Rules for counting significant figures in mathematical operations, Rounding off, Practice Problems, FAQs

Have you ever wondered how the billing amount at your local grocery shop is rounded off to a number which is an integer ? This type of rounding off is normal in science and engineering too. In the process of estimation and even in experiment rounding off is widely used.

The significant figures actually tell us about the uncertainties in measuring a physical quantity. In the process of calculations, involving two or more physical quantities, the result finally obtained can have different significant figures than the involved values of physical quantities. It seems to make no sense to have a more precise final value than the values on which it is based.

It is required to define rules for the calculation of significant figures in case of arithmetic operations.

Table of content:

Rules for counting significant figures in mathematical operations:
Rounding off
Practice Problems
FAQs

Rules for counting significant figures in mathematical operations:

For addition or subtraction:

Suppose two readings of a physical quantity are to be added or subtracted. In that case, the final value will have the same number of significant digits after decimal, as the constituting values with the least number of significant digits after decimal.

For multiplication or division:

If two or more values of the same or different physical quantities are multiplied or divided, the final calculated value should have as many significant figures as there are in the original constituting value with the least number of significant figures.

After calculations with the measured values, the result may contain one or more uncertain values which should be deleted. In this process of rounding off, the rules are as follows:

In case, the digit to be dropped off is less than 5, then its preceding digit remains unchanged.

E.g., $1.24 \approx 1.2$

Here the digit 4 is dropped off. It is less than 5. So, the digit 2 remains unchanged.

In case the digit to be dropped off is more than 5 then its preceding digit is increased by one.

E.g., $1.28 \approx 1.3$ Here the digit 8 is dropped off. It is more than 5. So, the digit 2 is increased by 1.

In case the digit to be dropped off is equal to 5 we have to consider two cases

If its preceding digit is an even number, then they should remain unchanged.

E.g., $1.25 \approx 1.2$

Here the digit 5 is dropped off. The preceding digit 2 is an even number. So, the digit 2 remains unchanged.

If its preceding digit is an odd number, then it should be increased by one.

E.g., $1.15 \approx 1.2$

Here the digit 5 is dropped off. The preceding digit 1 is an odd number. So, the digit 1 is increased by 1.

Practice Problems :

Q1. Calculate 5.8 + 2.035 .
Answer: 5.8 has 1 significant digit after decimal. 2.035 has 3 significant digits after decimal.

Here the final result must have the same number of significant digits after decimal, as the constituting values with the least number of significant digits after decimal i.e. 1 significant digit after decimal.

$5.8 (1 s i g n i f i c a n t d i g i t a f t e r d e c i m a l) + 2.0 (a f t e r r o u n d i n g o f f) = 7.8$

(2.035 rounded off to 1 significant digit after decimal)

Q2. Calculate $5.8 \div 2.035$ .
Answer: 5.8 has 2 significant digits and 2.035 has 4 significant digits.

Here, the final calculated value should have as many significant figures as there are in the original constituting value with the least number of significant figures, i.e. 2 significant digits.

$\frac{5.8 (2 s i g n i f i c a n t d i g i t s)}{2.035 (4 s i g n i f i c a n t d i g i t s)} = 2.85012285 \approx 2.8$ (Rounded off to 2 significant digits)

Q3. Round off 1.358500 to 2 significant digits.
Answer: The 3rd significant digit is equal to 5 which is to be dropped off. Its preceding digit, 3 is odd number so it is to be increased by 1.

i.e., $1.358500 \approx 1.4$

Q4. Round off 1.358500 to 4 significant digits.
Answer: The 5th significant digit is equal to 5 which is to be dropped off. Its preceding digit, 8 is an even number. So, it is to remain unchanged.

i.e., $1.358500 \approx 1.358$