Significant Figures: Definition, Rules, Algebra

Significant Figures: In physics, numbers are extremely important. We take readings during experiments and write them down in a certain order. An inaccuracy in the measurements can result in an error in the experiment’s outcome. As a result, we must utilise the proper equipment and appropriately record the observations before we can compute all of the data and arrive at our conclusion. All of the measurements we make when observing and computing values are subject to some mistakes. These mistakes must be recorded in a certain way in order for the measurement to be accurate. In general, we publish the result as a number that contains all of the digits in the specified number, as well as the first unknown digit.

Let us learn why some digits are more significant than others and the rules we need to follow while calculating them.

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Significant Figures: An Overview

Measurement is an important tool in physics. While experimenting, we make measurements of various quantities, and each measurement can be associated with a numerical value. It is quite critical to make a measurement that we carefully examine every digit in a given number. Therefore, we can conclude that not all digits in a number are equally significant. Accuracy and precision are the most reliable tools in any measurement.
Accuracy: In measurement, accuracy determines how close a measured value is to the given quantity’s true value.
Precision: For any measurement, precision tells us regarding the resolution or limit to which the given quantity is measured.
To understand this better, consider the following examples:
Imagine that you and your friend decide to meet sometime today in a park. The reported time can have three parts: the hour, the minute and the second. To finalise a time for the meeting, which is the most important part of the time? Well, the least information you need to set a time for the meeting is the hour. Now, if you include the exact minute of your meeting, you can set a much accurate time to arrive at the park and so on. However, the precision of your measurement will be decided by the timing at which you and your friend arrive in the park. The lesser is the time difference between your arrival, the more precise will be your measurement.
Similarly, suppose you wish to know how old is your mother. Her birthdate will have three parts: the date, month and year. But to calculate her age, all you need to know is her birth year. But if you include the month, it will increase the accuracy with which you can calculate her age. The exact date will further improve the accuracy and so on, and now if you use sources like your grandmother, your mother’s aadhar card and pan card to confirm all her birth details, you will be able to make a much precise measurement.
Thus, from the above examples, every measurement part can either help increase or decrease its accuracy. That effectively means that there is definite importance associated with every part of a reported number. The numbers that contribute to the accuracy of measurement are Significant figures.

What are Significant Figures?

The total number of digits in a given number, including the last digit whose value is uncertain, are called the Significant figures. These are the digits that are known with certainty and meaning to the value of a quantity. The reliability of a measured value depends on the number of digits used to express it.  To do so, we try and represent it with digits known with a certain level of certainty achieved by using Significant figures. They comprise all the definite digits plus one doubtful digit in a number. Further, we can mention here that the number of significant figures varies with the precision of a given instrument. Generally, greater precision means a greater number of significant digits.

Rules to Count the Significant Figures

1. All the stated non-zero numbers in measurement are significant.
   For example, \(44.123\) has \(5\) and \(44872317\) has \(8\) significant digits.
2. Zeroes between two non-zero digits are significant.
   For example, \(56900790000240000043\) has \(20\) significant digits, \(609.3\) has \(4\) and \(6.093\) also has \(4\).
3. Zeroes to the left of the first non-zero digit in the given number are not significant.
   For example, \(0.0019\) has \(2\) significant figures while \(0.0000407\) has \(3\).
4. Only the trailing zeroes or the zeroes to the right side of the decimal point are significant.
   For example, \(0.7000\) has \(4\), \(7.00\) has \(3\) and \(0.70\) has \(2\) significant figures.
5. When a number ends with zeros with no decimal point, then the trailing zeros are not significant.
   For example, \(78000\) has \(2\) significant figures, and \(780\) also has \(2\) significant figures.

