# 7.02 Significant digits

In the previous examples all the digits in the numbers were significant digits. Therefore the last significant digit was simply the last digit. This is not always the case.

A significant digit is one that contributes towards expressing the precision of a numerical value. (i.e. once we know how many significant digits there are in the number we identify the least significant digit and the uncertainty is within +/- half a unit of this digit.)

So if 140 has two significant digits then the least significant digit is 4 which lies in the tens column. Half of ten is five so the uncertainty is +/- 5 i.e the value is 140 +/- 5

Further examples
If the number 32500 has only 3 significant digits the value would be 32500 +/- 50
If the number 27000 has only 2 significant digits the value would be 27000 +/- 500
If the number 3600 has only 3 significant digits the value would be 3600 +/- 5

There are criteria for deciding whether digits appearing in numbers are regarded as significant or not These are

1. All non-zero digits are considered significant (1,2,3,4,5,6,7,8,9)
2. Zeros between two non-zero digits are significant (e.g 20105403 has 8 sig figs)
3. Leading zeros are not significant (e.g. 00034 has 2 sig figs)
4. Trailing zeros in a number containing a decimal point are significant (100.00 has 5 sig figs)
5. Trailing zeros in a number not containing a decimal point may or may not be significant!!

Criteria 5 is obviously ambiguous. To illustrate the problem of numbers with trailing zeros consider the following example.

The area of a field is measured giving a value of 1300 +/- 50 m2
A second field is measured with greater precision giving a value of 2500 +/- 0.5 m2
In the first example the number has only 2 significant digits (i.e. the uncertainty is within +/- half a unit of the second digit)
In the second example the number has 4 significant digits (i.e. the uncertainty is within +/- half a unit of the fourth digit)
When these numbers are written down without explicitly stating the uncertainty they both just appear as 4 digits numbers i.e. 1300 & 2500

So the problem is how would anyone know how many significant digits are in each number. One method is to underscore the last significant digit so the values could be written as 1 300 & 2500.

A better solution is to use scientific notation, this is explained in the next section