Approximations

Significant Figures: The number of digits in a measurement about which we are certain plus one additional digit which is uncertain are known as significant figures.

Thus, in $52.58$ cm, there are four significant figures. Similarly, $10.434$ have  $5$ significant digits.

Rules for Finding the Number of Significant Figures

Rule 1: All non-zero digits in a measurement are significant.

Example: $-546.13$ g has five significant figures. $307$ m has three significant figures.

Rule 2: In a decimal number which is greater than  $1$, all digits (including zeros) are significant.

Example:

• $205.002$ kg has six significant figures.
• $63.00$ has four significant figures.
• $2.50$ has three significant figures
• $92.0501$ km has six significant figures

Rule 3: In a decimal number which lies between 0 and 1, all zeros to the right of the decimal point but to the left of the· first non-zero digit are not significant

Example: $0.00686$ km has only  $3$ significant figures while  $0.001010$ kg has  $4$ significant figures.

Rule 4: If the measurement is a whole number, then the case becomes ambiguous if there are zeros to the left of an understood decimal point but to right of a non-zero digit.

Rule 5: All the final zeros in a decimal number obtained by rounding off a decimal to a given number of decimal places are significant.

Example: when  $7.896$ is rounded off to two decimal places, we get  $7.90$, which has three significant figures

Rounding off a Decimal to the Required Number of Significant Figures

Sometimes during measurements it is required to obtain the value of a decimal number correct to the required number of significant figures.

Example: Round $74.312$ kg to three significant figures:

Solution: The measurement  $74.312$ kg has  $5$ significant figures. To round it off to  $3$ significant figures, we are required to round it off to  $1$ place after the decimal. Therefore  $74.312$ kg =  $74.3$ kg rounded off to  $3$ significant figures.

Rounding off to a Specified Unit

Example: Round off: Rs.  $64326$ to the nearest  $10$ rupees. Rs.  $64326 =$ Rs.  $64330$ rounded to the nearest  $10$ rupees.