What are significant figures?

Rounding off a number to given significant figures follow some rules. Before studying those rules, it is important to learn some key terms.

The most significant figure is the left-most digit in a number. For example, the most significant figure in $$0.0057$$ is $$5$$ because $$0$$ in any digit works as a placeholder. If you round of $$k$$ significant figures, $$k^{th}$$ digit from most significant digit is the least significant figure, and it can be $$0$$. $$k+1^{th}$$ digit from the most significant digit is the first non-significant digit.

Rules for rounding off numbers:

If the value of the first non-significant figure is less than $$5$$, the least significant figure remains unchanged. Whereas, if it is greater than $$5$$, then the least significant figure increases by $$1$$. At last, replace all non-significant figures with $$0$$.

E1.9A: Make estimates of numbers and give an approximation to specified numbers of significant figures, and round off answers to reasonable accuracy in the context of a given problem.

Let us understand the rounding off the numbers to given significant figures with the help of examples.

Worked examples

Example 1: Round off the number $$0.567$$ to two significant digits.

Step 1: Take a look at the first and second significant digits.

$$5$$ and $$6$$ are the first and second significant figures, respectively.

Step 2: Check if the remaining numbers are above $$5$$ or below $$5$$.

Since the $$7$$ is greater than $$5$$, increase the least significant figure by $$1$$.

This gives rounding off the number $$0.567$$ to two significant digits, as $$0.57$$.

Example 2: Round the number, $$0.5671$$, to three significant digits.

Step 1: Take a look at the first, second, and third significant digits.

$$5$$, $$6$$, and $$7$$ are the first, second, and third significant digits, respectively.

Step 2: Check if the remaining numbers are above $$5$$ or below $$5$$.

Since the $$1$$ is smaller than $$5$$, the least significant figure remains the same.

This gives rounding off the number $$0.5671$$ to three significant digits, as $$0.567$$.

E1.9B: Make estimates of quantities and give approximation to specified numbers of significant figures, and round off answers to reasonable accuracy in the context of a given problem.

Quantities can be rounded off using the same rules given above. Let us understand with the help of examples.

Worked examples:

Example 1: Round the number, $$721\;\text{kg}$$, to 1 significant figure.

Step 1: Take a look at the first significant figure.

Digit $$7$$ is the first significant figure.

Step 2: Check the digit after the least significant figure.

Since the number $$2$$ is below $$5$$, change the digit $$7$$. But the non-significant figure should be changed to $$0$$.

Thus, the answer is $$700\;\text{kg}$$.

Example 2: Round the number, $$751\;\text{liters}$$, to two significant figures.

Step 1: Take a look at the first significant figure and the second significant figure.

$$7$$ and $$5$$ are the first and second significant figures, respectively. 

Step 2: Check the digit after the least significant figure.

Since the digit $$1$$ is below $$5$$, no need to change the digit $$5$$. But change the non-significant figure to $$0$$.

The answer is $$750\;\text{liters}$$.

E1.9C: Make estimates of Lengths and give an approximation to specified numbers of significant figures, and round off answers to reasonable accuracy in the context of a given problem.

Estimation and rounding off are used in everyday life. Not only the numbers but also the physical quantities, such as length, distances, etc., are estimated by rounding off the actual number to the nearest possible whole number.

Worked examples:

Example 1: Round the number $$22.37\;\text{inches}$$.

Step 1: Take a look at the least significant figure.

$$3$$ is the least significant figure.

Step 2: Check the digit after the least significant figure.

Since the number $$7$$ is greater than $$5$$, increase the least significant figure by $$1$$.

Thus, the final answer is $$22.4\;\text{inches}$$.

Example 2: Rounding the length of the rectangle $$186\;\text{cm}$$ to two significant figures.

Step 1: Take a look at the first significant figure and the second significant figure.

$$1$$ and $$8$$ are the first and second significant figures, respectively.

Step 2: Check the digit after the least significant figure.

Since the number $$6$$ is greater than $$5$$, increase the least significant figure by $$1$$, and replace all the non-significant digits by $$0$$. 

The final answer is $$190\;\text{cm}$$.