The Box-And-Dot Method

Counting Significant Figures

The Box-and-Dot Method

The “Box- and- Dot Method” is a simple strategy for determining the number of significant figures in a measurement. The method uses the device of “boxing” significant figure

based on two simple rules, then counting the number of digits in the box(es).

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.
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.

Step 2

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|.

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, 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.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.

Worked Examples of the Box-and-Dot Method

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.

Question 1

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.

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.

For numbers written in exponential notation, the box-and-dot 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?

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 Endingwith a Decimal Point

How would you write the number five thousand so that it would have four significant figures? Writing the number as “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). 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.

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).