Rules for Conversion and Rounding

General

The accuracy of measured quantities depends on many factors such as the quality and type of measuring instrument used and the degree of diligence exercised in making the measurement. Before a number can be used, its significance must be determined. Significance is the degree of uncertainty in the digits that are given. Usually this can be determined by examining the source of the number.

Some numerical values are exact (such as the result of an exact count) but all measured values are approximations. In any approximation, the true value falls within ±0.5 in the place of the last digit retained or measured unless a deviation or tolerance is stated. For example, a measurement of 3.4 kilograms falls between 3.35 and 3.45 kilograms. However, if the value is given as 3.40 ± 0.02 kilograms the true value falls between 3.38 and 3.42 kilograms.

Significant Digits

Any digit that is necessary to define the specific value or quantity is said to be significant. When measured to the nearest 1 meter, a distance may be recorded as 187 m; this number has three significant digits. If the measurement had been made to the nearest 0.1 meter (or the nearest decimeter), the distance may have been 187.3 meters; this number has four significant digits. In each of these cases the value of the right-hand digit was determined by measuring the value of an additional digit and then rounding to the desired degree of accuracy. Thus, 187.3 was rounded to 187 in the first case; in the second case, the measurement was rounded from 187.32 to 187.3

In general, the number of significant digits are those digits which are certain plus one more.

To determine which digits are significant, some simple rules are:

• All nonzero digits are significant.
• Zero digits between nonzero digits are significant.
• Zero digits at the extreme right of a decimal number are significant.
• Zeros preceding the first nonzero digit of a decimal fraction are not significant.
• Zeros at the end of a whole number, if used only to locate the decimal point, are not considered significant unless specifically identified as significant.

Examples

348,000 - Only 3, 4, and 8 are significant.

3.480 - All digits are significant.

3,048 - All digits are significant.

0.0348 - Only 3, 4, and 8 are significant.

348,000 (to nearest hundred) - Last two zeros are not significant.

348,000 (exact) - All digits are significant.

3.48 × 10³ in scientific notation clearly has three significant digits.

Derived Quantities

Derived quantities are quantities which are obtained from some mathematical combination of direct measurements (such as areas and volumes obtained from multiplication of measured lengths). In the determination of derived quantities as well as in all conversions, the number of significant digits retained should be such that the implied or stated accuracy is neither sacrificed nor exaggerated. The most accurate equivalents are obtained by multiplying the specified quantity by the most accurate conversion factor available and then rounding to the appropriate number of significant digits.

The number of significant digits to be retained after multiplication, division, addition or subtraction is determined as follows:

Multiplication and Division

A product or quotient shall contain no more significant digits than are contained in the number with the fewest significant digits used in the multiplication or division. For example,

113.2 × 1.43 = 161.876 must be rounded to 162

113.2 ÷ 1.43 = 79.161 must be rounded to 79.2

Addition and Subtraction

The answer of an addition or subtraction shall contain no significant digits farther to the right than occurs in the least accurate figure.

For example, assume that in the following three numbers to be added the zeros do not indicate a specific value but only indicate the magnitude of the number.

251,000

132,780

+ 4,762

388,542

This total implies an unrealistic precision. The numbers should first be rounded to one significant digit farther to the right than that of the least accurate number, and then added as follows:

251,000

132,800

+ 4,800

388,600

The answer is then rounded to 389,000.

More About Significant Digits

Zeros may be used either to indicate a specific value, like any other digit, or to indicate the magnitude of a number. The 1960 U.S. population figure rounded to thousands was 179,323,000. The six left-hand digits of this number are significant; each of them measures a value. The three right-hand digits are zeros which merely indicate the magnitude of the number rounded to the nearest thousand. This point may be further illustrated by the following list of estimates and measurements, each of which is of different magnitude but each of which has only one significant digit:

1000
 100
  10
   0.01
   0.001
   0.0001

It is also important to note that in the case of the first three numbers above, the identification of significant digits is only possible through knowledge of the circumstances. The number 1000 may be rounded from about 965, or it may be rounded from 999.7, in which case all three zeros are significant.

Numbers that are exact counts are treated as though they consist of an infinite number of significant digits. More simply stated, when a count is used in computation with a measurements the number of significant digits in the answer is the same as the number of significant digits in the measurement. If a count of 40 is multiplied by a measurement of 10.2, the product is 408. However, if 40 were an estimate accurate only to the nearest 10, and hence contained but one significant digit, the product would be 400.

Rounding Values

When a figure is to be rounded to fewer digits than the total number available, the procedure should be as follows:

When the first digit discarded is less than 5, the last digit retained should not be changed. For example, 3.46325, if rounded to four digits, would be 3.463; if rounded to three digits, 3.46.

When the first digit discarded is greater than 5, or if it is a 5 followed by at least one digit other than 0, the last figure retained should be increased by one unit. For example 8.37652, if rounded to four digit would be 8.377; if rounded to three digits, 8.38.

When the first digit discarded is exactly 5, followed only by zeros, the last digit retained should be rounded upward if it is an odd number, but no adjustment made if it is an even number. For example, 4.365, when rounded to three digits, becomes 4.36. 4.355 would also round to the same value, 4.36, if rounded to three digits.

References

Metric Practice Guide, E 380-72, American Society for Testing and Materials
SI-Metric, IBM Reference Manual
"American National Standard Practice for Inch-Millimeter Conversion for Industrial Use", ANSI B48.1-1933 (R1947), ISO R370-1964.