# Significant Figures

When we buy eggs, bananas, chairs etc. we ask for an exact number and get an exact number of items. However when we try to measure the height of a person with a measuring tape, it may not be possible to measure the height exactly.

The difference in two situations arises because whereas eggs are measured by a discrete variable height is measured by a continuous variable.

Let’s suppose that height of a person is in between 160 cm and 161 cm. Then a measuring tape with centimetre markings can only tell that height is greater than 160 cm and less than 161 cm which means uncertainty in value of value is ± 1 cm.

If we take a scale with millimetre markings, it will be possible to measure to nearest millimetre but not further. Here, the uncertainty in the height will be ± 1 mm.

So measurement of a continuous variable can only be as precise as the choice of measuring instrument, but no matter what we do, some uncertainty will always remains.

Measurement can be made to different degree of precision depending upon instrument used. It is therefore, important to include such information while reporting results.

According to accepted convention, a number expressing any measurement should include all digits which are certain and a last digit which is uncertain. Sum of number of certain digits and 1 is called number of significant figures.

The number of significant figures in a measured quantity is equal to number of digits whose values are known with certainty, plus the first uncertain digit.

Number of significant figures refers to precision of a measured quantity.

The concept of significant figures can be better understood by considering following example.

Suppose, height of a person has been reported in three different ways – 160 cm, 160.0 cm and 160.00 cm

Although, three ways may look equivalent, but their scientific significance is different.

Reported value of 160 cm indicates that measurement has been made by a scale which can read only up to cm unit. The true value may lie between 159 cm and 161 cm.

Thus in value of 160 cm, the digits 1 and 6 are certain, but 0 is uncertain.

So, measurement of 160 cm has three significant figures.

The reported value of 160.0 cm indicates that this measurement has been made by a scale which can read up to one-tenth of a centimetre (up to mm).

Thus true value may lie between 159.9 cm and 160.1 cm

In this case, in 160.0 cm first three digits are certain while fourth digit is uncertain.
So value of 160.0 cm has four significant figures.

Similarly reported value of 160.00 cm indicates that this measurement has been made by a scale which can read up to one-hundredth of a centimetre (up to 0.1mm)

Thus true value may lie between 159.99 cm and 160.01 cm
In this case, in 160.00 cm first four digits are certain while fifth digit is uncertain.
So, 160.00 cm has five significant figures

It is also important to realise that result of any measurement should reflect faithfully precision of measurement. To report more significant figures than is possible to measure in a given situation is misleading.

For example – reporting your body mass up to gram or milligram is not significant.

## Rules for determining number of Significant Figures

Rule 1 – All non-zero digits are significant
The position of the decimal point is irrelevant.

For example – Number 217 has 3 significant figures and 2.17 also has 3 significant figures.

Rule 2 – A zero having digits on its right is not significant
A zero at beginning of a number is not significant.

For example
0.124 has 3 significant figures
0.0124 also has 3 significant figures
0.000005 has 1 significant figures

Rule 3 – A zero lying in between the two digits is significant
For example
1.0004 has 5 significant figures
145.0002 has 7 significant figures

Rule 4 – A zero having digits on its left may or may not be significant
For example
250.0 has 4 significant figures
250.00 has 5 significant figures

In exact numbers, which have zeros at end, these zeros at end may or may not be significant.
For example – 2500 may have 2, 3 or 4 significant figures depending up uncertainty.

If uncertainty in 2500 is ± 100 then second digit (5) will be uncertain as it can be either 4 or 6.
So number of significant figures in 2500 will be two.

If uncertainty in 2500 is ± 10 then second digit (0) will be uncertain as it can be either 1 or 9.
So number of significant figures in 2500 will be three.

If uncertainty in 2500 is ± 1 then second digit (0) will be uncertain as it can be either 1 or 9.
So number of significant figures in 2500 will be four.

Rule 5 – In exponential numbers, numerical portion represents number of significant figures
For example
In number 2 × 103, number of significant figures is one
In number 2.0 × 103, number of significant figures is two
In number 2.00 × 103, number of significant figures is three

## Calculations involving Significant Figures

In order to express results of an experiment, we have to often add, subtract, multiply or divide numbers obtained in different measurements. Usually, these different numbers do not have same precision.

Therefore, it is difficult to say what should be precision of final result.

Common sense tells that when several numbers of different precision are combined (added, subtracted, multiplied or divided) final result cannot be more precise than least precise number involved in calculations

In order to understand how to handle significant figures while combining numbers, you need to first understand rules about Rounding off Numbers.

