- Measurements - Significant Figures
- Rules for Arithmetic Operations with Significant Figures
- Rounding off the Uncertain Digits
- Rules for Determining the Uncertainty in the Results of Arithmatic Calculations
Every measurement results in a number that includes reliable digits and uncertain digits. Reliable digits plus the first uncertain digit are called significant digits or significant figures. These indicate the precision of measurement which depends on least count of measuring instrument.
A choice of change of different units does not change the number of significant digits or figures in a measurement.
Example, period of oscillation of a pendulum is 1.62 s. Here 1 and 6 are reliable and 2 is uncertain. Thus, the measured value has three significant figures.
Rules for determining number of significant figures:-
All non-zero digits are significant.
All zeros between two non-zero digits are significant irrespective of decimal place.
For a value less than 1, zeroes after decimal and before non-zero digits are not significant. Zero before decimal place in such a number is always insignificant.
- Trailing zeroes in a number without decimal place are insignificant.
Cautions to remove ambiguities in determining number of significant figures
Change of units should not change number of significant digits. Example, 4.700m = 470.0 cm = 4700 mm. In this, first two quantities have 4 but third quantity has 2 significant figures.
Use scientific notation to report measurements. Numbers should be expressed in powers of 10 like a x 10b where b is called order of magnitude. Example, 4.700 m = 4.700 x 102 cm = 4.700 x 103 mm = 4.700 x 10-3 In all the above, since power of 10 are irrelevant, number of significant figures are 4.
Multiplying or dividing exact numbers can have infinite number of significant digits. Example, radius = diameter / 2. Here 2 can be written as 2, 2.0, 2.00, 2.000 and so on.
Rules for Arithmetic operation with Significant Figures
|Type||Multiplication or Division||Addition or Subtraction|
|Rule||The final result should retain as many significant figures as there in the original number with the lowest number of significant digits.||The final result should retain as many decimal places as there in the original number with the least decimal places.|
Density = Mass / Volume
if mass = 4.237 g (4 significant figures) and Volume = 2.51 cm3 (3 significant figures)
Density = 4.237 g/2.51 cm3 = 1.68804 g cm-3 = 1.69 g cm-3 (3 significant figures)
436.32 (2 digits after decimal),
227.2 (1 digit after decimal) & .301 (3 digits after decimal) is
Since 227.2 is precise up to only 1 decimal place, Hence, the final result should be 663.8
Rules for Rounding off the uncertain digits
Rounding off is necessary to reduce the number of insignificant figures to adhere to the rules of arithmetic operation with significant figures.
|Rule Number||Insignificant Digit||Preceding Digit||
Example (rounding off to two decimal places)
|1||Insignificant digit to be dropped is more than 5||Preceding digit is raised by 1.||
Number – 3.137
Result – 3.14
|2||Insignificant digit to be dropped is less than 5||Preceding digit is left unchanged.||
Number – 3.132
Result – 3.13
|3||Insignificant digit to be dropped is equal to 5||If preceding digit is even, it is left unchanged.||
Number – 3.125
Result – 3.12
|4||Insignificant digit to be dropped is equal to 5||If preceding digit is odd, it is raised by 1.||
Number – 3.135
Result – 3.14
Rules for determining uncertainty in results of arithmetic calculations
To calculate the uncertainty, below process should be used.
Add a lowest amount of uncertainty in the original numbers. Example uncertainty for 3.2 will be ± 0.1 and for 3.22 will be ± 0.01. Calculate these in percentage also.
After the calculations, the uncertainties get multiplied/divided/added/subtracted.
Round off the decimal place in the uncertainty to get the final uncertainty result.
Example, for a rectangle, if length l = 16.2 cm and breadth b = 10.1 cm
Then, take l = 16.2 ± 0.1 cm or 16.2 cm ± 0.6% and breadth = 10.1 ± 0.1 cm or 10.1 cm ± 1%.
On Multiplication, area = length x breadth = 163.62 cm2 ± 1.6% or 163.62 ± 2.6 cm2.
Therefore after rounding off, area = 164 ± 3 cm2.
Hence 3 cm2 is the uncertainty or the error in estimation.
1.For a set experimental data of ‘n’ significant figures, the result will be valid to ‘n’ significant figures or less (only in case of subtraction).
Example 12.9 - 7.06 = 5.84 or 5.8 (rounding off to lowest number of decimal places of original number).
2.The relative error of a value of number specified to significant figures depends not only on n but also on the number itself.
Example, accuracy for two numbers 1.02 and 9.89 is ±0.01. But relative errors will be:
For 1.02, (± 0.01/1.02) x 100% = ± 1%
For 9.89, (± 0.01/9.89) x 100% = ± 0.1%
Hence, the relative error depends upon number itself.
3.Intermediate results in multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.
Example:1/9.58 = 0.1044
Now, 1/0.104 = 9.56 and 1/0.1044 = 9.58
Hence, taking one extra digit gives more precise results and reduces rounding off errors.