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Significant figures can be defined as a method of indicating the precision of a measurement. These are digits in a numerical value which are known to carry meaning and contribute to the measurement's accuracy.
- Each of the non-zero digits is significant. For instance, the number 128 has three significant figures.
- Zeroes which can be seen between non-zero digits are significant. For instance, the number 504 has three significant figures.
- Leading zeroes (that is, the zeroes which are to the left of the first non-zero digit) are not significant. For instance, the number 0.065 has two significant figures.
Significant figures were introduced to compensate for the uncertainties in experimental measurements. Precision measures how close different measurements are to each other for the same quantity. Accuracy, on the other hand, refers to the agreement of the value of a particular value with the true value of a result.
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Key Terms: Units and Measurements, Significant Figures, Digits, Milimetre, Analytical Measurements, Average, Coefficient, Rounding Off, Pacific Rule, Atlantic Rule, Precision
What are Significant Figures?
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Significant figures can be defined as the digits in a numerical value which are considered to be meaningful and relevant. These figures help to indicate the precision of a measurement or calculation. Significant Figures play an important role in communicating the degree of uncertainty associated with a value.
- The number of significant figures in a value can be determined by the precision of the measuring tool used to get the value.
- For example, if a ruler has markings denoting millimetres, then measurements made with that ruler can be reported to the nearest millimetre.
- In case a measurement is made to be 3.45 millimetres, then there are three significant figures because the measurement can only be reported to the nearest hundredth of a millimetre.
To ensure precision and accuracy in measurements and get reliable data, a fixed method was required to compensate for these uncertainties, leading to significant figures. Here’s an example to understand the difference between accuracy and precision.
| Measurements | |||
|---|---|---|---|
| Student Type | 1 | 2 | Average (g) |
| Student A | 2.95 | 2.93 | 2.940 |
| Student B | 2.94 | 3.05 | 2.995 |
| Student C | 3.01 | 2.99 | 3.000 |
In the above example, the true value of an experiment is 3.00g. Student 'A' takes two measurements, and their results are 2.95g and 2.93g. These values are precise as they are close to each other but not accurate.
Student B discovers 2.94g and 3.05g from the same experiment. This is neither a precise nor an accurate result. The results of the experiment when repeated by student C were 3.01g and 2.99g. The result is precise and accurate in this case.
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Significant Figures Video Lecture
Rounding Significant Figures
A number can be rounded off to obtain the required number of significant digits by leaving one or more than one digit from the right. The process of rounding to significant figures involves changing a number to a new value that has fewer significant figures.
In case the first digit has a value greater than 5, then the last digit is further rounded up. However, when the number left is 5, the number which is held can be rounded up or down to obtain an even number. However, when more than a digit is left, rounding off must then be done as a whole instead of one digit at a time.
The rules of Rounding Significant Figures include:
- First, check up to which digit the rounding off is going to be performed.
- In case the rounding digit is lesser than 5, we need to exclude the number which is present on the right-hand side.
- However, if the digit after the rounding off digit is greater than 5, then we must add 1 to the rounding of digit, excluding the numbers on the right side.
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Rules for Significant Figures
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To calculate an accurate measurement of significant figures, certain rules must be followed.
Here are the basic elements of the law:
- All non-zero digits are significant.
- Zeros preceding the first non-zero digit are not significant.
- Zeros between two non-zero digits are significant.
- Zeros at the end or right of a number are significant, provided they are on the right side of the decimal point.
Explanation of upper mentioned laws with examples
- All non-zero digits are significant.
For example, in the case of 1432, here we have 4 significant figures and in 0.295, there are three significant figures.
- Zeros preceding to first non-zero digit are not significant.
For example, 0.06 has one significant figure but 0.0092 has two significant figures.
- It is also a significant figure when there is a zero between two non-zero digits.
For example, 3.002 has four significant figures.
- Zeros at the end or on the right side of a number are also significant.
For example, there are three significant figures in 0.600
- When counting the quantities of an object.
For example, 5 apples or 10 lemons, there are infinite significant figures since these numbers are exact. They can be represented by writing an infinite number of zeros after placing a decimal i.e., 5= 5.000000 or 10 = 10.000000.
Significant Figures Example:
- 5807 – 4 significant figures
- 30.05 – 4 significant figures
- 5.00 – 3 significant figures
- 0.00300 – 3 significant figures
Read more: Quantitative Analysis
Frequently Asked QuestionsQues. What are Exact Numbers? Ans. Exact numbers can be defined as quantities that are known with complete precision and do not have any uncertainty with them. They can be determined by performing simple arithmetic operations or defining numerical constants. Exact numbers are considered to have an infinite number of significant figures because they are not subject to any measurement error or uncertainty. For example, the number of students in a classroom is an exact number if it is counted precisely. Ques. What are Measured Numbers? Ans. A measured number is a number that has been acquired via a process of measurement. These numbers are often subject to uncertainties and errors in measurement, which can affect the number of significant figures and the precision of the number. |
Addition and Subtraction of Significant Figures
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The result cannot have more digits to the right of the decimal point than either of the original numbers.
