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The difference between the calculated values and the actual value of any physical quantity is known as error. Error in Physics can be of two types which are Random Error and System Errors. Those errors that occur in an instrument during the repeated calculation of a physical quantity is known as random error and system error is defined as an error that happens due to the poor calibration of an instrument. The Significant figure is a digit in a number that is reliable and helps in indicating the quantity of any physical measurement.
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Key Terms: Error, Significant Figure, Arithmetic Operations, random Error, System Error, Calculation, Measurement, Addition, Subtraction, Systematic Error
What is Error?
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As we all know in Science, measurements are the fundamental of all experimental science and technology. Every calculation has its own results. The term error can be defined as the uncertainty of all measurements by measuring any physical quantity. The difference between the actual value and the calculated values of any physical quantity is known as Error of Measurement.
Significant Figures Video Lecture
Thee measures error of any physical quantity can be classified into three types which are:
- Systematic Error: The errors of any physical quantity whose results can be either positive or negative and be in one direction only is known as systematic errors. However, the reasons for these errors are normally known.
- Random Error: These errors can be defined as an error that occurs randomly irrespective of their sign and size. It can mostly arise because of the random and unpredictable fluctuations of experimental conditions.
- Least Count Error: It is the smallest error that is associated with the resolution of any measuring instruments. It is a type of random error but within a limited space as it occurs with both systematic as well as random error.
Systematic Error and Random Error
Also Read: Precision of a Measurement
Definition of Significant Figure
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The significant figures can be defined as the figure or number between 0 to 9 that is mainly used in the coefficient of expression. It helps in explaining the precision of an expression. It can be obtained by rounding off the expression after calculations. The larger the number of the significant figures we get in the measurements the greater is its accuracy.
Below we have given a table of significant figures along with the significant notification to understand better:
| Decimal Notation | Scientific Notation | Significant Figures |
|---|---|---|
| 1,544,000.00 | 1.544 × 10? | 4 |
| 0.00007650000 | 7.65 × 10-? | 3 |
| -0.0000000100 | -1 × 10-? | 1 |
Rules for Determining Significant Figure
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The significant figures can be counted based on the digits in any physical quantity. Some of the common rules that are used for counting the significant figures in an expression are given below:
- Rule 1: Every non zero digits in an expression is significant. For example, the number 7.876 contains four significant figures.
- Rule 2: The zeros between two non zero numbers can be calculated in significant figures. For example, 2007 has four significant numbers.
- Rule 3: The number ending with zero can be significant if and only if it comes after the decimal. For example, the number 4.300 has four significant figures.
- Rule 4: All the zeros to the left and right of any number can't be a significant number.
- Rule 5: The number having zeros at the rightmost part of any decimal number can be counted into a significant number.
- Rule 6: The zeros that are on the rightmost part of any number is not significant number. For example, the number 7800 and 450 has only two significant figures.
- Rule 7: The power of any significant digit can't be a significant digit. For example, 4.5 × 10? has only two significant numbers.
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Rules for Arithmetic Operations of Significant Figure
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The rules for calculating the significant figures can be obtained using some rules. The accuracy of the measurements is limited by the accurate calculation of an expression. Some of the expressions that are used in the significant figures are
- Addition and Subtraction
- Division and Multiplication
Addition and Subtraction
In addition and subtraction of a number, the result obtained after the calculation should retain as many decimal numbers in the expression as it has with the least decimal places. To be more clear let's understand it with an example
Example 1: If we add three measurements of length numbers such as 3.2 cm, 1.88 cm and 1.056 cm the result we get after the calculation is 6.136 cm. However, according to the significant figures, we can write it as 6.1 cm.
Example 2: Similarly if we subtract two numbers 15.87 cm and 15.8 cm with each other such as 15.87 cm - 15.8 cm = 0.07 cm and after rounding off the number we get 0.1 cm.
Multiplication and Division
As we have discussed the significant figures of Addition and subtraction rules, the same rules are followed in the multiplication and subtraction of a number. In multiplication and division, the result we obtained after the calculation may get rounding off the number as the original number of the expressions.
