Concept of Rounding Off

Some scientific calculators can be set to round off all displayed numbers to a particular number of decimal places or significant digits. However, it’s easy enough to round off numbers without a calculator: Simply drop all unwanted digits to the right of the digits that are to be retained and increase the last retained digit by $$1$$ if the first dropped digit is $$5$$ or greater. It may be necessary to replace dropped digits by zeros in order to hold the decimal point; for instance, we round off $$5157.3$$ to the nearest hundred as $$5200$$.

Rounding off should be done in one step, rather than digit-by-digit. Digit-by-digit rounding off may produce an incorrect result. For instance, if $$8.2347$$ is rounded off  to four significant digits as $$8.235$$, which in turn is rounded off to three significant digits, the result would be $$8.24$$. However, $$8.2347$$ is correctly rounded off in one step to three significant digits as $$8.23$$.

Example:
            Round off the given number as indicated.
(a) $$1.327$$ to the nearest tenth
(b) $$ – 19.8735$$ to the nearest thousandth
(c) $$4671$$ to the nearest hundred
(d) $$9.22345 \times {10^7}$$ to four significant digits

Solution:
            (a) To the nearest tenth, $$1.327 \approx 1.3$$
            (b) To the nearest thousandth, $$ – 19.8735 \approx – 19.874$$
(c) To the nearest hundred, $$4671 \approx 4700$$
(d) To four significant digits, $$9.22345 \times {10^7} \approx 9.223 \times {10^7}$$

If approximate numbers expressed in ordinary decimal form are added or subtracted, the result should be considered accurate only to as many decimal places as the least accurate of the numbers, and it should be rounded off accordingly. To add or subtract approximate numbers expressed in significant notation, first convert the numbers to ordinary decimal form, and then round off the result as above. Finally, rewrite the answer in scientific notation to obtain the appropriate number of significant digits.