# Rules for significant figures in Calculations

Suppose you have calculated a value and the calculator is showing the value 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, we need the concept of significant digits. Significant figures or significant digits are very important in measurements to write the data in a systematic way. In this article, we are going to learn some rules of significant figures in calculations and a systematic way to present a number up to any significant figure.

1. What are the significant figures?
2. Importance of significant figures
3. Rules for significant figures
4. Rules for significant figures in calculations
5. Calculation of Significant figures for addition
6. Significant figures for subtraction
7. Significant figures in Multiplication
8. Division in significant figures

## What are the significant figures?

Significant figures are the digits in a number which are used to increase the precision of the calculation in a measurement. Significant figure is also known as significant digit. It has an important role to make the calculated data precision 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. We can measure the current through the circuit by an ammeter. Let the smallest division of ammeter is 0.01 ampere and it measures the current as 3.33 ampere. So, measured value is 3.33 ampere.

Again, we 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.

So, 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 calculated value also be 3.33 Ampere. Then calculated value becomes precision with measured value. Therefore, significant digits are important in measurements and calculations. Therefore we need to know the rules for finding of the number of significant figures in a given number.

## 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

1. 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 next rules.
2. 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.
3. 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.
4. Any zero at the end (at extreme right) after decimal is significant. So, in 0.00250 and in 2.00530 the green zero is significant. But the zero at extreme right of a whole number is not significant. Example: 1 and 2 are the only two significant figures in 1200.
5. Power of 10 are not significant. In the Avogadro number 6.022×1023 the power of 10 are not the significant figures.

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 we learn that significant figures are important thing in calculations. Now, we also need to know how to use the significant digits in calculations like addition, subtraction, Multiplication and division. I have discussed those in below. Before this, we need to know rounding off the numbers. Read this post on Rounding off significant figures.

For the addition or subtraction of two or more numbers we need to consider the significant digits after the decimal. In both cases, first we need to add or subtract the numbers as usual manner. Then we need to 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 in 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 of 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 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 smallest number of significant digit after decimal. So, answer will be 3.50 in significant figures after the rounding off. Because we cannot write 3.49 as next digit is greater than 5. So we have to add 1 with 9.

### Significant figures for multiplication

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

Example-1: Multiply 3 with 2.5 in significant figure.

Multiplication between 3 and 2.5 is 7.5.

Now the number 3 has smallest significant figure after decimal. So the result will not contain the digit after decimal. So, answer is 8 as 7 is odd any after 7 there is 5.

Emaple-2: Multiply 5 with 2.5 in significant figure.

Multiplication of 5 with 2.5 is 12.5. But among the given numbers 5 has no digit after decimal. So, 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 significant figure.

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

## Solved problems on significant figures

1. 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.
2. Find the number of significant figures in 0.06900. Answer: Here, the digits 6, 9 and the two zeros at extreme right are significant figures. Follow the rule-1, 2, 4 and 3 in above to figure it out. So, the number of significant figures in 0.06900 is four.
3. Find the number of significant figures in 100. Answer: 100 is a whole number. Rule-4 in 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:

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

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

Thank you!

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