How to Minimize Calculation Errors Using Significant Figures
What Are Significant Figures?
In Mathematics, we encounter questions where the solution comes out as 7.73.
We have a practice of rounding off the figures; here, we may get confused that can 7.73 be written as such or 7.7 or 7.730 or 7.70?
Let’s take another example:
Suppose your friend’s salary got incremented to ₹55,000. Now, he tells his salary as ₹55,000, however close to the exact amount.
Which means he told the rounded-off amount to you.
The above examples were to explain the significance of the figures or digits.
So, what is the definition of significant figures?
Significant figures in the measured value of a physical quantity which tells the number of digits in which we have a surety.
What Do You Understand by Significant Figures?
Let’s say you have a rod, and its length is measured 25.3 cm.
Here, the digit after the decimal point is called the ‘Doubtful digit’.
It’s because the measurement can be 25.2, or 25.28 or 25.345 cm as we are not confident about the value to be preferred, while the digits in the number ‘25’ are accurate because we have confidence upon these digits.
Significant Digits = Adding all the accurate digits + First doubtful digit.
There are common rules for counting significant figures that are discussed further.
What Are the Rules for Determining Significant Figures?
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Following are the common rules for counting significant figures in a given expression:
Rule 1: All non-zero digits are significant
For example, we have a number 7.744
It has four significant digits.
Rule 2: All zeroes occurring between two non-zero digits are significant
For example, 2007 has four significant digits, and 1.08079 has six significant digits.
Rule 3: Ending zeroes: The ending zero will be significant, if and only if, it is after the decimal.
For example, in a number, 3.400, the ending zero is after the decimal is significant. Now, according to rule 2, if the zeroes occur between the two significant digits, then they are also significant.
So, the number, 3.400, has four significant figures, i.e., 3, 4, 0, 0.
Rule 4 or Rule of Initial Zeroes: In a number less than one, all zeros to the right of the decimal point and the left of non-zero digits are NOT significant.
In simple words, the zeroes at the initial zeroes are not significant.
Let’s take a number, 0.00078
Here, we can see 0 at the initial (before the decimal point), which is not significant and the three non-ending zeroes.
So, 0.00078 has two significant digits.
Let’s say, a number, 0.9080
Here initial zero is insignificant, and 9080 has four significant digits.
Rule 5: All zeroes on the right of the last non-zero digits in the decimal part are significant.
For example, a number, 0.00700, has three significant figures 7, 0, 0.
Rule 6: All zeroes on the right of non-zero digits are NOT significant.
Here, we have numbers like 420, 4300; the ending zero is not after the decimal.
So, zero is not significant, and they have two significant digits only.
Rule 7: Power of 10 is insignificant
For example, 3.4 x 108 is the number, having 108 ending zeroes, but the power of 10 is always insignificant and having only two significant digits, i.e., 3 and 4.
Rules for Arithmetic Operation of Significant Figures
Addition
Let’s say the sum of three measurements of length; 3.2 m, 1.88 m, and 1.056 m is 6.136 m, rounded off to 6.1 m.
Subtraction
If we take x = 15.87 m, and y = 15.8 m, then,
x - y = 15.87 - 15.8 = 0.07 m, which is rounded off to 0.1 m.
In the subtraction of quantities, the magnitude and accuracy are almost destroyed.
Here, 15.87 has four significant figures and 15.8 has three. So, on subtracting these two numbers, we get 0.01 m which has one significant digit.
Multiplication
For example, x = 3.5 and y = 2.125, then xy = 7.4375
Here, out of 3.5 and 2.125, 3.5 has the least significant digits, i.e., 2.
So, rounding off 7.4375 to two significant digits, which is 7.4.
Division
If x = 9600, y = 11.25, then
x/y = 9600/11.25 = 853.33..
Here, 9600 has the least significant digits, i.e., 2. So, the answer will also have two significant digits.
Therefore, 853.33 is rounded off to 830.
Error Arithmetic Operations of Significant Figures
Relative error or fractional error and percentage error
Relative error or fractional error is calculated by the formula:
= mean absolute error/ mean value = Δμ’/μ
Percentage error is calculated by the formula:
= Δμ’/μ x 100 %
The measurements of different rods have got these values 4.2, 3.5, 4.7, 4.4, and 5.4, respectively. Find the mean value, relative, and the percentage error.
