Bhardwaj Method | More Accurate | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| F | Original √ | Prev. √ | Square (P) | Next √ | Square (N) | Answer | Deviation | ²√P + G1/G2 | Deviation | Method |
Conditional formatting (as in the Excel file): Perfect square — exact Traditional deviation ≥ 0.5 Aniket Bhardwaj deviation ≤ 0.25 · Bars = deviation size (data bars).
How far each formula deviates from the true value, across the selected range — computed with the exact formulas used in the research Excel file.
Quick answers about the square root shortcut method.
Use the Aniket Bhardwaj Method: √F = √P + (F−P)/(N−P), where P is the previous perfect square and N is the next perfect square. Example: √48 = √36 + 12/13 = 6.92 (calculator value 6.928).
²√F = ²√P + G1/G2, where G1 = F − P (gap from the previous perfect square) and G2 = N − P (gap between the two perfect squares).
Verified on every whole number from 1 to 225, the method's average deviation is about 52% lower than the traditional shortcut, and its worst-case deviation is 0.25 versus 1.00 for the old method. It is exact for perfect squares.
The Aniket Bhardwaj Method was discovered by Mr. Aniket Bhardwaj on 2 October 2024 in Dwarka, New Delhi, India, and published on 2 February 2025.
Only the perfect squares. Identify which two perfect squares your number lies between — for example, 78 lies between 64 and 81 — then apply √78 = √64 + 14/17 = 8.82.
Download the complete verification files and presentation.