Scientific Notation

Scientific notation involves representing a number in \(N × 10^n\) form.
\(N:\) is the number with a single non-zero digit to the left of the decimal point.
\(n:\) corresponding integer value
For example, we have a number \(71000\). We can represent it in the following ways:
\(7.1 × 10^4\): Two significant digits
\(7.10 × 10^4\): Three significant digits
\(7.100 × 10^4\): Four significant digits
\(0.71 × 10^5\): Two significant digits
\(0.071 × 10^6\): Two significant digits

Let Us Practice

How many significant figures are there in the number \(100\)?
To count the number of significant figures in \(100\), we will have to apply the rules that we have learned above.
The trailing zeros to the right side of a number without a decimal point are not significant.
Thus, \(100\) has one significant figure.
Zeroes between the last non zero number and decimal point on the left are significant. Trailing zeros on the right side of the decimal point is significant. Thus \(100.0\) has four significant figures.
Now we can write it in scientific notation as \(1 \times 10^2\) has only one significant digit.
It can also be written as \(10.0 \times 10^1\) has three significant digits.

Rules for Rounding Off

The rules used for rounding off a number to a required number of significant digits are:

1. If the last digit or the digit which has to be removed is greater than \(5\), then the digit to be retained or the digit preceding the last digit is increased by one.
   For example, \(1.478\), If we have it off to two decimal places, our answer will be \(1.48\).
2. If the last digit which has to be removed is less than \(5\), then the digit to be retained, or the digit preceding the last digit, remains unchanged. For example, \(5.334\) If we have to round it off to two decimal places, our answer will be \(5.33\).
3. If the rightmost digit to be removed is \(5\), then the number before it remains unchanged if this number is an even number. But, if the preceding number is odd, it is increased by one. For example,
   We are asked to round off:
   (i) \(5.625\) to two decimal places.
   Since \(2\) is an even number, our answer will be \(5.62\).
   (ii) \(7.875\) to two decimal places.
   Since \(7\) is an odd number, our answer will be \(7.88.\)

Algebra of Significant Figures

In physics, we deal with equations, and several of them involve arithmetic operations. The result of a calculation that involve an approximate measured value of quantities (i.e. values with a limited number of significant figures) must mirror the uncertainties in the original measured values. So while performing such algebraic operations on the measured values, we ensure that the result should not be more precise than the least precise value reported. So the final answer should not have more significant figures than the original data from which it was attained.
Thus, when we report our result, the number of significant figures in our calculation should be the same as the number of significant figures in the least precise value.

Addition and Subtraction

While performing the addition and subtraction over some measured values, the answer should have the same number of significant figures after rounding off as the value has the least number of significant digits.
For example:
\(1.2 + 3.66 + 7.777 = 12.637\)
But here \(12.637\) is not the correct answer; we have to account for the number of significant figures.
Here, \(1.2\) has the least precise value, and it contains two significant digits. So the answer will be rounded such that it also has \(2\) significant digits.
Thus, the answer is \(12.6.\)

Multiplication and Division

While performing the multiplication and division over some measured values, the answer should have the same number of significant figures after rounding off as the value has the least number of significant digits.
For example:
\(2.43 × 1.22 = 2.9646\)
But here, \(2.9646\) is not the correct answer; we have to account for the number of significant figures.
Each digit in the calculation has three significant digits. Therefore, the answer will be rounded off such that it also has three significant digits.
Thus, the answer is \(2.96.\)

Summary

The total number of digits in a given number, including the last digit whose value is uncertain, is called the Significant figures. These are the digits that are known with certainty and meaning to the value of a quantity.
The following diagram represents the rules for counting significant figures:

So, do important figures have anything to do with precision or accuracy? Hopefully, you can see that it is only focused on accuracy. Consider using a ruler to measure the length of a metal rod multiple times. With a slight reading error equivalent to around 1/10 of the smallest division on the ruler, you’ll receive virtually the same measurement again and over. You calculated the length with great accuracy. However, you have no way of knowing whether or not the ruler was accurate to begin with. Perhaps it was a stretched plastic ruler that had been left in the hot Texas heat. You won’t know how precise your measuring equipment is unless you calibrate it, which means comparing it to a known-to-be-accurate ruler.

Frequently Asked Questions

Q.4. Give three most important rules for counting significant figures.

Ans: (1) All non-zero digits in a given number are significant.
(2) The zeros between two non-zero digits are significant.
(3) The rightmost zeros in the case of a decimal number are considered significant.

Q.5. Is it correct to say that number of significant figures in \(1,\,10\) and \(100\) are the same?

Ans: Yes, the number of significant figures in each of these values is one. The trailing zeros in a non-decimal number are considered to be insignificant.