Let’s first see what are these rules and then move on to handling of significant figures while combining numbers.

### Rules for Rounding Off numbers

Rounding off a number means that figures (digits) that are not significant are dropped. The rounding off is done to retain only the significant figures.

Rule 1
If digit to be dropped is greater than 5, then add 1 to the last remaining digit.

For example – 795.138 will become 795.14 if it’s rounded off to two places of decimal to retain only 5 significant figures instead of 6 in 795.138

Rule 2
If the digit to be dropped is less than 5, then last remaining digit is not changed.

For example – 859.732 will become 859.73 if it’s rounded off to two places of decimal to retain only 5 significant figures instead of 6 in 859.732

Rule 3
If the digit to be dropped is 5, then last remaining digit is left unchanged if it is even, whereas 1 is added to the last retained digit, if it is odd.

For example
5.4825 will become 5.482 (last retained digit is even, hence no change in last retained digit)
5.8835 will become 5.884 (last retained digit is odd, hence it will be raised by 1)

Rule 4
If during rounding off, more than one digit is to be dropped, then these are dropped one at a time following the above rules.

For example
2.12456 if rounded off to 3 decimal places (four significant figures) would not becomes 2.124

Instead rounding off done in 2 stages would give 2.125

1st step – 2.1246 (last digit to the dropped is greater than 5)
2nd step – 2.125 (last digit to be dropped is 6, greater than 5)

So if 2.12456 is rounded off to 3 decimal places then it will become 2.125

Now let’s move on to discussing rules for combining numbers if they have significant figures.

### Significant Figures rules for Addition and Subtraction

In addition and subtraction, the final results should be reported to the same number of decimal places as that of the term with least number of decimal places.

The reported number should be rounded off to the desired place of decimal as per rule described above.

It is to be noted that total number of significant digits in the final result would depend upon the number of decimal places required in final result.

Let’s understand this rule by considering some examples.

Example 1
11.1
2.11
0.111

Then report the final result to correct place of decimal.

Firstly we need to break down number of decimal places, significant figures in each of given number.

As per rule addition of these numbers should be reported such that final number will have least number of decimal places amongst numbers which are being added.

Out of 11.1, 2.11, 0.111 number of decimal places is 1

11.1 + 2.11 + 0.111 = 13.321 but we need to report it to 1 decimal place only so we need to round this number off such that it will contain only 1 digit after decimal.

Thus final sum of 11.1, 2.11, 0.111 will be reported as 13.3

Example 2
Subtract 4.2563 from 16.182 and report the correct result

Firstly we need to break down number of decimal places, significant figures in each of given number.

As per rule subtraction of these numbers should be reported such that final number will have least number of decimal places amongst numbers which are being subtracted.

Out of 16.182, 4.2563 least number of decimal places is 3

16.182 – 4.2563 = 11.9257 but we need to report it to 3 decimal places so we need to round this number off such that it will contain only 3 digits after decimal.

As per Rounding Off Rules, which I have discussed above 11.9257 will be written as 11.926 when rounded off to 3 decimal places.

Thus subtraction of 4.2563 from 16.182 will be as 11.926

## Significant Figures rules for Multiplication and Division

Results of multiplication or division are reported to the same number of significant figures as least number of significant figures amongst significant figures of all numbers involved in multiplication or division.

Let’s understand this rule by considering some examples.

Example 1
Multiply numbers
0.256
0.08205
298.16
Then report the results correctly

Firstly we need to break down number of significant figures in each of given number.

Least number of significant figures is 3
Hence final number from multiplication should be reported such that it contains 3 significant figures.

0.256 × 0.08205 × 298.16 = 6.26279

6.26279 have 6 significant figures but we need to report this number to 3 significant figures only.

Thus rounding off this number to 3 significant figures

6.26279 rounded off to 3 significant figures will be 6.26

So sum of 0.256, 0.08205, 298.16 will be reported as 6.26

Example 2
Divide 0.36 by 2.487 and report the result correctly

Firstly we need to break down number of significant figures in each of given number.

Least number of significant figures is 2

Hence final number from division should be reported such that it contains 2 significant figures only.

0.36 ÷ 2.487 = 0.14475

0.14475 have 5 significant figures but we need to report this number to 2 significant figures only.

Thus rounding off this number to 2 significant figures

0.14475 rounded off to 2 significant figures will be 0.14

So division of 0.36 by 2.487 will be reported as 0.14