1Cademy - Estimating Square Roots Between Consecutive Whole Numbers

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Perfect Square Numbers

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Estimating Square Roots Between Consecutive Whole Numbers

When a number is not a perfect square, its square root is not a whole number. However, it can be estimated by identifying the two consecutive perfect squares that the number falls between. Because the square-root function preserves order, if a number lies between two consecutive perfect squares, then its square root must lie between the corresponding consecutive whole numbers.

For instance, since the perfect squares $4$ and $9$ are consecutive ( $2^2 = 4$ and $3^2 = 9$ ), any number between $4$ and $9$ — such as $5$ , $6$ , $7$ , or $8$ — has a square root that falls between $2$ and $3$ . In particular, for $\sqrt{5}$ we can write:

$2 < \sqrt{5} < 3$

The estimation technique works as follows: given a number, find the two consecutive perfect squares it lies between, then take the square roots of those perfect squares to establish the two consecutive whole numbers that bound the answer. This approach provides a quick way to approximate square roots without a calculator.

Updated 2026-05-01

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