## SIGNIFICANT FIGURES

**Description:
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Engineers and surveyors communicate a great deal of their professional information using numbers. It is important, therefore, that the number of digits used, correctly indicates the accuracy with which the field data were measured. This is particularly important since the advent of pocket calculators, which tend to present numbers to as many as eight places of decimals, calculated from data containing, at the most, only three places of decimals, whilst some eliminate all trailing zeros.

This latter point is important, as 2.00 m is an entirely different value to 2.000 m. The latter number implies estimation to the nearest millimetre as opposed to the nearest 10 mm implied by the former. Thus in the capture of field data, the correct number of significant figures should be used. By definition, the number of significant figures in a value is the number of digits one is certain of plus one, usually the last, which is estimated. The number of significant figures should not be confused with the number of decimal places. A further rule in significant figures is that in all numbers less than unity, the zeros directly after the decimal point and up to the first non-zero digit are not counted. For example:

Two significant figures: 40, 42, 4.2, 0.43, 0.0042, 0.040

Three significant figures: 836, 83.6, 80.6, 0.806, 0.0806, 0.00800

Difficulties can occur with zeros at the end of a number such as 83 600, which may have three, four or five significant figures. This problem is overcome by expressing the value in powers of ten, i.e. 8.36 × 104 implies three significant figures, 8.360×104 implies four significant figures and 8.3600×104 implies five significant figures. It is important to remember that the accuracy of field data cannot be improved by the computational processes to which it is subjected. Consider the addition of the following numbers:

155.486

7.08

2183.0

42.0058

If added on a pocket calculator the answer is 2387.5718; however, the correct answer with due regard to significant figures is 2387.6. It is rounded off to the most extreme right-hand column containing all the significant figures, which in the example is the column immediately after the decimal point. In the case of 155.486 7.08 2183 42.0058 the answer should be 2388. This rule also applies to subtraction.

In multiplication and division, the answer should be rounded off to the number of significant figures contained in that number having the least number of significant figures in the computational process. For instance, 214.8432 × 3.05 = 655.27176, when computed on a pocket calculator; however, as 3.05 contains only three significant figures, the correct answer is 655. Consider 428.4 × 621.8 = 266379.12, which should now be rounded to 266 400 = 2.664 × 105, which has four significant figures. Similarly, 41.8 ÷ 2.1316 = 19.609683 on a pocket calculator and should be rounded to 19.6. When dealing with the powers of numbers the following rule is useful. If x is the value of the first significant figure in a number having n significant figures, its pth power is rounded to:

n − 1 significant figures if p ≤ x

n − 2 significant figures if p ≤ 10x

For example, 1.58314 = 6.28106656 when computed on a pocket calculator. In this case x = 1, p = 4 and

p ≤ 10x; therefore, the answer should be quoted to n − 2 = 3 significant figures = 6.28.

Similarly, with roots of numbers, let x equal the first significant figure and r the root; the answer should be rounded to:

n significant figures when rx ≥ 10

n − 1 significant figures when rx < 10

For example:

3612 = 6, because r = 2, x = 3, n = 2, thus rx < 10, and answer is to n − 1 = 1 significant figure.

415.3614 = 4.5144637 on a pocket calculator; however, r = 4, x = 4, n = 5, and as rx > 10, the answer is rounded to n = 5 significant figures, giving 4.5145.

As a general rule, when field data are undergoing computational processing which involves several intermediate stages, one extra digit may be carried throughout the process, provided the final answer is rounded to the correct number of significant figures.

**ROUNDING NUMBERS:**

It is well understood that in rounding numbers, 54.334 would be rounded to 54.33, whilst 54.336 would become 54.34. However, with 54.335, some individuals always round up, giving 54.34, whilst others always round down to 54.33. Either process creates a systematic bias and should be avoided. The process which creates a more random bias, thereby producing a more representative mean value from a set of data, is to round to the nearest even digit. Using this approach, 54.335 becomes 54.34, whilst 54.345 is 54.34 also.