Here 10.0 has only one digit to the right of the decimal point. Therefore the result should also have only one digit after the decimal point. So the answer is 37.2.
359.62 - 67.3 = 292.3
In this case, 359.62 has two digits after the decimal point and 67.3 has one digit right to the decimal point. So here the answer 292.3 also has one digit after the decimal point.
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Multiplication and Division of Significant Figures
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In this case, the result must not have more significant figures than the original numbers used in the measurement with few significant figures.
3.52.12= 7.42
Since 3.5 has two significant figures, the result should not exceed significant figures more than two. Therefore the answer is 7.4.
Pacific Rule and Atlantic rule
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Pacific Rule should be applied to numbers that have decimal Present (note the double P). Start at the left side of the number, since the Pacific Ocean is on the left side of the United States. Count the significant figures starting at the first non-zero number and continuing to the end.
As an example, start from the left side of 0.000530 because it contains a decimal. The number of significant figures should not be counted until the first non-zero number (5). Continue counting until the end of the number. Consequently, this number has three significant digits (5,3,0).
Use the Atlantic Rule if you find a number without a decimal (the decimal is absent). Note again the double-A once more. Starting on the right side of the number, begin counting significant figures at the first number that is not zero since the Atlantic ocean is on the right side of the United States.
As we don't have a decimal in 3200, we should start from the right and count significant figures at the first non-zero number (2). Therefore, there are two sig figs in this number (3,2).
Things to Remember
- To round off the numbers, one must consider the following points for limiting the result to the required number of significant figures as per the above mathematical operations:
- If the rightmost digit to be removed is greater than 5, the preceding number is increased by one. For example, 2.257. If we have to remove 7, we have to round it to 2.26.
- The preceding number is not changed if the rightmost digit to be removed is less than 5. For example, 5.342 if 2 is to be removed, then the result is rounded up to 5.34.
- When removing the rightmost digit, if it is 5, the preceding number stays the same if the number is even, but is increased by one if it is odd. For example, if 7.35 is to be rounded by removing 5, we have to increase 3 to 4 giving 7.4 as the result. However, if 4.65 is to be rounded off it is rounded off to 4.6.
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Sample Questions
Ques. Why are Zeros at the End of the Number not Considered Significant? (1 Mark)
Ans. The first zero that is not present between any two significant figures, and is not present towards the right of the decimals, is not considered significant.
Ques. What is 18.5 multiplied by 21.62, taking into account Significant Digits? (2 Marks)
Ans. The calculator speaks that the product Is 399.97. Here, 185 have three significant digits, and 21.62 have a tour. So, we have to use three significant digits as our answer. That makes the answer 399.
Ques. Perform the following calculation and give your answer with the correct number of significant figures. (2 Marks)
\(\frac {(0.2920)(0.15)}{2.28}\)= ___
Ans. \(\frac {(0.2920)(0.15)}{2.28}\)=\(\frac{4.38}{2.28}\)=1.921
The correct answer to the above calculation with the correct number of significant figures is 1.9.
There are a total of two significant figures in the answer. This is because the answer is always written with that number of significant figures that have the least precise magnitude.
Ques. What are the Three Most Crucial Significant Figure Rules? (2 Marks)
Ans. These are the three most important rules of significant figures:
- Non-Zero figures can always be significant.
- Any zeros that are present between two significant digits are still significant.
- A final or trailing zero is significant only when it's in the decimal portion.
Ques. The respective number of significant figures for the numbers 23.023, 0.0003
and 2.1×10are: (5 Marks)
A)5,1,2
B)5,1,5
C)5,5,2
D)4,4,2
Ans. Significant figures in a particular number refer to the necessary numbers, required to represent that number accurately. To count the number of significant figures in a number, we have a certain set of rules, as given below:
- All non-zero figures are significant
- All zeroes between two non-zero figures are significant
- Leading zeroes (before non-zero figures) are not significant
- Trailing zeroes (after non zero figures) on the right side of the decimal are significant
- Trailing zeroes (after non-zero figures) on the left side of the decimal are significant. However, in the case of a non-decimal number, trailing numbers (after non zero figures) are not significant
Now, let us count the number of significant figures in the given numbers using these rules.
- 23.023
According to the rule, ‘All zeroes between two non-zero figures are significant’, all the given figures in the number are significant and hence, the total number of significant numbers is 5.
