"Man is nothing but a subject full of natural error that cannot be eradicated except through grace." - Blaise Pascal
Every single measurement you make, is tainted by error; no matter how nimble you are with your measuring instrument, the is some uncertainty associated with your measurement.
Imagine if you had to weigh an empty flask for a calorimetry experiment; you set the flask on the triple-beam scale and obtain the following mass as seen below:
How would you record that mass in your laboratory notebook? You can see from the enlarged image that the flask must weigh about 96 or so grams, but how many exactly? Can you estimate the value of the mass with a higher resolution? Note that the scale's beam in front represents grams and the lines in between each number represent a tenth of a gram. Knowing this, we can with some certainty state the beaker weighs between 96.0 and 96.1 grams; the last digit - within the hundredths place - is the one we will have to estimate, the one that is uncertain. We would record in our laboratory notebook the flask weighs about 96.08 grams.
For anyone reading your laboratory notebook, they would know from reading the mass, that you must have used a scale with graduated lines up to the tenth of a gram, and you by necessity, had to estimate the hundredths place.
If you were working in a laboratory at a research university - chances are you would use scales that are much more expensive; a commonly used analytical balance can reflect resolutions up to one thousandth of a gram. It is assumed the thousandths place value is estimated. If you worked in such a laboratory, you might record the mass of the same flask at 96.077 grams.
Uncertain or estimated figures follow not only mass measurements, but all measurements. Consider the temperature reading from the thermometer at right: with certainty, we know that it reads somewhere between 19 and 20 degrees Celsius. To report the temperature reflecting the resolution of the thermometer, we are going to have to estimate the last digit in tenths of a degree Celsius. Since the red line is so close to 20, we might state the thermometer reads 19.9 degrees Celsius.
The main function of significant digits then, is to reflect to any readers of our work, the degree of resolution of our instruments, or the degree of error within our measurements.
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