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Significant figures provide an indication of the precision with which a quantity is measured or known. The last digit represents, in a qualitative sense, some degree of uncertainty or error. For example, a measurement of 8.32 inches implies that the actual quantity is somewhere between 8.315 and 8.325 inches. This applies to calculated and measured quantities; quantities that are known exactly (e.g. pure integers) have an infinite number of significant figures. Note, however, that there is an upper limit to the accuracy with which physical measurements can be made.

The significant digits of a number are the digits from the first nonzero digit on the left to either (a) the last digit (whether it is nonzero or zero) on the right if there is a decimal point, or (b) the last nonzero digit of the number if there is no decimal point. For example:

370	has 2 significant figures
370.	has 3 significant figures
370.0	has 4 significant figures
28,070	has 4 significant figures
0.037	has 2 significant figures
0.0370	has 3 significant figures
0.02807	has 4 significant figures

Whenever quantities are combined by multiplication and/or division, the number of significant figures in the result should equal the lowest number of significant figures of any of the quantities. In long calculations, the final result should be rounded off to the correct number of significant figures. When quantities are combined by addition and/or subtraction, the final result cannot be more precise than any of the quantities added or subtracted. Therefore, the position (relative to the decimal point) of the last significant digit in the number that has the lowest degree of precision is the position of the last permissible significant digit in the result. For example, the sum of 3702., 370, 0.037, 4, and 37 should be reported as 4110 (without a decimal). The least precise of the five numbers is 370, which has its last significant digit in the tens position. The answer should also have its last significant digit in the tens position.

Unfortunately, engineers and scientists rarely concern themselves with significant figures in their calculations. However, it is recommended that the reader attempt to follow the calculation procedure set forth in this section.

In the process of performing engineering calculations, very large and very small numbers are often encountered. A convenient way to represent these numbers is to use scientific notation. Generally, a number represented in scientific notation is the product of a number (< 10 but > or = 1) and 10 raised to an integer power. For example,

and

A positive feature of using scientific notation is that only the significant figures need to appear in the number.

When approximate numbers are added or subtracted, the results should also be represented in terms of the least precise number. Since this is a relatively simple rule to master, note that the answer follows the aforementioned rule of precision. Therefore,

These numbers have two, one, and three decimal places, respectively. The least precise number (least number of decimal places) is 2.8, a value carried only to the tenths position. Therefore, the answer must be calculated to the tenths position only. Thus, the correct answer is 4.7 L (the last 6 and 7 are dropped from the full answer 4.667, and the result is rounded up to provide 4.7 L).

In multiplication and division of approximate numbers, finding the number of significant digits in the appropriate numbers is used to determine how many digits to keep (i.e. where to truncate). One must understand significant digits in order to determine the correct number of digits to keep or remove in multiplication and division problems. As noted earlier in this section, the digits 1 through 9 are considered to be significant. Thus, the numbers 123, 53, 7492, and 5 contain three, two, four, and one significant digit, respectively. The digit zero must be considered separately.

Zeros are significant when they occur between significant digits. In the following examples, all zeros are significant: 10001, 402, 1.1001, and 500.09 (five, three, five, and four significant figures, respectively). Zeroes are not significant when they are used as place holders. When used as a holder, a zero simply identifies where a decimal is located. For example, each of the following numbers has only one significant digit: 1000, 500, 60, 0.09, and 0.0002. In the numbers 1200, 540, and 0.0032, there are two significant digits, and the zeroes are not significant. Once again, when zeros follow a decimal and are preceded by a significant digit, the zeros are significant. In the following examples, all zeroes are significant: 1.00, 15.0, 4.100, 1.90, 10.002, and 10.0400. For 10.002, the zeroes are significant because they fall between two significant digits. For 10.0400, the first two zeroes are significant because they fall between two significant digits; the last two zeroes are significant because they follow a decimal and are preceded by a significant digit. Finally, when approximate numbers are multiplied or divided, the result is expressed as a number having the same number of significant digits as the expression in the problem having the least number of significant digits.

When truncating (removing final, unwanted digits), rounding is normally applied to the last digit to be kept. Thus, if the value of the first digit to be discarded is less than 5, one should retain the last kept digit with no change. If the first digit to be discarded is 5 or greater, increase the last kept digit’s value by one. Assume, for example, only the first two decimal places are to be kept for 25.0847 (the 4 and 7 are to be dropped). The number is then 25.08. Since the first digit to be discarded (4) is less than 5, the 8 is not rounded up. If only the first two decimal places are to be kept for 25.0867 (the 6 and the 7 are to be dropped), it should be rounded to 25.09. Since the first digit to be discarded (6) is 5 or more, the 8 is rounded up to 9.

In some desalination studies, a measurement in gallons or liters may be required. Although a gallon or liter may represent an exact quantity, the measurement instruments that are used are capable of producing approximations only. Using a standard graduated flask in liters as an example, can one determine whether there is exactly 1 liter? Likely not. In fact, one would be pressed to verify that there was a liter within the ± 1/100 of a liter. Therefore, depending upon the instruments used, the precision of a given measurement may vary.

If a measurement is given as 16.0 L, the zero after the decimal indicates that the measurement is precise to within 1/l0 L. Given a measurement of 16.00 L, one has precision to the 1/l00 L. As noted, the digits following the decimal indicate how precise the measurement is. Thus, precision is used to determine where to truncate when approximate numbers are added or subtracted.