  ## When Is a Zero Significant? Since we use zeros as place holders at the end of a number to denote insignificant digits, some confusion can arise if a significant digit has a value of zero. A zero is obviously more than a place holder if it doesn't fall at the end of a number. For example, the zero in 405 is significant. Thus,

 If a zero is found between significant digits, it is significant.

Zeros can be used as (insignificant) place holders to the left of significant digits if the number is a decimal. For example, a mass of 42 g has two significant digits. Expressed in kilograms, the mass of 0.042kg should still have two significant digits. The zeros to the left of the 4 are place holders and not significant:

 Zeros located to the left of the first non-zero digit are not significant.

If the last significant figure(s) is (are) zero, life gets a whole lot more complicated. Suppose I used a metric ruler with millimeter markings to measure the width of a skateboard. The skateboard, a precision model, measures exactly 20 centimeters, and I report the width as 20cm 0.02mm. The uncertainty in this number is in the fourth digit (the hundredth's place), so it has 4 significant digits, not just 1. If I write out the 4 sig-digs, the width is 20.00cm. Since the zeros to the right of the decimal place are not necessary as place-holders, their inclusion indicates they are significant.

 Zeros to the right of the decimal place that are NOT merely place holders are significant; these significant zeros will be to the right of non-zero significant digits.

The zero to the left of the decimal place in 20.00cm is also significant since it now falls between significant digits.

If the last significant digit is zero and to the left of the decimal place, we have a problem. Our standard way of writing numbers has no real way of differentiating between 4500 to 2 significant digits, and 4500 to 3 or 4 significant digits.  Let's go back to and look at the earlier example when you counted football fans.  You might want a better count than your rough estimate, so you could decide to count filled rows in the stadium.   Each row holds 10 people, and most rows were packed.  When rows weren't completely filled, you could combine them and find the equivalent number of filled rows.  When you were done, you found 450 filled rows.  Your counting might have overlooked a few empty seats, but you are fairly sure the number of people in the stadium is between 4490 and 4510.  So when you report 4500 people attended the game, you really have three significant figures.  If you had counted individual people as they entered the stadium and found that exactly 4500 people attended, you might also report the attendance as 4500.  We need a way to distinguish these three cases apart.

Enter scientific notation. Scientific notation, sometimes called exponential notation, allows us to express all significant zeros to the right of the decimal place by multiplying by factors of ten. For example, we could write the attendance counted at the gate as 4.500 x 102. This has four significant digits, as it should. For the row-counting scenario with three significant digits, the attendance is 4.50 x 102.  And the rough estimate could be reported as either 4.5 x 102 or as 4500.

Scientific notation will be used to indicate significant zeros, so zeros at the end of a number and to the left of the decimal place will be considered insignificant place holders. This convention, however, is not uniformly followed, so you must be careful using it.

 Conventions about zeros at the end of a number and to the left of the decimal place vary. In these modules, such zeros will always be considered insignificant. You should, however, look in your textbook to determine that author's convention.  