SIGNIFICANT DIGITS
Significant digits are any measured digits. Since all measuring instruments are calibrated to a certain decimal
place, the last digit in all measurements should be an estimated digit. For example, if the numbers on the ruler in
the diagram below is in cm, we know for sure the length of the piece of wood is something between 34.4 cm and
34.5 cm. The reported measurement should be 34.45 cm since the wood looks to be halfway in between the two
marks.
Furthermore, since the last digit in a measurement is assumed to be estimated, the precision of the measurement
depends on the decimal place of that estimated digit. Consequently, any calculations involving measurements
should not produce answers that are “better” than the measurements taken. Thus, you must be able to determine
which digits of a measurement are significant. For example, the length of the wood piece above could be reported
in any of the following manners:
34.45 cm
344.5 mm
0.03445 m
344500 μm
The number of significant digits does not change in any of these quantities—they all have 4 significant digits.
There are 3 basic rules in determining which digits in a measurement are significant:
1) All digits are considered to be significant except if the next two rules apply.
2) Trailing zeroes (zeroes at the end of a number) without a written decimal point are NOT significant.
a. 344500 μm → the two zeroes are NOT significant since there is no decimal point written.
b. 300. cm → both zeroes in this case ARE significant because there is a decimal written.
c. 10.000 mg → all four zeroes ARE significant because there is a decimal written.
3) Leading zeroes (zeroes at the beginning of a number less than 1) ARE NEVER significant.
Some examples:
0.003230 kg → 4 significant digits (the trailing zero is significant because the decimal is written)
20000 g → 1 significant digit (the trailing zeroes are not significant because the decimal is not written)
20000. g → 5 significant digits (trailing zeroes are significant because of the decimal)
2020.1040 mm → 8 significant digits
2.250 x 1019 km → 4 significant digits (only significant digits go into the first part of a number written in
scientific notation)
MULTIPLICATION / DIVISION CALCULATIONS
When multiplying or dividing, the answer must have the same number of significant digits as the quantity in the
calculation that has the least number of significant digits.
23.3 g
 9.70833333
2.5 mL
g
mL
Since 23.3 g has three significant digits and 2.5 has only two, the answer can only have two significant digits. So
you will round off the answer to achieve this. Therefore the reported answer in this case is:
g
9.7 mL
Suppose you have the following calculation:
65.000
mi
h
 1.609 km  1000 m  1 h



 1 mi  1 km  3600




  29.0513889
s 
m
s
This calculations brings up another aspect to consider—some quantities are exact numbers when doing conversions.
This means that the quantity has an infinite number of significant digits. The exact numbers in the above
calculation are:
1 mi, 1 km, 1 h → when in a conversion factor, any quantity of 1 is considered to be exact
1000 m, 3600 s → when a conversion factor involves two quantities of the same system, the quantities in
the conversion factor are considered to be exact (think about it, there are exactly
1000 m in 1 km and 3600 s in 1 hour)
We do NOT count exact numbers when determining the number of significant digits in a measurement. Therefore,
the only two quantities we will consider in this problem are:
65.000 mi
(5 significant digits)
h
1.609 km (4 significant digits)
Since 1.609 km has lesser number of significant digits, the answer must have the same number of significant digits
(4). Therefore, the reported answer in this case is:
29.05 ms
ADDITION / SUBTRACTION CALCULATIONS
When adding and/or subtracting measured quantities, we DO NOT count the number of significant digits in them.
Instead, we look at the precision of the measurements. As mentioned before, the precision of the measurement
depends on the decimal place of the estimated digit. So…we find the quantity that has its last significant digit in
the largest decimal place of any quantities in the problem. In other words, find the quantity whose estimated digit
is in the decimal place furthest to the left.
234.40 m
+ 34
m
268.40 m
Since 34 m has its last significant digit furthest to the left (the larger decimal place), we will round off the answer
at this decimal place.
234.40 m
+ 34
m
268.40 m
So the reported answer here is 268 m.
Example:
420 cm + 15.2 cm + 0.125 cm = 435.325 cm
reported answer = 440 cm
When doing calculations involving a combination of addition/subtraction and multiplication/division, you follow the
order of operation and determine the number of significant digits following each step.
 32.4 cm2  22 cm2


25.25 s

  (10.00 cm)(22.00 cm)   54 cm2   (10.00 cm)(22.00 cm) 
2



  2.1 cms  0.516



 


426 s
426 s
  25.25 s  

 
cm2
s
 2.6
cm2
s