Decimal Approximations of Square Roots

The approximation of an irrational number by a decimal number introduces a margin of error. This error propagates through subsequent calculations and can significantly reduce the reliability of the final result.

For this reason, it is often preferable to carry out calculations directly with radicals and avoid replacing irrational numbers with their decimal approximations.

A practical example

Consider two irrational numbers written in radical form.

$$ \sqrt{29} $$

$$ \sqrt{87} $$

For each number I compute a decimal approximation to two decimal places, both from below and from above, with a margin of uncertainty of 0.01 in the last digit.

$$ 5.38 < \sqrt{29} < 5.39 $$

$$ 9.32 < \sqrt{87} < 9.33 $$

Note. In general, lower approximations, those on the left, are often preferred because they preserve the same first two decimal digits even when the approximation is refined with additional decimal places. However, choosing a lower approximation does not eliminate the problem of error propagation.

I now multiply the two inequalities term by term.

$$ 5.38 \cdot 9.32 < \sqrt{29} \cdot \sqrt{87} < 5.39 \cdot 9.33 $$

The result therefore lies between 50.1416 and 50.2887.

$$ 50.1416 < \sqrt{29} \cdot \sqrt{87} < 50.2887 $$

Although the product is expressed with four decimal places instead of two, it is still far from precise.

After the multiplication the margin of uncertainty is 0.1471, which is much larger than 0.01.

$$ 50.2887 - 50.1416 = 0.1471 $$

Thus, the product has a greater uncertainty than its factors.

The margin of uncertainty would continue to increase if further operations were performed.

Consequently, the reliability of the result gradually decreases as the calculations proceed.

Note. The propagation of uncertainty caused by decimal approximations of irrational numbers is not the same in every mathematical operation. It is greater in multiplication and division, and smaller in addition and subtraction. For example, if I add the two radicals above $$ 5.38 + 9.32 \cdot < \sqrt{29} + \sqrt{87} < 5.39 + 9.33 $$ $$ 14.7 < \sqrt{29} + \sqrt{87} < 14.72 $$ the margin of uncertainty between the lower and upper approximations increases from 0.01 to 0.02. This value is therefore much smaller than in the multiplication case (0.1471).

For this reason, it is preferable to perform calculations involving radicals using the rules for operations with radicals rather than replacing them with decimal approximations.

$$ \sqrt{29} \cdot \sqrt{87} = \sqrt{29 \cdot 87} = \sqrt{2523} $$

In this way the forward propagation of error is avoided.

Note. This discussion concerns only the approximation of irrational numbers. Not every radical represents an irrational number. For example $$ \sqrt{29} \cdot \sqrt{81} = \sqrt{29} \cdot \sqrt{ 9^2 } = \sqrt{29} \cdot 9 $$

And so on.