Significant Figures, Rules and Rounding off

The result of any measurement should be reported in a way that depicts the precision of the measurement. This gives rise to the concept of significant figures or digits. It can be defined as the digits carrying meaningful contributions to its measurement resolution. It can be represented as:

Significant digits = Certain & reliable digits + First uncertain digit

Including any more uncertain digits would convey misleading information. Certain rules are there to find out the number of significant digits a result contains:

1. All non-zero numbers are significant. Ex: 25.98 contains 4 significant digits
2. All zeroes between two non-zero digits are significant. Ex: 2002 has 4 significant digits
3. All the trailing zeroes to the right of the decimal point are significant. Ex: 2.50 has 3 significant digits
4. The trailing zeros in a number without a decimal point are insignificant. Ex: 250 contains 2 significant digits, whereas 250. contains 3 significant digits.
5. Leading zeroes are not significant. Ex: 0.25 and 0.0025 have 2 significant figures.
6. Exact numbers have an infinite number of significant digits. This rule applies to the numbers that are defined.  Ex: Circumference = 2*π*r, here 2 is an exact number and it can be written as 2.00000000.....so on.
7. For a number written in scientific notation - n * 10^[m] - all the digits of n are significant in accordance with rule 1 to 5. The exponent of 10 is insignificant. EX: 3*10^[8] has just 1 significant digit.

Arithmetic operations rules:

1. For addition and subtraction: Retain as many decimal points as present in the number with the least decimal points. Ex: 40.3+52.895+688.92 = 782.115 = 782.1 (final result), as only one digit is present in 40.3 after the decimal point.
2. For multiplication and division: Retain as many significant digits as present in the number with the least significant digits. Ex: 52.895/40.3 = 1.31 as there are 3 significant digits present in 40.3.

Rules for rounding off the uncertain digits:

1. If the insignificant digit to be dropped is greater than 5: Raise the preceding digit by 1. Ex: 15.489 ~ 15.49
2. If the insignificant digit to be dropped is less than 5: No change in the preceding digit. Ex: 15.482 ~ 15.48
3. If the insignificant digit to be dropped is equal to 5: 
• Case 1: Raise the preceding digit by 1 if the preceding digit is odd. Ex: 15.475 ~ 15.48
• Case 2: No change in the preceding digit if the preceding digit is even. Ex: 15.485 ~ 15.48

Note: Rounding off should only be done in the final result for a multi-step calculation.