1.2 Significant Figures and Scientific Notation
Significant Figures
Since many of the sciences are done in imperfect conditions, there needs to be a way to determine the best way to report experimental numbers. This is done with Significant Figures (sig-figs) or Significant Digits, a way of determining how to report measurements from devices, as well as how to perform calculations with these rounded numbers. Depending on the type of calculation, the amount at which you can “trust” or accept a value, is imperative to the sig-fig system.
A good example to better understand significant figures is with cutting bread. Say you have 1 loaf of bread and need to cut it in half. If you have ever cut a loaf of bread before, you may have realized that there is often a lot of crumbs left behind. If we bought a loaf marketed at 500 g and cut it in half we may have 2 halves one weighing 245.7321 g and another 252.4834 g. Together our new total of bread is 245.7321 g + 252.4834 g = 498.2155 g. Clearly 498.2155 g does not equal 500 g, and in this scenario we rounded to 4 decimal places, but maybe someone else would have rounded to 1 decimal place and have gotten 245.7 g + 252.5 g = 498.2 g which is a different number than before. And perhaps someone else decided to round to the whole gram and get 246 g + 252 g = 498 g. So now we are all getting different weights for our bread, how can we know who is right?
This is where significant figures come into play. A set of rules to determine exactly how to round and perform calculations to get the same answer as everyone else.
Defining Significant Digits
Firstly, we need to know WHAT counts as a significant digit.
In this table below, the digits in blue and underlined are digits which apply to the specific rule
The digits in pink are significant digits, but do not apply to the specific rule
The digits in black are not significant digits
| Non-zero digits are significant | 43 92.31 |
| Zeros between two significant digits are significant | 102.3 200304.08 |
| Zeros to the left of the first significant digit are not significant | 0.092 0.000430633 |
| Decimal trailing zeros are significant digit are significant (if they are within the precision of measurement) | Rounded to 1 digit 3.000 Rounded to 2 digits 3.000 Rounded to 3 digits 3.000 |
| Integer trailing zeros can be significant (if they are within the precision of measurement, it is best to use scientific notation to avoid ambiguity) | Rounded to 1 digit 4000 Rounded to 2 digits 4000 Rounded to 3 digits 4000 |
| Exact numbers and constants have infinite significant digits | $\pi$ 6 cars |
As you can see here, there are specific rules on defining what is and is not significant. It takes some practice to get a hang of it! Practice some problems in the practice section.
Determining Significant Figures
When reading measurements you will report to the precision of the tool you are using plus one estimated digit
In Image 1, the ruler reports to half inches, however, the red line which we are measuring, is between 4.5 inches and 5 inches. So we are able to report to the half inch + one digit of estimation. For this red line we could say it measures to 4.56 inches.
Even here, in image 2, it seems like the delineations of the ruler are too small to make a reasonable guess for an additional significant figure, but you should still try to assume as best as possible. This could be read as anything between 12.41 cm and 12.49 cm.
Rounding to Significant Figures