Significant Figures
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.
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.
- Trailing zeroes in a number with decimal place are significant.
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. |
|
Example |
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) |
Addition of 436.32 (2 digits after decimal), 227.2 (1 digit after decimal) & .301 (3 digits after decimal) is = 663.821
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 |