Significant Figures & Scientific Notation
One of the most baffling subjects for students is frequently significant figures. The reason for
this is simple: Nobody ever seems to know what they’re supposed to be used for. Why should I
care if “100” has one significant figure or “100.0” has four?
Fortunately, your nice teacher Mr. King is here to help. Let’s take a look at the joyous
excitement produced by significant figures:
Why do we need significant figures?
Significant figures are important because they tell us how good the data we are using are. For
example, let’s consider the following three numbers:
100 grams
100. grams
100.00 grams
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•
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The first number has only one significant figure (namely, the “1” in the beginning). Because
this digit is in the “hundreds” place, this measurement is only accurate to the nearest 100
grams (i.e. the value of what we’re measuring is closer to 100 grams than it is to 200 grams
or 0 grams).
The second number has three significant figures (the decimal makes all three digits
significant, as we’ll discuss later). Because the last significant figure is in the “ones” place,
the measurement is accurate to the nearest gram (i.e. the value of what we’re measuring is
closer to 100 grams than it is to 101 grams or 99 grams).
The third number has five significant figures (as we’ll talk about later). Because the last
significant figure is in the “hundredths” place, the measurement can be considered to be
accurate to the nearest 0.01 grams (i.e. the value of what we’re measuring is closer to
100.00 grams than it is to 100.01 or 99.99 grams).
In short, when you plug these three numbers into your calculator, there’s no difference in how
the calculator will manipulate them – your calculator neither knows nor cares about how good
the numbers it’s working with are. However, to you, the taker of data, these three numbers tell
you whether or not your data is good enough to pay attention to.
How do we find the correct number of significant figures?
Now that you know why significant figures are important, the following rules will help you
determine how many significant figures are present in your data.
Rule 1: Non-zero integers are ALWAYS significant.
Example: 1, 2, 3, 4, 5, 6, 7, 8, 9
Next we’ll talk about zeros. They can be confusing, so stay with me…
Rule 2: Leading zeros precede all non-zero digits and NEVER count as significant figures.
Example: 0.00059 contains 2 significant figures
Adapted from: http://misterguch.brinkster.net/sigfigs.html
Rule 3: Captive zeros are any zeros between two non-zero digits and ALWAYS count as
significant figures.
Example: 508 has 3 significant figures
7.004 has 4 significant figures
Rule 4: Trailing zeros are zeros found at the end of the number. They are significant only if the
number is written with a decimal point.
Example: 100 has 1 significant figure
100. has 3 significant figures
0.0050 has 2 significant figures
What exactly is Scientific Notation then?
Scientific work often involves using numbers that are either very large or very small. For
example, the speed of light is approximately 30,000,000,000 cm/s, the mass of the Earth is
approximately 6,000,000,000,000,000,000,000,000,000 g. The mass of an electron is
0.000000000000000000000000000910953 g. As you can see, these numbers are tough to
write, even tougher to read and cumbersome to use in calculations.
Scientific notation is a way to express these numbers in a simpler, friendlier manner. To
accomplish this, scientific notation makes use of the power of 10. Scientific notation is written in
the form:
M x 10n
In this form, “M” is any number between 1 and 10 (only 1 digit left of the decimal point) and “n”
is a positive or negative number. If the number you are converting to scientific notation is
greater than 1, then “n” is a positive number. If the number being converted is less than 1, then
“n” is a negative number.
To convert a number to scientific notation, simply move the decimal point until you have 1 digit
to the left of the decimal – that is your “M” number.
Next, count the number of spaces you moved the decimal – that is your “n” number.
For example: 15,000 = 1.5 x 104
0.0097 = 9.7 x 10-3
This brings up another significant figures rule…
Rule 5: When you write numbers in scientific notation, only the part before the “x” is counted in
the significant figures. So the number 2.39 x 104 has 3 significant figures because we only
worry about the “2.39” part.
Example: 1 x 102 has 1 significant figure
1.00 x 102 has 3 significant figures
5.0 x 102 has 2 significant figures
Keep in mind that you must keep the same number of significant figures in your scientific
notation number as in the original number.
Example: 345,000,000,000 has 3 significant figures so it must be converted to a number
with 3 significant figures – 3.45 x 1011
561,090,000,000 has 5 significant figures so it must be converted to a number
with 5 significant figures – 5.6109 x 1011
Adapted from: http://misterguch.brinkster.net/sigfigs.html