Significant Figures: Definition, Rules, and Examples

In this blog post, you will learn “What are significant figures and their rules?” with examples. After that, we will discuss why we round off significant figures and how to do it?. 

What are Significant Figures? 

As you know, the word significant is used for something that is essential and important. The term Significant figures is used for those digits in a number or measurement that contribute to its accuracy.

According to Wikipedia: 

“Significant figures of a number in positional notation are digits in the number that are absolutely necessary to indicate the quantity of something.”

Use of Significant Figures:

Significant figures have great importance in every field like science, medical e.t.c. 

These digits help to know the accurate measurements. Scientists use it while measuring substances, doctors use it to measure medicines and drugs or to know the measurements of different body-related masses and weights.

Similarly, engineers also determine significant figures to know the required precise value of various physical quantities in different devices and calculations. 

Rules of Significant Figures:

Significant figures are determined by taking a number of rules under consideration. You can learn these rules in the portion below with examples.

• Rule of Non-zero digits:

All the non-zero digits (0 through 9 inclusive) are considered significant. These numbers play a role in the precision of a measurement.

Example: 

In the numbers 1234 and 2.98, all the values are significant. But in 0.18 only 1 and 8 are significant. “Why 0 is insignificant?”, you will learn this later.

• Zero between two non-zero digits:

The zero which is present between two significant non-zero values is considered significant. 

Example: 

In example 2.09, the zero between 2 and 9 is significant because both 2 and 9 are significant digits. 

• Zero at the end (trailing zero):

When a number ends at zero, there can be two situations. The first is that the number is a whole number value e.g 2300. The second scenario is that the number has a decimal point e.g 2.40.

In the first case, the zeros are insignificant while in the second case zeros are significant.

Example: 

A number 230000 has only two significant values, the non-zero 2 and 3. But the 23.0000 has 6 significant values, all the zeros included.

• Other zeros:

Don’t you think these are more of “whether ‘this’ zero is significant or not rules” rather than “significant figure rules” ;)? Anyway, now we will discuss those zeros that did not come under any of the above headings.

• Zeros that are used as space holders in values less than one such as 0.001. In these numbers, the bolded zeros are insignificant.

• Zero that is before the decimal point in values less than one is also insignificant. For example first zero in 0.001.

These zeros are also called leading zeros. 

• Exact Numbers: 

There are some measurements that we know for sure e.g one cubic meter is equal to 1000 liters. Similarly, it’s a fact that there are 12 inches in a foot.

Such values are called exact numbers. These values come from actual counting and not from experiments. In such numbers, all of the digits are significant.

For example, the value of 1000 millimeters in a meter has 4 significant digits although it contradicts the sig fig rules.

Did you know that now the speed of light is also an exact number? Its value is 299,792,258 ms-1.

How to determine significant figures?

For the shortcut, you can use the significant figures calculator. It can round off the numbers, tells the individual sig figs, and gives different notations for the number. But if you want to know manually, see the example.

Example:

How many significant figures are in 0.02030?

Solution:

We will examine each digit separately.

• Zero before the decimal point is insignificant. (leading zero)
• Zero after the decimal point is also insignificant. (leading zero)
• 2 is significant because it’s non-zero.
• 0 after 2 is significant as it is present between two two sig figs.
• 3 is also significant.
• The last zero is significant because it is the trailing zero after the decimal point.

There are 4 significant figures (2, 0, 3, 0).

Rounding off significant figures:

Consider you measured a piece of cloth with an ultimate device and the value you got is 38.871 meters. A friend of yours also measured a cloth but he used a normal scale. The value he got was 51 meters.

Now if you want to add these two values, how would you do it? Simple addition?

It will be a little problematic. The answer will be 89.871 meters. One can think after looking at this value that the clothes are exactly this meter long. 

But there is a high possibility that there are some more digits in the value of the length your friend measured. It could be 51.23, 51.678, or any other number. You will never know unless you measure. 

In such cases, the significant figures are rounded off so that they can be easily used in math calculations. In other words, rounding off is used to remove uncertainty. 

In our example, the value 38.871 would be rounded off to two significant figures (because there are two sig figs in 51 meters). The rounded value of 38.871 is 39. 

Now, the sum of the two values will be 90.

If a whole number like 92999 is rounded off to 2 digits the answer will be 93000. Such values are then often represented in scientific notation i.e 9.3 x 104. (you can use the standard form calculator for this conversion.)

Rules of rounding off:

There are two basic rules to round off significant figures. Count up to the required number of significant values and take the first insignificant value (x).

1. If x < 5, leave the last significant figure in its original form.
2. If x > or equal to 5, add 1 to the last significant digit.

Now if the value is a whole number then replace all the insignificant values with 0’s. For example, rounding off 3987 to one sig fig gives 4000. 

But if the value has a decimal point then the insignificant values after the decimal point are dropped. And if there is an insignificant value before the decimal point, it is replaced with ‘0’. 

For example, if you want to round off 906.09 to two significant values, then the answer will be 910. The 0 and 9 after the decimal are dropped while 6 before the decimal is replaced with 0 after adding 1 to the last significant figure.

How to round off Significant figures?

The best way to learn anything is through examples and practice. So, here are a few examples of rounding off significant figures.

Example 1:

Round off 8.900 to one significant figure.

Solution:

The only significant digit we need is 8 and it is also the last significant figure. The first insignificant number is 9 which is greater than 5. This means we will have to add 1 to the 8.

= 8 + 1 = 9

Since all of the insignificant values come after the decimal place, we will drop them. Hence, 8.900 rounded off to 1 sig fig is 9.

Example 2:

What will be the rounded-off form of 40.950 to three significant figures?

Solution:

Step 1: Identify the first three significant digits.

= 40.9

Step 2: The first insignificant digit is 5. Add 1 to the last sig fig.

= 40.9 + 1

= 41.0

This is the answer.

Example 3:

How will you round off 56 to four significant figures?

Solution: 

The number itself contains only two significant figures. So how can we round it off to 4 places?

You remember that the trailing zeros after the decimal are significant, right!? We will use this rule here.

Step 1: Add decimal point to the number.

= 56.

Step 2: Add zeros to the required number. In our example, we need two extra significant places. This means we have to add 2 zeros.

= 56.00