In the Classroom
The Box-and-Dot Method:
A Simple Strategy for Counting Significant Figures
W. Kirk Stephenson
Division of Science and Mathematics, Angelina College, Lufkin, TX 75901; [email protected]
When test time arrives many students struggle to correctly
assign significant figures to numbers. They confuse (or forget
altogether) simple rules previously memorized, making the rules
difficult or impossible to apply under pressure situations.
This situation is largely remedied with the use of a visual
approach I refer to as the box-and-dot method. The method uses
the device of “boxing” significant figures based on two simple
rules, then counting the number of digits in the box(es).
The three steps of the box-and-dot method are detailed
below; additional examples show the method’s ease of use.
Box-and-Dot Basics
The box-and-dot method consists of three simple steps for
determining the significant figures in any real number (except
zero). The steps must be followed explicitly.
Step 1
Draw a box around all nonzero digits, beginning with the
leftmost nonzero digit and ending with the rightmost nonzero
digit in the number.
the expression 0.0|1230123||00| reveals nine digits surrounded
by boxes. Therefore, there are nine significant figures. Specifically, the significant figures are: 1, 2, 3, 0, 1, 2, 3, 0, and 0.
Note that the box-and-dot method does not expressly address leading zeros.2 There is no need. Leading zeros, which are
never significant, always lie to the left of the box drawn in Step
1 and are therefore excluded from consideration as significant
figures by virtue of the explicitness of Steps 1 and 2. Those steps
provide criteria for boxing (and thus rendering significant)
trapped and trailing zeros only, to the exclusion of all other
zeros.
In general, at least one, but not more than two, boxes will be
associated with any given number. Step 1 always requires that a
box be drawn. Step 2 only allows for a (second) box to be drawn
when two conditions are met: (i) a decimal point is present, and
(ii) one or more trailing zeros are present.
Typically, none of the steps requires more than a few seconds to complete.
Worked Examples of the Box-and-Dot Method
For example, drawing a box around the nonzero digits in
the number 0.0123012300 gives 0.0|1230123|00. Any zero(s)
trapped or “sandwiched” between nonzero digits will necessarily
be included in the box.
For convenience, a digit or number surrounded by a box
may be referred to as a “boxed” digit or “boxed” number, respectively.
Several examples follow, in the form of sample questions.
For convenience, each of the three steps discussed above is
represented with a chevron (»). The number prior to the first
chevron in each example is the number for which significant
figures are to be determined. To demonstrate the ease of use and
visual nature of the box-and-dot method, little or no explanation
is provided. Deriving the correct answer becomes second nature
after working only a few problems.
Step 2
Question 1
If a dot is present, draw a box around any trailing zeros.
Continuing with the above example, a dot (decimal point)
is present in the expression 0.0|1230123|00; therefore, trailing
zeros1 are boxed, which gives 0.0|1230123||00|.
Step 2 uses the term dot in lieu of decimal point for reasons
of brevity and ease of recall. “Box-and-decimal point method”
possesses neither the pith nor lilt of “box-and-dot”.
The position of a decimal point within a number is
irrelevant—the only test for boxing trailing zeros is the mere
presence of a dot.
Because method steps are followed explicitly, it is understood that trailing zeros should not be boxed when a dot is not
present. In other words, draw a box around trailing zeros if and
only if a dot is present.
Step 3
Consider any and all boxed digits significant.
Boxed digits are significant, whereas digits that are not
boxed are not significant. Continuing the example from Step 2,
How many significant figures are in the number 123.01230?
Answer: 123.01230 » |123.0123|0 (nonzero digits are
boxed) » |123.0123||0| (dot present, so the trailing zero is
boxed) » eight numbers are boxed, therefore the number has
eight significant figures.
Question 2
How many significant figures are in the number 12301230?
Answer: 12301230 » |1230123|0 » no dot is present (so
the trailing zero is not boxed) » seven significant figures.
Question 3
How many significant figures are in the number 123.0123?
Answer: 123.0123 » |123.0123| » no trailing zeros » seven
significant figures.
Question 4
How many significant figures are in the number 10100?
Answer: 10100 » |101|00 » no dot (so no trailing box) »
three significant figures.
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 86 No. 8 August 2009 • Journal of Chemical Education
933
In the Classroom
Question 5
How many significant figures are in the number 10000.0?
Answer: 10000.0 » |1|0000.0 » |1||0000.0| (dot present)
» six significant figures.
Question 6
How many significant figures are in the number 0.00010?
Answer: 0.00010 » 0.000|1|0 » 0.000|1||0| » two significant figures.
