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Estimating Square Roots Between Consecutive Whole Numbers

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Approximating Square Roots with a Calculator

A calculator's $\sqrt{x}$ key can be used to find decimal approximations of square roots. For perfect squares, the calculator returns an exact value — for instance, $\sqrt{4} = 2$ and $\sqrt{9} = 3$ . For numbers that are not perfect squares, however, the calculator display shows only an approximation, not the exact square root. The display is limited by the number of digits it can show.

For example, a 10-digit calculator displays $\sqrt{5} \approx 2.236067978$ . To confirm this is an approximation and not the exact value, square the result: $(2.236067978)^2 = 5.000000002$ , which is close to $5$ but not exactly $5$ . Similarly, rounding to two decimal places gives $\sqrt{5} \approx 2.24$ , and $(2.24)^2 = 5.0176$ — again close, but not equal to $5$ .

Using the calculator and rounding to two decimal places, the square roots of $4$ through $9$ are:

Expression	Value
$\sqrt{4}$	$= 2$ (exact)
$\sqrt{5}$	$\approx 2.24$
$\sqrt{6}$	$\approx 2.45$
$\sqrt{7}$	$\approx 2.65$
$\sqrt{8}$	$\approx 2.83$
$\sqrt{9}$	$= 3$ (exact)

Notice that $\sqrt{4}$ and $\sqrt{9}$ use the equal sign because $4$ and $9$ are perfect squares, while the others use the approximation symbol $\approx$ because $5$ , $6$ , $7$ , and $8$ are not perfect squares.

Updated 2026-05-01

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