The Number of Significant Digits

Significant digits are all the numbers that are certain, and one digit that contains some uncertainty.

When you record a measured number from an experiment, you must make sure that you write down the digits correctly.  When doing calculations, we need to know the number of significant digits.  In most numbers this is easy.  Just count the digits (not the number of decimal places) in the recorded measured number.

Value Number of significant digits
3468 4
34.68 4
3.468 4

Almost every bit of confusion in counting significant digits comes from zeroes.  Sometimes zero is a measured number.  Sometimes though it is a place holder.

Here are some numbers containing zeroes.  Can you determine why some of them are counted, and some are not?
 
Value Number of significant digits
34608 5
0.3468 4
3.4680 5
0.03468 4
0.034680 5
3.4608 5
0.034608 5
0.0003468 4
0.00034680 5
0.00034608 5
Which of the following are correct? Check all that apply.
a) a zero between two non-zero digits always counts
b) zeroes only count if they precede a non-zero number
c) all zeroes to the left of the first non-zero digit never count
d) a zero after a decimal point and to the right of a non-zero digit always counts
e) zeros always count if they are to the right of a decimal point

Check your list of rules for zeros.  How close are your rules to these?

Hopefully that doesn't seem too bad, so far.  Now here's the problem.  Suppose you asked me to tell you the population of Prince Albert, my home town.  I'd probably answer "about 30,000."  You probably wouldn't think that is the exact number of people resident here, though it might be.  Ok, just how many significant digits are there in that number?  In fact, the 2001 census lists the population of Prince Albert at 34,291 persons.  If you know this, then obviously, 30,000 wasn't an exact answer.  How many significant digits are there in this number?  The 3 is certain. The first 0 after the 3 isn't right though, it is a rounded off estimate. This lets me know that there are just two significant digits in 30,000, because it is uncertain at the thousands place. 

We can figure this out because we know what the real answer is.  However, there is no way you can tell this just looking at the number 30,000.   If you made the measurement there is never any problem – for you.  You know whether you measured the zeroes or not.  But there well may be for someone else who is just looking at your data. 

Zeros can be terribly confusing.  So, to avoid confusion, it is always best to write numbers in scientific notation.  In fact, this is absolutely essential when writing numbers with zeroes.   All the digits in a number written in scientific notation are considered significant (of course, don't write down the digits if they are not!)

Here's how we'd write the population of Prince Albert in scientific notation:

Value Significant digits
3.4 x 104 2
3.4181 x 104 5

 

Always write numbers in scientific notation if there is any confusion about the number of significant digits they contain.  Every digit in a number in scientific notation is a significant digit.


In the following, assume the numbers are the result of experimental measurements.

1. Which of the following numbers must be written in scientific notation to avoid confusion with zeroes as significant digits?  Check all that apply.

a) 5001    b) 50.00  c) 5000  d) 0.5000  e) 0.0005000    


2. How many significant digits are there in each of the following numbers?
50000 

  uncertain     1    2    3     4    5 

5.00 x 104

  uncertain     1    2    3     4    5 

5 x 104

  uncertain     1    2    3     4    5 

5.0000 x 104

  uncertain     1    2    3     4    5 

5005

  uncertain     1    2    3     4    5 

Remember, the idea of significant digits applies only to experimental values, and the calculations done with them.