Significant Figures tell us about the precision of a measurement, instead of its magnitude.
When you make a scientific measurement, you HAVE to report your data using the correct
number of significant figures - which gives an indication of
how precise your measurement was.
You can’t write down some data that is more precise than the tools you use.
And you can’t go on to do calculations and then report an answer that is more precise
than the data you started with.
That would be misleading.
We’ll start with some quick examples of significant figures,
and then explain the rules.
21 has 2 sig figs - all nonzero numbers are significant.
2001: 4 sig figs - all 4 of these digits are significant,
including the zeroes in between non-zero numbers.
2.1 x 10^3 2 sig figs - in scientific notation,
the digits of the coefficient on the left are significant.
Trivia: this part of the number is called the “significand” ...and it’s significant!
2.100 x 10^3 4 sig figs - again, in scientific notation,
all the digits of the coefficient on the left are significant.
Where did these ideas of significance come from?
From measurement.
If you have a thermometer with each degree marked, you can easily read it
to the nearest degree.
But you can also estimate the next digit.
We are estimating the temp to the nearest 10th of a degree.
Here, it’s between 37 and 38.
It’s closer to 37.
I would read this as about 37.2 degrees C.
This estimated temperature has 3 digits.
The first 2 digits are known with certainty, while the last digit has been estimated, and
so it has some uncertainty.
But all 3 digits are considered “significant figures.”
The significant figures in a scientific measurement include all the digits that you know for certain,
plus a last digit that is estimated.
Similarly, when you weigh a sample on a balance capable of measuring to the nearest
0.01 g, you can report the mass as 345.22 plus or minus 0.01 g The plus or minus notation
expresses the uncertainty of the measurement.
Now let’s turn to the rules for significant figures, and we’ll apply each one.
RULE 1: Non-zero digits are ALWAYS significant.
In the number 459,494,832, there are 9 digits.
All of them are non-zero, so according to rule 1, all 9 digits are significant.
RULE 2: A zero is only significant if it is At the right end of a number
AND after a decimal.
74.0 has 3 significant figures.
Between digits that are significant according to rules 1 and 2.
So in the example above, 2001, the zeros in the middle of the number are significant.
Zeros to the left of nonzero digits or at the end of a quantity written as a whole number
are “placeholders” and are not significant.
For example, 0.128 has 3 significant figures.
That first zero is just a placeholder.
Let’s see some more examples of Rule 2.
596,494,832.0 has 10 significant figures.
If the number was 590,494,832, there are 9 significant figures.
Same with 90,464,830.0
Remember, as long as the zero is between two
significant figures, it counts as a significant figure.
But in 0.04463, there are 4 significant figures.
The zeros to the left of the nonzero digits are not significant.
RULE 3: If a quantity is known to be exact, it has an unlimited number
of significant figures.
If you count 12 eggs in that box of eggs you just bought,
there is no uncertainty about this measurement.
Counting individual items does not include an uncertain digit at the end.
It’s sort of like 12.00000000000000 and continue with an infinite number of zeroes.
This is important in calculations, because these values will NOT limit the number of
sig figs in your answer.
RULE 4: When a number ends in zeros but contains no decimal point,
the zeros may or may not be significant.
This is SO confusing and it’s a real problem when you’re recording data.
BUT!
This is a real benefit of writing data in scientific notation.
If a quantity is written in scientific notation, all the digits of the coefficient are significant.
It can be argued that a number like 43000 may have 2, 3, 4, or 5 significant figures.
It depends on how many of those zeroes at the end were measured.
If you are trying to say that you actually measured the 43000, you might try to denote
it with a decimal place at the end, like this: 43000.
Or sometimes you’ll see people put a line above the digits that were actually measured.
This is really hard to do if you’re typing, however.
There just isn’t a real standard for indicating this kind of measurement ending in a zero,
so the best solution is to put your data in scientific notation.
That way, if you want to show that you have 2 significant figures, you write like this:
4.3 x 10^4 If you want to show you have 3 significant
figures in your measurement, you write it like this:
4.30 x 10^4 And so on:
4.300 x 10^4 4 sig figs 4.3000 x 10^4 5 sig figs
Let’s next discuss how to deal with significant figures when you do calculations.
At the very END of a calculation involving significant figures, you will round up to
the final significant digit, only if the next digit is 5 or higher.
So let’s look at the rules for rounding after
1)Adding or Subtracting OR 2) Multiplying or Dividing.
When you ADD or SUBTRACT, you round the result to the same number of DECIMAL PLACES as the
measurement with the fewest decimal places.
For example: 4.96 + 1.3 = 6.26,
but you need to round to 1 decimal place (the same as in 1.3).
So your answer is 6.3 (rounding up, because 6 is greater than or equal to 5).
When you MULTIPLY or DIVIDE, you round the result to the same number of SIGNIFICANT FIGURES
as the measurement with the fewest significant figures.
For example, if you were asked to find the area of this rectangle:
2.1m x 2.336m = 4.9056m^2 Rounded to 2 sig figs, because there are 2
sig figs in 2.1, you get 4.9 m^2. This kind of raises the question - why are there more
sig figs in one side of the rectangle than the other - it implies you used different
kinds of measuring sticks to get that value, and these measuring sticks had different precision.
Here’s a special case that only applies when you do calculations with logs. In that
case, the number of digits after the decimal point = the number of sig figs in the original
number. For example, 81.5 is a measured quantity. Then log(81.5) = 1.911
(3 sig figs after the decimal point).
Here’s another piece of Trivia: The decimal part of a logarithm is called the mantissa.
The digits in the mantissa are the only significant figures in a logarithm. The number in front
of the decimal place indicates order of magnitude, and is not significant.
The opposite of taking the log of a number is to raise 10 to the power of that number.
As you might expect,the sig fig rule for this function is the opposite of the sig fig rule
for logs. The number of digits in the mantissa gives the number of significant figures in
the resulting number. So let’s do the opposite of our previous
example. We’ll raise 10 to the power of 1.911, and we should get 81.5 back. We get
81.470, and when we round to 3 sig figs, we get 81.5
I have a useful tip for doing multiple calculations: When a calculation involves 2 or more steps
and you are writing down answers to intermediate steps, keep at least one additional digit
past the # of sig figs. If you’re using a calculator, this doesn’t really matter,
because you just re-use your intermediate answers without rounding, but if you’re
working things out on paper, and writing down the rounded answers to the intermediate steps,
this way you can still round off at the end to the right number of significant figures
without losing any significant data.
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