Regarding Numbers Written in Scientific Notation
The box-and-dot method also works for numbers written
in exponential notation—numbers that take the form a × 10n,
where a is any real number and n is an integer. Here, the term
exponential notation is used in the most general sense, including
both “standard” (for example, 1.20 × 109) and “nonstandard”
(for example, 120 × 107) forms.3 Students may be tasked with
converting numbers written in nonstandard exponential notation to standard form, so both forms are addressed.
For numbers written in exponential notation, the box-anddot method is used as described above, only the power term
(10n) is neglected. It can be neglected because zeros associated
with the power term are never significant—all significant digits
are expressed in the coefficient.
In classroom parlance, when assigning significant figures
to a number expressed in exponential notation, invoke the rule:
“skip the ten-to-the-n term”.
Question 7
How many significant figures are in the number 6.022 × 1023,
known as Avogadro’s number in some countries, and as Loschmidt’s number in other countries.
Answer: 6.022 × 1023 » |6.022| (skip the ten-to-the-n
term) » no trailing zeros » four significant figures.
Question 8
How many significant figures are in the number associated with
1.60 × 10‒19 C, the charge on a single electron?
Answer: 1.60 × 10‒19 » |1.6|0 (skip the ten-to-the-n term)
» |1.6| |0| (dot present) » three significant figures.
Question 9
(a) How many significant figures are in the number 500.0 ×
1010? (b) Write the number in standard scientific notation.
Answer: (a) 500.0 × 1010 » |5|00.0 » |5||00.0| » four
significant figures.
(b) 500.0 × 1010 = 5.000 × 1012 (the number has four
significant figures, all of which must be expressed in the coefficient).
An Area of Ambiguity: Numbers Ending
with a Decimal Point
How would you write the number five thousand so that
it would have four significant figures? Writing the number as
934
“5000” indicates the presence of one significant figure only.
Writing the number as “5000.” is not recommended because the
ACS Style Guide states that one should “use numbers before and
after a decimal point” (1). Attempting to remedy the problem
by adding a zero after the decimal point produces the number
“5000.0”, which possesses five—rather than the requisite four—
significant figures.
Ambiguity can be removed by expressing the number in
scientific notation (5.000 × 103).4 It should be noted that not
all publications adhere to ACS style guidelines, and numbers
ending with decimal points (such as “5000.”) are sometimes
encountered.
Regardless of convention, the box-and-dot uses the presence of a decimal point to indicate whether or not trailing zeros
are significant. It is assumed the originator recorded the number
in proper, unambiguous form.
Question 10
How many significant figures are in the numbers (a) 5000, (b)
5.0 × 103, and (c) 5.000 × 103?
Answer: (a) 5000 » |5|000 » no dot » one significant
figure.
(b) 5.0 × 103 » |5.|0 » |5.||0| » two significant figures.
(c) 5.000 × 103 » |5.|000 » |5.||000| » four significant
figures.
Teaching the Box-and-Dot Method in the Classroom
To determine the significant figures in any real number,
execute the following three steps explicitly. For numbers written
in exponential notation, skip the ten-to-the-n term.
For Step 1, show students a number with multiple digits.
Demonstrate drawing a box around all the nonzero digits,
beginning with the leftmost nonzero digit and ending with the
rightmost nonzero digit in the number. Ask students to do the
same thing and point out to them that the box will include any
zeros “sandwiched” between nonzero digits.
For Step 2, explain to students that if a “dot” (a decimal
point) is present, they should draw a box around any trailing
zeros. Conversely, if a dot is not present, they should not draw
a box around any trailing zeros.
For Step 3, confirm with students that they should consider
any and all boxed digits significant, while digits that are not
boxed are not significant.
Acknowledgments
I am grateful to Tom Dudley and the referees for their valuable comments and recommendations.
Notes
1. The term trailing zeros refers to any and all zeros present to the
right of the rightmost nonzero digit (i.e., to the right of the box drawn
in Step 1).
2. The term leading zeros refers to any and all zeros present to
the left of the leftmost nonzero digit (i.e., to the left of the box drawn
in Step 1).
Journal of Chemical Education • Vol. 86 No. 8 August 2009 • www.JCE.DivCHED.org • © Division of Chemical Education In the Classroom
3. Regarding numbers written in exponential notation—numbers
which take the general form a × 10n—standard form refers to cases in
which n is an integer, and the absolute value of the coefficient is greater
than or equal to one, but less than ten (1 ≤ |a| < 10); nonstandard form
refers to cases in which the absolute value of the coefficient falls outside
this range.
4. Although less common, ambiguity may also be removed by
using the convention in which an overbar is placed above the rightmost
significant figure (in this case, the third zero from the decimal point).
Literature Cited
1. ACS Style Guide, 2nd ed., Dodd, J. S., Ed.; American Chemical
Society: Washington, DC, 1997; p 148.
Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2009/Aug/abs933.html
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