Significant Figures Counting Significant Figures Multiplication

The term is also sometimes used to describe some rules-of-thumb, known as significance arithmetic, which attempt to indicate the propagation of error in a scientific experiment or in statistics when perfect accuracy is not attainable or not required.

The concept of significant figures is derived from the method of measuring a value so that the smallest accurately known decimal place is next to last and only one further is estimated; for example, if an object is measured with a ruler that is marked by millimeters and is known to be between six and seven mm and appears to the measurer to be approximately two-thirds of the way between them, an acceptable measurement for it could be 6.6mm or 6.7mm, but not 6.666666... mm. This rule based upon the principle of not introducing false accuracy into measurements taken in this manner.

1 Counting significant figures

Before calculations can be done according to the rules-of-thumb of significance arithmetic, one must determine the number of significant digits in each number being used in the calculations.

Note that because of rounding, a number to n significant figures is not necessarily the same as the first n digits of that number. The rules for determining the significance of digits are as follows:

Conventionally, a number with value zero is considered to have one significant digit.

In order to correctly show which digits are significant, figures such as 2000 should be expressed in scientific notation to the correct number of significant figures. If two digits — the '2' and the first '0' — are significant (i.e., the true value could be anywhere from 1990 to 2010), the correct representation is 2.0x10³; if three are significant (from 1999 to 2001) then it's 2.00x10³; if four are significant (from 1999.5 to 2000.5), then it could be either 2000. (two, zero, zero, zero, decimal point) or 2.000x10³. A plain 2000 indicates that only the '2' is significant (from 1900 to 2100)

2 Multiplication and division using significance arithmetic

When multiplying and dividing numbers together, the product or quotient is rounded to the number of significant figures of that of the factor with the least. For instance, using significant figures rules:

In the above, all numbers are assumed to be measurements (therefore potentially inexact). For example: the answer yielded from 8x8 is actually 64, but because 8 is treated as a measurement, it only has one significant figure, and so the answer must be rounded to 60. Exact numbers are treated as having a limitless number of significant figures.

When squaring or taking the square root of a value, the number of significant figures decreases by one using some systems of significant digits.

3 Addition and subtraction using significance arithmetic

When you add or subtract significant figures, limit to, and round your answer to the least number of decimal places in any of the numbers that make up the problem. For instance, using significant figures rules:

(The answer in Significant figures is 2, because 1 has no decimal place, so the answer can have no decimal place)

(The answer in Significant figures is 2.1, since 1.0 and 1.1 both have one decimal place, so the answer must have one decimal place also)

(The answer in Significant figures is 200, since 100 and 110 have no decimal places, so the answer cannot have any decimal places either)

(The answer in Significant figures is 210, since 111 and 10² have no decimal places, so the answer cannot have any decimal places either)

(The answer in Significant figures is 255.5, because 46.0 only has one decimal place, so the answer can only have one decimal place)

4 The even-odd rule, also known as bankers' rounding

As with all rounding procedures, if the number directly to the right of the digit to be rounded to is less than five, the digit stays the same; if more than five, the digit is rounded up. However, to always round up or down if the digit is equal to exactly five would skew data in one direction or the other. Thus, when using the significant figures system and rounding in such situation, the even-odd rule is used: round in whichever direction would make the last digit of the final product even. For example:

If 3.5 had to be rounded to one significant figure, it would become 4, since four is even
If 2.5 had to be rounded to one significant figure, it would become 2, since two is even

In this way, the even-odd rule avoids skewing data either upwards or downwards.

5 Measuring with significant figures

As illustrated in the above example involving the length measurement in millimeters, the significant figures method teaches that when measuring using a non-electronic instrument, the observer should estimate within the nearest tenth of a division marked on the instrument. For example, if a graduated cylinder was marked off at every milliliter (ml), the observer should measure the amount of volume contained in the cylinder to the nearest tenth of a milliliter. In order to express the degree of precision to which a value was measured, decimals are used. When using significant figures rules, it should be assumed that the last significant digit of every value was estimated. Using the previous example, if the observer read the amount of liquid in the cylinder to be exactly at the 12 ml mark, the observer would write the value as 12.0 ml, which would indicate that the tenths place was the precision obtained. If the cylinder was marked off to every tenth of a ml, the observer would write the value as 12.00 ml. Note that exact numbers obtained by counting should not be subject to the rules of significant figures.

6 References

See the decimal arithmetic FAQ for a discussion of why significance arithmetic results in grossly inaccurate error bounds. The classic paper on this is: Computation with Approximate Numbers, Daniel B. Delury, The Mathematics Teacher 51, pp521-530, November 1958.

7 External link

Worksheets for Practicing Rounding

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