Numbers without Uncertainty
Now, after all this discussion about uncertainty, it may seem odd but there actually are numbers that don’t have any uncertainty whatsoever. Roll a die, for example, and get a five. That is a data point without any uncertainty. Another example is the conversion factor between centimeters and inches: there are exactly 2.54 cm/in. This number is an exact definition with no uncertainty. In the language of significant figures, this number has an infinite number of sig figs. We will encounter other such values in the lecture part of this course. One particularly famous example is the speed of light. The speed of light is exactly 299 792 458 m/s: an infinite number of significant figures and no uncertainty because this is actually how the meter is defined.
Those examples were a little trivial. Now, let’s consider something a little more complex. Consider the following chemistry problem:
Balance the following chemical equation (i.e. solve for a, b, x, and y):
a C8H18 + b O2 → x CO2 + y H2O
You of course, give the correct answer:
a = 2, b = 25, x = 16, y = 18
a real reaction, however, will never have these ratios. You will never get 16 moles of CO2 from 2 moles of C8H18 and 25 moles of O2. In real reactions there are non-ideal effects: you will get CO and even C60. If the source of the oxygen is air, you will also get nitrogen oxides NOX. However for most applications you can ignore these effects and treat these numbers (2, 25, 16, and 18) as having zero uncertainty and that’s good enough.
- There are values without uncertainty (the speed of light).
- Sometimes, we treat a value as exact if the uncertainty is too small to care about.