A NEW SHORTCUT FOR SQUARE ROOTS
OF NON-PERFECT SQUARE NUMBERS
RESEARCHED & INVENTED BY MR. ANIKET BHARDWAJ
▶ Watch the full research presentation on YouTube — youtu.be/GuqiDf9E5Zk
The Research
How this discovery was made, and what it improves.
The Aniket Bhardwaj Method: Approximating Square Roots Mentally
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.
The Discovered Formula
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.
Worked Examples — See the Gaps
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.
Live Research Calculator
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.
Verification Data — Numbers 1 to 225
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).
Deviation Comparison of Both Formulas
How far each formula deviates from the true value, across the selected range — computed with the exact formulas used in the research Excel file.
Frequently Asked Questions
Quick answers about the square root shortcut method.
How do I find the square root of a non-perfect square number without a calculator?
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).
What is the formula of the Bhardwaj square root shortcut?
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).
How accurate is this square root shortcut?
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.
Who invented this square root method?
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.
What do I need to know to use this shortcut?
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.
Research Materials
Download the complete verification files and presentation.