Here, mean (μ) = (4.2 + 3.5 + 4.7+ 4.4 + 5.4)/5 = 4.44 ≈ 4.4
Absolute errors in measurement are:
Δμ1 = 3.5 - 4.2 = - 0.7
Δμ2 = 4.7 - 3.5 = 1.2
Δμ3 = 4.4 - 4.7 = - 0.3
Δμ4 = 5.4 - 4.4 = 1.0
So, mean absolute error, Δμ’ = (- 0.7 + 1.2 - 0.3 + 1.0)/4 = 0.3
Relative error: Δμ’/μ = 0.3 / 4.4 = +/- ≈ 0.068 ≈ 0.1
Percentage error: 0.1 x 100% = +/- 10 %
FAQs on Error Arithmetic Operations and Significant Figures Explained
1. What are significant figures and why are they important for measurements in Physics?
In Physics, significant figures are the digits in a measured value that are known with certainty, plus one additional digit that is uncertain or estimated. For example, in a measurement of 25.3 cm, '2' and '5' are certain, while '3' is uncertain. They are crucial because they communicate the precision of a measurement, ensuring that the results of calculations don't appear more precise than the original measurements used.
2. What are the two primary rules for performing arithmetic operations with significant figures?
The rules for arithmetic operations depend on the type of calculation:
- For Addition and Subtraction: The final result should be rounded to the same number of decimal places as the measurement with the fewest decimal places.
- For Multiplication and Division: The final result should be rounded to the same number of significant figures as the measurement with the least number of significant figures.
3. How do you apply the rule for addition or subtraction with a practical example?
When adding or subtracting, you must focus on the number with the fewest decimal places. For instance, if you add three lengths: 15.11 m, 12.0 m, and 1.342 m.
Raw Sum: 15.11 + 12.0 + 1.342 = 28.452 m.
The value 12.0 m has only one decimal place, which is the least in the set. Therefore, the result must be rounded to one decimal place, making the final answer 28.5 m.
4. Can you provide an example of the rule for multiplication or division?
For multiplication or division, the result is limited by the number with the fewest total significant figures. For example, if you calculate the area of a rectangle with length = 4.52 cm (3 significant figures) and width = 2.1 cm (2 significant figures).
Raw Product: 4.52 cm × 2.1 cm = 9.492 cm².
Since the least precise measurement (2.1 cm) has only two significant figures, the final answer must be rounded to two significant figures, resulting in 9.5 cm².
5. How do you determine the number of significant figures in a measurement like 0.07080 g?
To determine the significant figures in 0.07080, you apply these rules:
- Leading zeros (the two zeros after the decimal) are not significant.
- Non-zero digits (7 and 8) are always significant.
- The zero between 7 and 8 is significant because it's a captive zero.
- The trailing zero at the end is significant because it is to the right of the decimal point.
Therefore, the number 0.07080 has four significant figures: 7, 0, 8, and 0.
6. How does error propagate when adding or subtracting quantities as per the CBSE Class 11 syllabus?
When two quantities are added or subtracted, their absolute errors add up. If a result Z is obtained from A and B, such that Z = A + B or Z = A - B, the absolute error in Z (ΔZ) is the sum of the absolute errors in A (ΔA) and B (ΔB). The formula is: ΔZ = ΔA + ΔB. This means the uncertainty in the final result is always greater than or equal to the uncertainty in the individual measurements.
7. Why do the rules for significant figures differ for addition/subtraction versus multiplication/division?
The rules differ because the operations handle error differently. Addition and subtraction are about aligning values of the same scale, so the limiting factor is the position of the least certain digit, which is determined by absolute error (decimal places). In contrast, multiplication and division involve scaling quantities, where the overall uncertainty is determined by the relative error. The number of significant figures is a direct indicator of this relative error, making it the limiting factor in these calculations.
8. How do exact numbers, like the '2' in the formula for a circle's circumference (C = 2πr), affect calculations involving significant figures?
Exact numbers, which include numbers from definitions (e.g., 100 cm in 1 m) or pure numbers in formulas (like the '2' in C = 2πr), are considered to have an infinite number of significant figures. This means they do not limit the precision of a calculation. The final answer's precision is determined solely by the measured quantities involved, such as the radius 'r' in this example.
9. What is the key difference between accuracy and precision, and how do significant figures relate to them?
Accuracy refers to how close a measured value is to the true or accepted value. Precision refers to how close multiple measurements of the same quantity are to each other. Significant figures are a direct indicator of the precision of a measuring instrument or a stated value. A high number of significant figures implies high precision, but it does not guarantee accuracy, as the instrument could have a systematic error.
