# Significant Figures

#### Topics

• Order of magnitude
• Significant figures
• Addition and subtraction of significant figures
• Multiplication and division of significant figures
• Rules for limiting the result to the required number of significant figures
• Rules for arithmetic operations with significant figures
• Rounding-off in the measurement

## 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 the least count of measuring instruments.

A choice of change of different units does not change the number of significant digits or figures in a measurement.

For example, the 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, the first two quantities have 4, but the 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. For example, 4.700 m = 4.700 x 102 cm = 4.700 x 103 mm = 4.700 x 10-3 In all the above, since the power of 10 is irrelevant, number of significant figures are 4.

• Multiplying or dividing exact numbers can have infinite number of significant digits. For 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 the 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 the 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, the below process should be used.

• Add the 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 percentages 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.

## Rules:-

1. For a set experimental data of ‘n’ significant figures, the result will be valid to ‘n’ significant figures or less (only in the case of subtraction).
Example 12.9 - 7.06 = 5.84 or 5.8 (rounding off to the lowest number of decimal places of the 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.
For example, the 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.

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