Square Root Calculator — Guess, Check & Learn
Get the root immediately, then test your estimate or watch a guaranteed interval shrink around it.
Learn by estimating the root
1. Guess & check
Test your own estimate. Exact decimal arithmetic decides whether its square is below, above, or equal to the number.
2. Bisection walkthrough
This independent walkthrough starts with a guaranteed integer bracket, checks an exact midpoint, and keeps the half that still contains the root.
One symbol and one equation are not the same statement
√n names the non-negative principal square root. The equation x² = n has a negative and positive solution when n is positive, but only one distinct solution when n is zero. The result cards keep those meanings separate.
Check your own estimate before seeing a method
Guess & check answers one focused question: is your chosen value below, above, or exactly at the root? The entered decimal is squared and compared with the target using scaled-integer arithmetic rather than a rounded display value.
Follow a bracket that cannot lose the root
The bisection walkthrough starts between consecutive integers that contain the root. Each midpoint comparison keeps one half. Visible interval endpoints are exact terminating decimals, and the final width is the stated error bound.
Bounded real-number scope
Inputs are non-negative plain decimals no greater than 1 trillion with up to six decimal places. Output rounding uses half-up at 2, 4, 6, 8, or 10 decimal places. The walkthrough is explanatory; the direct answer is calculated independently with exact rational comparisons.
Frequently asked questions
Does √9 mean 3 or ±3?
The radical symbol √9 means the principal, non-negative square root, which is 3. The separate equation x² = 9 has two solutions: -3 and 3. This calculator keeps those statements apart.
Is 2.25 a perfect square?
Its square root is exactly 1.5, so it is an exact decimal square. Under the traditional integer definition, however, a perfect square is an integer whose square root is an integer; 2.25 is therefore not labeled an integer perfect square here.
How does guess and check work?
Enter a non-negative guess. The calculator squares that exact entered decimal and compares it with the original number. A smaller square means the guess is low; a larger square means it is high.
What is the bisection method for square roots?
Start with a low value whose square is no greater than the target and a high value whose square is no less than it. Test the midpoint, keep the half that still contains the root, and repeat until the interval is narrow enough.
Why does an approximate answer still show ≈ when it rounds to a whole number?
Rounding can make an inexact root look like a whole number. The approximation symbol remains so the displayed digits are not mistaken for an exact square root.
Does this calculator simplify radicals or calculate complex roots?
No. This bounded page handles the real principal square root of non-negative decimals, equation solutions, guess checking, and bisection intervals. It does not simplify radical expressions, evaluate formulas, or calculate complex roots.