Rules for significant figures in Calculations

Suppose you have calculated a value and the calculator is showing it as 10.333333…. Now, what will you write to express the number? Up to what decimal number will you take the data? To know this, the concept of significant digits is essential. Significant figures or significant digits are very important in measurements to write the data in a systematic way. In this article, we’re going to learn some rules of significant figures in calculations and a systematic way to present a number up to any significant figure.

Contents in this article:

What are the significant figures?

Importance of significant figures
Rules for significant figures
Rules for significant figures in calculations

Calculation of Significant figures for addition
Significant figures for subtraction
Significant figures in Multiplication

Division in significant figures

What are the significant figures?

Significant figures are the digits in a number that are used to increase the precision of the calculation in a measurement. The Significant figure is also known as significant digit. It has an important role to make the calculated data precise with the measured data.

Importance of significant figures in Calculations

Significant figures in calculations have an important role to match the calculated values with the measured values with greater precision. Therefore, we need to know the rules to find the number of significant figures in a given number. One can understand this by the following example –

Let in a circuit a 3 ohm resistance is connected with a 10 Volt battery. The current through the circuit can be measured with an ammeter. Let the smallest division of the ammeter is 0.01 ampere and it measures the current as 3.33 ampere. So, the measured value is 3.33 ampere.

Again, one can calculate the value of current flow through the resistance by Ohm’s law of current electricity. Current, I=V/R which gives the calculated value of the current as 3.333333333… ampere.

Clearly, the calculated value is not precise with the measured value. If we take the calculated value in two significant digits after the decimal then the calculated value also be 3.33 Ampere. Then it becomes precise with the measured value. Therefore, significant digits are important in measurements and calculations.

Rules with significant figures

There are some rules to find the number of significant figures in a given number. Here, we are going to discuss those rules with examples. We will write the significant digits in green color and non-significant digits in red color. To understand the whole things we take three random numbers – i) 0.00250 ii) 2.00530 and iii) 6.022×10²³

All the Non-zero digits are significant. So, in 0.00250 the digits 2 and 5 are significant figures. In 2.00530 the digits 2, 5 and 3 are significant figures. But there may be more significant digits among the zeros. we will learn those in the next rules.
All the zeros at the beginning (left side) of a number are not significant. So, in the number 0.00250, the first three zeros are not significant. But in other two numbers, there are no zeros at the beginning.

The zero between two significant digits is significant. So, in the number 2.00530, the green zeros between 2 and 5 are significant digits. Also in 6.022, the zero is between two significant digits 6 and 2. So, zero in 6.022 is also significant.
Any zero at the end (at the extreme right) after the decimal is significant. So, in 0.00250 and in 2.00530 the green zeros are significant. But the zero at the extreme right of a whole number is not significant. Example: 1 and 2 are the only two significant figures in 1200.
Power of 10 is not significant. In the Avogadro number 6.022×10²³, the power of 10 is not a significant figure.

So the three numbers can be expressed in significant form as i) 0.00250 ii) 2.00530 and iii) 6.022×10²³

Clearly, the number 0.00250 has three significant digits, 2.00530 has six significant digits and there are four significant digits in 6.022×10²³.

Homework problems: Find the number of significant figures in the numbers i) 0.250800 ii) 100 iii) 110.0070

Rules for significant figures in calculations

From the first paragraph, you learn the importance of significant figures in calculations. But how to use significant digits in calculations like addition, subtraction, Multiplication and division. These are discussed below. Before this, you should have the concept of rounding off the numbers. Read this post on Rounding off significant figures.

Significant figures for addition

For the addition or subtraction of two or more numbers, one should concentrate on the significant digits after the decimal. In both cases, add or subtract the numbers in the usual manner. Then rounding off the result up to the lowest number of significant figures that any of those numbers has.

Example: Add the numbers 2.30, 2.578 and 2.5 to the significant figure.

General addition of those numbers is = (2.30+2.578+2.5)= 7.378

Now among the three numbers 2.5 has one, 2.30 has two and 2.578 has three significant figures respectively after the decimal. So the result of the addition should be up to one significant digit (lowest one) after the decimal. So, the answer to the addition in significant figure will be 7.4 (after rounding off).

Rules for Significant figures in Subtraction

Rules: First, do the usual subtraction. Then round off the result in the smallest significant figures that any of the given numbers have. Focus on significant digits after the decimal only.

Example: Subtract 1.72 from 5.218 in significant figures.

Subtraction of 1.72 from 5.218 is = (5.218 – 1.72) = 3.498

Now 1.72 has the smallest number of significant digits after the decimal. So, the answer will be 3.50 in significant figures after the rounding off.

Significant figures for multiplication

Rules: Do the usual multiplication and then write the result in the form of the smallest number of significant figures that the given numbers have.

Example-1: Multiply 3 by 2.5 in the significant figure.

The multiplication between 3 and 2.5 is 7.5.

Now the number 3 has the smallest significant figure after the decimal which is nothing. So the result will not contain any digit after the decimal. Therefore, the answer is 8 as 7 is odd any after 7 there is 5.

Emaple-2: Multiply 5 by 2.5 in the significant figure.

The multiplication of 5 with 2.5 is 12.5. But among the given numbers 5 has no digit after the decimal. So, the answer will be 12.

Rules for significant figures when dividing

Rules: Perform the usual division and then write the result in the form the smallest number of significant figures that the given numbers have.

Example: Divide 5.5 by 3.11 in a significant figure.

The division of 5.5 by 3.11 is 1.7684… Now, among the given numbers, 5.5 has one digit after the decimal and 3.11 has two digits after the decimal. So, the answer should contain the minimum digit after the decimal which is up to one. Hence the answer is 1.8 after rounding off.

Solved problems on significant figures

Find the number of significant figures in the number 1.065. Answer: In this case, all the digits are significant. Check rule-1 and 3 in above. So, there are 4 significant figures in this number.
Find the number of significant figures in 0.06900. Answer: Here, the digits 6, 9 and the two zeros at the extreme right are significant figures. Follow rule-1, 2, 4 and 3 above to figure it out. So, the number of significant figures in 0.06900 is four.
Find the number of significant figures in 100. Answer: 100 is a whole number. Rule-4 above says that zeros at the extreme right in a whole number are not significant. So, the number 100 has only one significant figure which is 1.

Homework problems:

Add 2.76 and 4.995 in significant figures
Multiply 10.2 with 0.9 in significant figures.

In this article, we learned the rules for significant figures in calculations. This is all from this article. If you have any doubts on this topic you may ask me in the comment section.

Thank you!

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