### 4.2 Significant figures

You may also come across the related concept of 'significant figures'. This is based on the same ideas about precision, but extends to the entire decimal number rather than just the decimal part. Let's look at a real-life example. Looking on Wikipedia, I find that the Earth's orbital period (or 'siderial' year) is 365.25636042 days (see http://en.wikipedia.org/wiki/Sidereal_year). This value means that the Earth does not go once around the Sun in an exact number of days - the journey takes about a quarter of a day longer than 365 'whole' days. To keep our calendars in pace with this, we have leap years every four years, where we pretend that it takes 366 years to orbit the Sun once. However, if we did this every four years we would gradually creep ahead, which is why we miss out leap years occasionally (eg at the start of a century).

So the 'real' value is 365.25636042 years (where that last '2' is 0.00000002 of a day or about one millisecond). In practical terms, we take the value of a year to be 365 days, which is using the first three digits of the value, which we also call 'expressing the value to three significant figures'. If we want to include the adjustment for leap years in most four year intervals, we would describe the length of a year as 365.26 years. This value has five significant figures, and better describes the length of the year. That there is a 6 rather than a 5 or a 7 in the second decimal place implies that we can describe the length of the year with a certainty or precision of one part in one hundred thousand - equivalent to a clock that gains or loses five minutes in the space of a year. If we include eight significant figures the value is 365.25636, and the value in the last decimal place implies a precision of one part in one hundred million - a clock that gains or loses about a third of a second over one year.

The number of decimal places and the number of significant figures are clearly related, and both are fundamentally concerned with the precision of the value. The number of significant figures takes into account the size of the whole-number part, as these examples show:

Value | Number of decimal places | Number of significant figures | |||||||||

1 | . | 4 | 6 | 7 | 3 | 4 | 5 | ||||

6 | 1 | 3 | . | 4 | 6 | 7 | 3 | 4 | 7 | ||

6 | 1 | 3 | . | 4 | 7 | 2 | 5 | ||||

3 | 7 | 6 | 1 | 3 | . | 4 | 7 | 2 | 7 | ||

3 | 7 | 6 | 1 | 3 | . | 4 | 7 | 0 | 0 | 4* | 9* |

** Note that the value in the last row is numerically identical to that in the row above, but is expressed to an additional two decimal places and two significant figures.*