- 0.0003
According to the rule, ‘Leading zeroes (before non-zero figures) are not significant, all the numbers except 3 are not significant and hence, the number of significant figures is 1.
- 2.1×10−3
The number can be written as 0.0021. According to the rule, ‘Leading zeroes (before non-zero figures) are not significant, all the numbers except 2 and 1 are not significant and hence, the number of significant figures is 2.
Therefore, from the above explanation, it is clear that the correct answer is option A.
Ques. How many significant figures are there in the measurement of 0.020 km? (5 Marks)
Ans. To determine the significant figures in a suitable quantity, the following rules are to be applied:
(1) All non-zero digits are significant. For example, 175 cm, 0.175 cm, and 1.75 cm all have three significant figures.
(2) Zeros to the left of the first non-zero digit in the number are not considered as significant. Such zero indicates the position of the decimal point.
For example, 0.0165 cm has three significant figures and 0.0027 cm has two significant figures.
(3) Zeros between two non-zero digits are significant.
For example, 1.007 cm has four significant figures. 1.07 cm has three significant figures.
(4) Zeros at the right or end of a number are considered to be significant based on the condition that they belong to the right side of the decimal point. For example, 7.00 cm has three significant figures and 0.080 cm has two significant figures.
But if the terminal zeros are not significant if there is no decimal point e.g., 100 has only one significant figure.
(5) Exact numbers possess or show an infinite number of significant figures.
For example, in 5 pens or 50 copies, there are infinite significant figures present because these are exact numbers and can be shown or represented by writing an infinite number of zeros after placing a decimal i.e. 5 = 5.0000 or 60 = 60.00000
The ambiguity in the last point can be removed by expressing the number in scientific notation.
For example, we can express 4500 m in scientific notation in the following forms depending upon whether it has two, three, or four significant figures.
4.5×103m (Two significant figures)
4.50×103m 4.50 x 103 m (Three significant figures)
4.500×103m (Four significant figures).
In these expressions all the zeros to the right of the decimal point are significant.
So from these rules, we can conclude that there are 2 significant figures in 0.020 km, which is our required answer.
Ques. How many significant figures are in the measurement 0.0034 kg? (4 Marks)
Ans. We know that significant figures are sometimes referred to as the significant digits or precision of a number. They are digits that carry meaningful contributions to its measurement resolution.
Let us now look at the rules of significant figures:
All non-zero digits are considered significant. For example, 77 has two significant figures.
Zeros appearing anywhere between two significant figures are significant. For example, 101 has 3 significant figures. Zero is counted as a significant figure as it lies between 2 significant digits (1).
Zeros which are present on the left side of the significant figures are not significant. In other words, leading zeros are insignificant. For example, 0.0037 has two significant figures, that is, 3 and 7.
Zeros present to the right of the non-zero digits, that is, the trailing zeros are significant if they are to the right of the decimal point as these are necessary to indicate precision.
In the above question, the number given to us is 0.0034. Since leading zeros are insignificant, hence, the significant digits are 3 and 4.
Hence, 2 significant digits are present in 0.0034 kg.
Ques. State the number of significant figures in the following: (5 Marks)
a) 0.007m2
b) 2.64×1024 kg
c) 0.2370gcm−3
d) 6.320J
e) 6.032Nm−2
f) 0.0006032m2
Ans. a) 0.007m2: In this case, the number 0.007 can be written as 7×10−3. Hence, the only significant figure here is 7 i.e. there is only one significant figure in this number.
b) 2.64×1024kg: In this case, the number 2.64 has three figures which are non-zero figures, which makes all of them to be significant numbers. i.e. there are 3 significant figures in this case.
c) 0.2370gcm−3: Here, the number 0.2370 has three non-zero digits and one zero to the right of the decimal point, which follows the 3 non-zero digits. Hence, here we have four significant digits.
d) 6.320J: Here, just like the previous case we have one such zero that follows non-zero numbers and lies to the right of the decimal point. So, three non-zero figures and one significant zero make the total count of significant figures be 4.
e) 6.032Nm−2: The number 6.032 can be written as 6032×10−3, we know that all non-zero digits are significant and a zero that lies between two significant digits is also a significant figure. Hence, here we have 4 significant figures.
f) 0.0006032m2: The given number can be written as 6032×10−7, hence, similar to the above case the total number of significant figures is 4.
Ques. Identify the number of significant digits/figures in the following given numbers.
45, 0.046, 7.4220, 5002, 3800 (4 marks)
Ans. The following can be represented as:
| Number Given | Significant digits or figures |
|---|---|
| 45 | Two |
| 0.046 | Two |
| 7.4220 | Five |
| 5002 | Four |
| 3800 | Two |
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