Example 1: Let's take two numbers x = 3.5 and y = 2,125. After multiplying the above two numbers we get xy = 7.4375. As we have seen that in the numbers x and y, x has the least digits. That's why the result after rounding off is 7.4.
Example 2: If we take two different numbers such as x = 9600 and y = 11.25. After dividing x by y we get the result 853.3333. Now as per the significant figures rules the above results can be rounded off to 850.
Things to Remember
- The accuracy of any number is the extent to which the calculated values may approach the true values of any physical quantity. As the error of any measurement decreases, its accuracy may increase.
- The precision of the number is the degree of exactness or refinement of the measurements of a physical quantity.
- The systematic error of any physical quantity can be obtained by three different errors such as instrumental error, personal error and imperfections in experimental techniques.
- The wrong results or errors in any calculation can happen because of some slightly different positions and wrong reading.
Sample Questions
Ques. What are the three most important significant figures rules? (3 Marks)
Ans. The significant figures can be explained based on its three most important rules:
- The zeros that exist between two non-zeros significant digits can be a significant
- Non zero digits always be a significant figure
- The final zero can be called a significant digit if it lies in the decimal portion
Ques. How does error propagate in addition and subtraction? How can we combine errors? (3 Marks)
Ans. The error in the result of any expression is equal to the square root sum of the squares of the individual errors in quantities being added or subtracted in the expression.
The errors can be combined whenever two physical quantities can be added or subtracted, the absolute error we get in the final result is equal to the sum of the absolute errors in the individual quantities.
Ques. What is the volume of a cylinder in cm³ having a diameter is 0.55 cm and its length of 1.35 cm? Calculate the volume of the block having length, breadth and thickness of the block is 12 cm, 6 cm and 2.45 cm? (3 Marks)
Ans. The volume of a cylinder = =π d²/4 × l =0.32057 cm³
= 0.32 cm³ as it has two significant figures.
The volume of a block can be calculated using the volume of a cuboid formula that is
Volume of block = l × b × h
= 12 × 6 × 2.45 cm³
= 2 × 10² cm³ ( since it has one significant figure).
Ques. Is relative error and fractional error the same? (3 Marks)
Ans. The relative can be obtained by dividing the absolute error with the quantity by the quantity itself. The relative error is mainly more significant than the absolute error. Relative errors are dimensionless and when reporting relative errors, it is usual to multiply the fractional error by 100 and report it as a percentage.
Example: 1 mm error in the diameter of a skate wheel is more serious than a 1 mm error in a truck tire.
Ques. What is the difference between random error and systematic error? What do you mean by the Absolute Uncertainty Formula? (3 Marks)
Ans. The systematic error can be referred to as a constant error as it remains the same in all measurements whereas random error can occur in an experiment which shows uncertainty in every result.
Absolute uncertainty is a type of uncertainty that we usually get in certain experiments. For example, if the height of a table is 236 ± 6 mm, then the absolute uncertainty is 5 mm.
Ques. What is a significant figures error and how many significant figures are in 100? (3 Marks)
Ans. The number written in the result of any expression or the number of meaningful digits implies the error in the measurements. For example, the physical quantity 0.438 has three significant figures.
As we know that zeros at the rightmost part of any number cannot be counted in the significant figures. Hence, the number 100 has only one significant figure.
Ques. What is the product of the number is we multiply 18.5 multiplied by 21.62, taking into account Significant Digits? (2 Marks)
Ans. The product of 18.5 and 21.62 is equal to 399.97. Here we see that the number 185 have three significant figure and the number 21.62 has four significant digits. So if we multiply the above two numbers the answer based on the least significant figure can be written as 399.
Ques. What do you mean by Pacific rule in significant figures? (3 Marks)
Ans. The Pacific rule can be defined as the rule number should be present in double decimal points. To be more clear suppose you are starting from the left side of the number, as the Pacific Ocean is situated on the left side of the United States. Now we count the significant figures from starting at the first non-zero number and continuing it to the end.
For Example- 0.0030
Count from the first non-zero number on the Pacific side, So from the first non-zero number from the Pacific side, 3, counting to the ‘Atlantic’, would be two significant figures.
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