You can't make imprecise numbers more precise just by combining them with ones that already are. That's why rules exist for mathematical operations with numbers of different precision, and these rules are based on significant digits. However, the rule for addition and subtraction is not the same as for multiplication and division. Also, the rule for addition and subtraction is sometimes easier to understand in terms of decimal places.
Addition and Subtraction
Suppose you have two scales. One reads in increments of 0.1 g, and the other in increments of 0.001 g. If you measure 2.3 g of salt on the first scale, and combine this with 0.011 grams of salt weighed on the second scale, what is the combined mass? Well, it depends on which scale you weigh it on. On the first scale it still comes in at 2.3 g, but on the second it could be 2.311 or 2.298 or 2.342. If all you know are the two original masses, then you can only assume a precision of 0.1 g. So, the precision of the final result is determined by the least number of decimal places in the two numbers, and you round to that number of decimal places. In this case, 2.3 + 0.011 → 2.3. Other examples: 100.19 + 1 → 101, 100.49 + 1 → 101, 100.51 + 1 → 102, and 0.034 + 0.0154 → 0.050. The trailing zero is because we maintain precision to three decimal places. However, 0.0340 + 0.0154 → 0.0494. We keep four decimal places because the 0 after the four in -.0340 is significant.
About the Author
Ariel Balter started out writing, editing and typesetting, changed gears for a stint in the building trades, then returned to school and earned a PhD in physics. Since that time, Balter has been a professional scientist and teacher. He has a vast area of expertise including cooking, organic gardening, green living, green building trades and many areas of science and technology.