Significant figures and rounding off rules play a crucial role in measurements by ensuring accuracy and precision in scientific calculations. In physics and other sciences, every measured value contains a certain degree of uncertainty, and significant figures help represent this reliability properly. Understanding how to identify significant digits and apply correct rounding rules is essential for expressing results in a meaningful and standardized form.
Significant Figures
The significant figures express the accuracy with which a physical quantity may be expressed.
Define significant figures
The digits, whose values are accurately known in a particular measurement, are called its significant figures.
Greater the number of significant figures obtained when making a measurement, more accurate is the measurement. Conversely, a measurement made to only a few significant figures is not a very accurate one. For example, a recorded figure of 1.21 means the quantity can be relied on as accurate to three significant figures and a figure of 1.212 is said to be accurate to four significant figures. Thus, significant figures in a measured quantity indicate the number of digits in which we have confidence.
In the example given above, one can easily find the number of significant figures. It is because, each digit used to express the magnitude of the measurement was significant. However, when we use the different systems of notation, this is not always the case. Suppose that we write the average distance $S$ from the earth to the moon as:
$$S = 377,000\ \text{km} = 3.77 \times 10^5\ \text{km}$$
Does this number have six significant figures? No, this measurement is given to a accuracy of only three significant figures. The 3, the 7 and the 7 are significant. The last three zeros merely indicate the correct location of the decimal point. Similarly, 0.000123 cm contains only three significant figures.
Rules for determining Significant Figures
For counting significant figures, we make use of the rules listed hereunder:
(a) All Non-Zero Digits are Significant
Any number that is not zero counts as a significant figure.
- Example: x = 2567 has four significant figures.
Examples :
- 42.3 has three significant figures.
- 243.4 has four significant figures.
- 24.123 has five significant figures
(b) Zeroes Between Non-Zero Digits are Significant
The zeroes appearing between two non-zero digits are counted in significant figure
- Example: 6.028 has four significant figures.
Examples :
- 5.03 has three significant figures.
- 5.604 has four significant figures.
- 4.004 has four significant figures
(c) Leading Zeroes are Not Significant
The zeroes located to the left of the last non-zero digit are not significant.
- Example: 0.0042 has two significant figures.
Examples :
- 0.543 has three significant figures.
- 0.045 has two significant figures.
- 0.006 has one significant figures
(d) Trailing Zeroes in Numbers Without a Decimal are Not Significant
In a number without decimal, zeroes located to the right of the non-zero digit are not significant. However, when some value is assigned on the basis of actual measurement, then the zeroes to the right non-zero digit become significant.
- For example, L = 20 m has two significant figures but x = 200 has only one significant figure.
- Example: x=200 has only one significant figure, however, if written as 20.0, it has three significant figures.
(e) Trailing Zeroes in Numbers with a Decimal are Significant
If a number has a decimal, all zeroes to the right of the last non-zero digit are significant.
- Example: x=1.400 has four significant figures.
Examples :
- 4.330 has four significant figures.
- 433.00 has five significant figures.
- 343.000 has six significant figures.
(f) Powers of Ten are Not Counted as Significant Figures
The exponent in scientific notation does not contribute to the significant figures.
- Example: 1.4 x 10−7 has two significant figures i.e., 1 and 4.
Examples :
- 1.32 × 10–2 has three significant figures.
- 1.32 × 104 has three significant figures.
(g) Change in Units Does Not Affect Significant Figures
Change in the units of measurement of a quantity, however, does not change the number of significant figures. For example, suppose the distance between two stations is 4067 m. It has four significant figures. The same distance can be expressed as 4.067 km or 4.067 ×105 cm. In all these expressions, however, the number of significant figures continues to be four.
Table of Significant Figures Examples
| Measured Value | Number of Significant Figures | Rule |
|---|---|---|
| 12376 | 5 | a |
| 6024.7 | 5 | b |
| 0.071 | 2 | c |
| 410 m | 3 | d |
| 720 | 2 | d |
| 2.40 | 3 | e |
| 1.6 × 10¹⁴ | 2 | f |
Problem : Write down the number of significant figures in the following:
(i) 6729 N (ii) 0·024
(iii) 6·0023 g cm$^{-3}$ (iv) 2·520 $\times$ 10$^{7}$ m
(v) 0·08240 N m$^{-2}$ (vi) 4200
(vii) 4·57 $\times$ 10$^{8}$ (viii) 91·000 m
Sol. (i) 6729 N has four significant figures. [Rule a]
(ii) 0·024 cm has two significant figures. [Rule c]
(iii) 6·0023 g cm$^{-3}$ has five significant figures. [Rule b]
(iv) 2·520 $\times$ 10$^{7}$ m has four significant figures. [Rule g]
(v) 0·08240 N m$^{-2}$ has four significant figures. [Rule b]
(vi) 4200 has two significant figures. [Rule d]
(vii) 4.57 $\times$ 10$^{8}$ has three significant figures. [Rule f]
(viii) 91·000 has five significant figures. [Rule e]
Rules for Rounding Off Measurements
Rounding off measurements is an essential concept in physics, chemistry, and mathematics, ensuring precision and ease in calculations. It follows specific rules based on the value of the digits that are removed.
- If the digit to be dropped is less than 5, the preceding digit remains unchanged.
- Example: 7.82 → 7.8
- Example: 3.94 → 3.9
- If the digit to be dropped is more than 5, the preceding digit is increased by one.
- Example: 6.87 → 6.9
- Example: 12.78 → 12.8
- If the digit to be dropped is 5 followed by nonzero digits, the preceding digit is increased by one.
- Example: 16.351 → 16.4
- Example: 6.758 → 6.8
- If the digit to be dropped is exactly 5 or 5 followed by zeros, and the preceding digit is even, it remains unchanged.
- Example: 3.250 → 3.2
- Example: 12.650 → 12.6
- If the digit to be dropped is exactly 5 or 5 followed by zeros, and the preceding digit is odd, it is increased by one.
- Example: 3.750 → 3.8
- Example: 16.150 → 16.2
Very Short Frequently Asked Questions (FAQs) and Answers
Why are significant figures important in scientific calculations?
Significant figures indicate the precision of a measurement and help in reducing errors in calculations.
Why does 1000 have only one significant figure while 1000.0 has five?
In 1000, the trailing zeroes are not considered significant unless there is a decimal. 1000.0 explicitly indicates a precise measurement with five significant figures.
Why do we use rounding off in measurements?
Rounding off ensures precision, reduces complexity in calculations, and minimizes insignificant variations in measurement values.
What happens if we round off multiple times?
Repeated rounding may lead to cumulative errors, hence it is recommended to round off only at the final step of calculations.
Why is the even-odd rule used for rounding off 5?
The even-odd rule helps minimize statistical bias in rounding large datasets.
How does rounding off affect precision and accuracy in experiments?
Rounding off reduces precision but maintains reasonable accuracy. It simplifies calculations while ensuring consistency in measured values.
If we round off 14.755 to two decimal places, what will be the result? Explain.
The answer is 14.76. Since the last digit to be dropped is 5 and the preceding digit (5) is odd, it is increased by one.
Strengthen your understanding with Class 11 Physics Notes for complete theory and concepts.
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