\displaystyle \textbf{Approximations}

\displaystyle \textbf{Significant figures}

\displaystyle \text{The number of digits in a measurement about which we are certain, plus one}
\displaystyle \text{additional digit which is uncertain, are known as significant figures}.

\displaystyle \text{Thus, in }52.58\text{ cm, there are four significant figures. Similarly,}
\displaystyle 10.434\text{ has five significant figures}.
\displaystyle \\

\displaystyle \textbf{Rules for finding the number of significant figures}

\displaystyle \textbf{Rule 1: }\text{All non-zero digits are significant}.
\displaystyle \text{Examples:}
\displaystyle -546.13\text{ g has five significant figures}.
\displaystyle 357\text{ m has three significant figures}.

\displaystyle \textbf{Rule 2: }\text{Zeros between non-zero digits are significant}.
\displaystyle \text{Examples:}
\displaystyle 307\text{ m has three significant figures}.
\displaystyle 5.007\text{ kg has four significant figures}.
\displaystyle 205.002\text{ kg has six significant figures}.

\displaystyle \textbf{Rule 3: }\text{Trailing zeros to the right of a decimal point are significant}.
\displaystyle \text{Examples:}
\displaystyle 63.00\text{ has four significant figures}.
\displaystyle 2.50\text{ has three significant figures}.
\displaystyle 92.0500\text{ km has six significant figures}.

\displaystyle \textbf{Rule 3: }\text{In a decimal number between }0\text{ and }1,\text{ zeros after the}
\displaystyle \text{decimal point and before the first non-zero digit are not significant}.

\displaystyle \text{Example: }0.00686\text{ km has three significant figures, while }0.001010
\displaystyle \text{kg has four significant figures}.

\displaystyle \textbf{Rule 4: }\text{In a whole number ending in one or more zeros, the trailing}
\displaystyle \text{zeros may or may not be significant unless the accuracy is specified}.

\displaystyle \textbf{Rule 5: }\text{Final zeros in a decimal number obtained by rounding to a}
\displaystyle \text{specified number of decimal places are significant}.

\displaystyle \text{Example: When }7.896\text{ is rounded to two decimal places, we get }7.90,
\displaystyle \text{which has three significant figures}.
\displaystyle \\

\displaystyle \textbf{Rounding off a decimal to the required number of significant figures}

\displaystyle \text{Sometimes during measurements, it is required to obtain the value of a}
\displaystyle \text{decimal number correct to the required number of significant figures}.

\displaystyle \text{Example: Round }74.312\text{ kg to three significant figures}.

\displaystyle 74.312\text{ kg has five significant figures}.
\displaystyle \text{To round it to three significant figures, retain the first three}
\displaystyle \text{significant digits and examine the next digit}.
\displaystyle \text{The first three significant digits are }7,4\text{ and }3.
\displaystyle \text{Since the next digit is }1<5,\text{ the retained digits remain unchanged}.
\displaystyle \therefore 74.312\text{ kg}\approx74.3\text{ kg, correct to three significant figures}.
\displaystyle \\

\displaystyle \textbf{Rounding off to a specified unit}

\displaystyle \text{Example: Round off Rs. }64326\text{ to the nearest }10\text{ rupees}.

\displaystyle \text{The tens digit is }2\text{ and the digit to its right is }6.
\displaystyle \text{Since }6\geq5,\text{ increase the tens digit by }1.
\displaystyle \therefore \text{Rs. }64326\approx\text{Rs. }64330 \text{, rounded to the nearest }10\text{ rupees}.
\displaystyle \\

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