SIGNIFICANT
FIGURES....Who cares?
Now
that youŐre done wasting time checking little bubbles, onward ho!
One of the most often overlooked concepts in
science classes is that of significant figures. They are relatively easy, but
are easily forgotten. Students usually get so passionate about and involved in
a problem that they neglect to use them. Either that, or they never really learned
why it was so important to use them in the first place.
Confused
yet? DonŐt ask DAY (heŐll probably just confuse you more), just pay attention
here.
According
to http://en.wikipedia.org/wiki/Significant_figures,
ŇThe
hypothetical idea of significant figures (sig figs or sf), also called
significant digits (sig digs) is a method of expressing error in measurement.
Sometimes
the term is used to describe some rules-of-thumb, known as significance
arithmetic, which attempt to indicate the propagation of errors in a scientific
experiment or in statistics when perfect accuracy is not attainable or not
required. Scientific notation is often used when expressing the significant
figures in a number.
The
concept of significant figures originated from measuring a value and then
estimating one degree below the limit of the reading; for example, if an
object, measured with a ruler marked in millimeters, is known to be between six
and seven millimeters and can be seen to be approximately 2/3 of the way
between them, an acceptable measurement for it could be 6.6 mm or 6.7 mm, but
not 6.666666 mm as a recurring decimal. This rule is based upon the principle
of not implying more precision than can be justified when measurements are
taken in this manner.Ó
What
an excellent definition!
Sig figs ultimately help determine the
legitimacy of a number. Precision is more of a concern here than accuracy. The
calculator cannot be depended on to do all of the work regarding sig figs. It
doesnŐt care how many numbers it gives you, so you donŐt know if the teacher
wants 3.14159265358979323846264338327950288419716939937510582 or 3.14159265 or
3.14 or simply 3 as a value for pi; itŐs a good idea to ask!
Rounding also comes into play when using sig
figs; actually, itŐs a main part of the idea. Generally, if the last digit of a
number is 6 or greater, you round up (e.g. 3.14159 becomes 3.1416). If the last
digit of a number is less than 5, you round down (e.g. 3.141 becomes 3.14).
However, if the last digit of a number is 5, some more complex rules come into
play. Since you canŐt settle on a rule to either round 5 up or down (though
most people round up) because that will severely alter your data, it is common
practice to round up half of the time and round down half of the time. Other
sources say to round up if the number before the 5 is odd, and let it be (not
the song) if the number before the 5 is even. This may cause some problems if
you do not have many calculations to do, in which case you should probably
round at the very end of a problem. But if you have many calculations to do,
rounding up half of the time and down half of the time will, statistically,
make your answer more valid.
Significant figures arenŐt very important in everyday measurements. For
example, the speedometer in Figure 1 measures speed in miles per hour and
kilometers per hour. However, it is only precise to a small number of sig figs:
1. It is impossible to say, for instance, that you are going 40.5 miles per
hour or even 41. In this case, you would have to round to the nearest 5 mile
per hour mark.
Figure 1: A speedometer
Another
instance in which significant figures are used is when you take your temperature.
The thermometer (see Figure 2) could say 99ˇF, but what if it is 99.1ˇ, or
98.9ˇ? If it was truly 99ˇ, it should say 99.0ˇ, should it not? 99ˇ means that
the temperature could actually be plus or minus 1ˇ of 99ˇ. Luckily, in these
cases, a more precise number isnŐt usually needed.
Figure
2: A thermometer
Now
that youŐre catching the hang of it, hereŐs some practice on significant
figures.
How
many sig figs do these numbers have? Type your answers in the boxes.
Done?
Ok, so you didnŐt really have to type them in. Just for fun.
Answers:
1)
1
2)
2
3)
3
4)
4
5)
5
6)
6
One
thing that may have tripped you up were the decimals. Maybe you just need the
rules!
Briefly
(you can look up the longer-winded rules in the links at the bottom of the
page):
á
All non-zero numbers ARE significant (3492 has 4 sig figs).
á
Any zeros that are between non-zeros are also considered
significant (3,400,008 has 7 sig figs).
á
Any zeros that follow immediately to the right of the decimal
place in numbers smaller than one ARE NOT considered significant (0.000000348
has 3 sig figs).
á
Trailing zero digits that fall to the left of the decimal place
in a number with no digits provided that fall to the right of the decimal place
is less clear, but these are typically not considered significant unless the
decimal point is placed at the end of the number to indicate otherwise (9000
has 1 sig fig, while 9000. has 4).
á
Any zeros that follow the last non-zero digit to the right of
the decimal point are significant, e.g.: 0.002400 has four significant figures.
(0.0000345.00 has 5 sig figs)
á
A number with value 0 is usually considered to have one
significant figure.
A good tool to use is the SciTools Application for the TI-83,
TI-83+, TI-83+ SE, and TI-84.
100
has 1 sig fig:
These
are just a few rules, but itŐs a good start.
In calculations, the number of sig figs you
use in your answer relates to how many were used in the question. If you are
given 4 numbers in a problem: 4.35 m/s, 10. kg, the speed of light (c), and 1
J/m/s*slug/cm^2, your answer would have just 1 sig fig in it; always go with
the least number of sig figs given in the problem. The speed of lightŐs Ôsig
figgynessŐ is irrelevant since it is a constant. The same would go for gravity,
etc.
A thought: ŇAlthough the idea of significant
digits attempts to deal with the real problem of expressing measurement and
calculation error, the system itself leads to further (and unnecessary) error
in expressing a measurement or calculation. The basis of significant figures is
that of rounding, and rounding in itself reduces the accuracy of the
measurement (this is because rounding is a technique that fundamentally uses
addition or subtraction - thus creating artificial error by the amount added or
subtracted during rounding)Ó (again,
http://en.wikipedia.org/wiki/Significant_figures).
In
conclusion, be careful, and donŐt forget!
Additional
Information and Resources:
http://www.fayar.net/east/teacher.web/Math/young/Physics/SigFigs.doc
http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html
http://ostermiller.org/calc/sigfig.html
http://soulcatcher.chem.umass.edu/chemland/SigFig.html
http://www.ric.edu/bgilbert/s2sigfig.htm
http://www.foundationcoalition.org/resources/first-year/tamu/course-materials/sig-figs.html
http://science.widener.edu/svb/tutorial/sigfigures.html
http://www.staff.vu.edu.au/mcaonline/units/numbers/numsig.html
http://library.thinkquest.org/3310/higraphics/textbook/u01sigfg.html
http://en.wikipedia.org/wiki/Rounding
http://www.towson.edu/~ladon/sigfigs.html
http://lectureonline.cl.msu.edu/~mmp/applist/sigfig/sig.htm
And
if you get completely fed up with all of this, try:
But
not during class. . .
(Artwork
not by SI or DAY)
Contact
information:
Page
designer—Stephen Ironside, 3rd Period AP Physics C, ©2005. s_ironside@fayar.net
Advisor,
teacher, cool guy—David A. Young, Fayetteville High School. dyoung@fayar.net or dyoung7@prodigy.net.
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Best viewed in Camino 0.8.4, Safari 2.0.2, or Internet Explorer. Anything
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*Get
it? Webdings arenŐt significant at all.