Figure 1 measurement of a piece of paper
To understand the physical world, physicists attempt to find relationships between quantities. For example, how does the stopping distance of a car depend upon its initial speed? Or, how does the length of time it takes a kettle of water to boil depend on the amount of water in the kettle? Usually these relationships are expressed as mathematical equations. To verify a relationship, measurements must be made of the relevant quantities.
Of course, it is important to make measurements as precise as possible. However, no measurment is ever absolutely precise. Every measurement has some uncertainty associated with it. (We are not talking here about outright mistakes in measuring, for example, misreading the scale of a ruler.) For example, the precision of a measurement is limited by the ability (or inability) to read between the smallest divisions.
Let's say you are measuring the distance between two lines on a piece of paper. (Figure 1) The ruler is accurate to around 0.1 cm ( or, 1 mm), which is the distance between the smallest divisions. The measured distance has an uncertainty of about 0.1 cm, since it's difficult to interpolate between the lines on the ruler. The distance between the lines could be stated as 7.7 ± 0.1 cm (or, 77±1 mm).
Figure 1 measurement of a piece of paper
The distance between the smallest divisions, in this case 0.1 cm, is sometimes called the precision of the measurement. When using this particular ruler, it doesn't matter how wide the piece of paper is, the precision is still 0.1 cm because that is the smallest unit on the ruler. The precision of many bathroom scales is 1 pound, although digital scales may be precise to 0.1 pound. If you have a thermometer (outdoor, oven, candy, etc) can you determine its precision?
Often, the uncertainty in a measurement isn't stated but is implied by the way the number is written. The measurement 7.7 cm implies that the ruler had a smallest division of 0.1 cm so the uncertainty is about 0.1cm. The odometer of my car reads 23568 miles, so the uncertainty is about one mile (the smallest unit in the measurement).
Suppose I need to know the width of the paper in Figure 1 with more precision. I could use a vernier caliper and find the width to the nearest 0.01 cm ( or tenth of a cm). I do that and find the width is 7.72 cm. Now, I think I could get a better idea of the true width if I averaged the two measurements together. The first step in finding the average is to add the two measurements: 7.7 cm + 7.72 cm. Is the answer 15.42 cm? NO!!
The problem is that the first measurement wasn't precise enough to provide a digit in the 0.01 cm place. That is, the number is 7.7? where the ? indicates I have no idea what the digit should be (my ruler wasn't good enough). What do you get when you add ? to 2? I don't know!
THE RULE: When adding measurements of different precisions, the sum cannot be any more precise than the least precise measurement.
EXAMPLE: My odometer reads 23123 miles and I set the trip meter to 0. After I drive to school the trip meter reads 4.2 miles. What does the odometer read?23123 miles + 4.2 miles = 23127 miles (the answer is rounded to the nearest mile)
While this may not look like "correct arithmetic", it is correct if the numbers being added (or subtracted) are measurements. To state an answer to excess precision is to pretend to know the sum of numbers you never measured!