Significant Figures
In rounding off significant figures or significant digits we will be very useful because it enables us to simplify complex calculation.
We will about……..
● Rules to find the number of significant figures
● Rounding off a decimal to the required number of significant figures
● Round off to a special unit.
What are Significant figures?
If we have to find the speed of car with the help of speedometer, we observe that the speed lies between 92.4 km/hr and 92.5 km/hr. By close observation, we can say approximately that the speed of car is 92.46 km/hr. The last digit 6 is however uncertain.
Thus, significant figures are all certain digits in a measurement plus one uncertain digit. The use of significant figures in measurement and value is a form of rounding.
Thus, in the above example, we say that 92.46 km/hr has four significant figures.
Rules for finding the number of significant figures:
1. All non–zero numbers (1, 2, 3, 4, 5) are always significant.
For example;
● 1325 has four significant figures
● 235.14 has four significant figures
2. All zeros between non-zero numbers are always significant.
For example;
● 204.003 has six significant figures.
● 50.00 has four significant figures.
● 51.02010 has seven significant figures.
3. In a decimal number which lies between 0 and 1, all zeros which are to the right of the decimal point but to the left of non-zero number are not significant.
For example;
● 0.00247 has only three significant figures.
● 0.002030 has four significant figures.
4. In a whole number if there are zeros to the left of an understood decimal point but to right of a non-zero digit the case becomes doubtful.
In the number 402000, there is a understood decimal point after the given six digits. There are 3 zeros that lie to the left of the understood decimal point but to the right of a non - zero number so the case becomes doubtful.
If it is expressed as 4.02 × 105, it becomes clear and we can say that 4.02 × 105 has 3 significant figures. It is expressed as 4.020 × 105, then the number of significant figures is 4.
5. When a decimal is round off to a given number of decimal places, all the final zeros in a decimal number are significant.
For example;
If we round off 6.785 to two decimal places, we get 6.80 which has 3 significant figures.
Rounding off decimals to the required number of significant figures:
Rounding off the number correct to three significant figures
(1) 53.214 → It has 5 significant figures.
To round off it to 3 significant digits, we required to round it off to 1 place after the decimal.
Therefore, 53.214 = 53.2 rounded off to 3 significant figures.
[The digit in hundredths place is 1 which is less than 5. So, the digit in the tenths place remains 2 and the digits 1 and 4 disappear.]
(2) 4.3062 → It has 5 significant figures. To round it off to 3 significant figures we round it off to the 2nd place after the decimal point.
So, 4.3062 = 4.31 correct to three significant figures.
(3) 30.002 → It has 5 significant figures. To round it off to 3 significant figures, we required to round it off to 1 decimal place after the decimal point.
30.002 = 30.0 correct to 3 significant figures.
(4) 0.0001378 → It has 4 significant figures. To round it off to 3 significant figures, we require to round it off to 6 decimal places after the decimal point 0.0001378 = 0.000138 correct to 3 significant figures.
(5) 0.0001366 → It has 4 significant figures. To round it off to 3 significant figures, we require to round it off to 6 decimal places after the decimal point.
0.0001366 = 0.000137 correct to 3 significant figures.
(6) 7.304 → 7.30 correct to 3 significant figures.
(7) 4.888 → 4.89 correct to 3 significant figures.
(8) 5.999 → 6 correct to 3 significant figures.
Examples to round off the following measurements:
(i) 1273.866 kg correct to 6 significant figures. It has 7 significant figures. To round it off to 6 significant figures, we round it off to 2 decimal place after the decimal point.
Therefore, 1273.866 = 1273.87 correct to 6 significant figures.
(ii) 203.102 g correct to 4 significant figures. It has 6 significant figures. To round it off to 4 significant figures, we round it off to 1st decimal place after the decimal point 203.102 = 203.1 correct to 4 significant figures.
(iii) 1.0718 mg correct to 2 significant figures. It has 5 significant figures. To round it off to 2 significant figures, we round it off to 1st decimal place after the decimal point.
1.0718 = 1.1 correct to 2 significant figures.
(iv) 0.003674 km correct to 1 significant figure. It has 4 significant figures. To round it off to 1 significant figure, we round it off to 3 places after the decimal point.
Therefore, 0.003674 = 0.004 correct to 1 significant figure.
Examples to rounding off to a specified unit
(i) Round off $ 65437 to the nearest 10 dollars.
$ 65440 to the nearest dollar
(ii) Round off $ 198.287 to the nearest 10 cents.
To round off 198.287 to the nearest 10 cents, we have to round it off to 1 place of decimal.
= $ 198.30
(iii) Round off 782.58 to the nearest dollar.
To round off 782.58 to the nearest dollar, we have to round it off to the nearest whole number.
Therefore, $ 782.58 = $ 783 rounded to the nearest rupee dollar.
(iv) Round off 475.095 to the nearest cents.
To round off 475.095 to the nearest cents, we have to round it off to 2 places of decimal. 475.095 rounded off to the nearest cents is 475.10.
(v) Round off 18.066 cm to the nearest mm.
Since, 1 mm = 0.1 cm, so to round it off to the nearest mm, we have to round it off to one decimal places.
18.066 cm = 18.1 cm rounded off to the nearest mm or 181 mm.
(vi) Round off 53.4278 m to the nearest cm.
Since, 1 cm = 0.01 m, we round off 53.4278 m to the nearest cm. We have to round it off to two places of decimal.
53.427 m = 53.43 m rounded off to the nearest cm or 5343 cm.
(vii) 0.00737 kg to nearest g.
Since, 1 g = 0.001 kg so to round off 0.00737 kg to the nearest g, we have to round it off to three places of decimal.
Therefore, 0.00737 kg = 0.007 kg rounded off to the nearest g.
(viii) 17.2262 g to the nearest mg.
Since, 1 mg = 0.001 g so to round off 17.2262 g to the nearest mg, we have to round it off to three places of decimal.
Therefore, 17.2262 g = 17.226 g rounded off to the nearest mg.
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