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 to find the number of significant figures, systematic ways to present a number up to any significant figure. Also, we will learn to round off a value.

**Contents in this article:**

*What are the significant figures?**Importance of significant figures**Rules for significant figures**Rounding off the significant figures**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 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**

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 for 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**×**10 ^{23}**

**All the Non-zero digits are significant**. So, in 0.00**25**0 the digits 2 and 5 are significant figures. In**2**.00**53**0 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.**All the zeros at the beginning (left side) of a number are**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.*not significant*.**The zero between two significant digits is significant.**So, in the number 2.**00**530, the green zeros between 2 and 5 are significant digits. Also in 6.**0**22 the zero is between two significant digits 6 and 2. So, zero in 6.022 is also significant.**Any zero at the end (at extreme right) after decimal is significant.**So, in 0.0025**0**and in 2.0053**0**the green zero is significant.**But the zero at extreme right of a whole number is**. Example: 1 and 2 are the only two significant figures in 1200.*not significant***Power of 10 are not significant.**In the**Avogadro number**6.022×**10**the power of 10 are not the significant figures.^{23}

So the three numbers can be expressed in significant form as i) **0.00250** ii) **2.00530** and iii) **6.022×10 ^{23}**

Clearly, the number 0.00250 has three significant digits, 2.00530 has six significant digits and there are four significant digits in 6.022×10^{23}.

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

**Rounding off the significant figures**

To get better precision we need to rounding off the numbers up to some significant figures. There are also some rules for rounding off the significant figures. Let we need to rounding off a number **up to three significant figures**.

**If the fourth significant digit is less than 5, then third significant digit will remain same.**The rounding off value of 2.03234 up to three significant digit is 2.03. Because the fourth significant digit is 2 which is less than 5. So the third significant digit will remains same (here it is 3).**If the fourth significant digit is grater than 5 then in rounding off value we need to add 1 with third significant digit.**In 2.03634, the rounding off value is 2.04 as fourth significant digit is 6.**If the fourth significant digit is 5 and there is a non-zero digit after 5, then again we need to add 1 with third significant digit.**In 2.035134 the rounding off value up to three significant digit is 2.04.**If the fourth significant digit is 5 and there is 0 or there is no digit after this 5 then i) if the third significant digit is odd then we need to add 1 with third significant digit and ii) if the third significant digit is even then it will remain as it is.**For example: in the number 2.035 or 2.0350 the rounding off value is 2.04. Again, in 2.025 or 2.0250 the rounding off value is 2.02.

**Homework Problems:** Find the rounding of values up to two significant digits for the numbers i) 2.04 ii) 2.27 iii) 0.589

**Rules to find significant figures in calculations**

From the first paragraph we have learn that significant figures are important 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.

**Significant figures for addition**

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.

**Rules to find significant figures in 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 in Division**

**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**

**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 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.**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:**

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

So, in this article we learned the rules to find the number of significant figures in calculations and rounding off the numbers. 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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