Tips for teaching maths skills to our future chemists, by Paul Yates of Keele University

* In this issue: significant figures*

When measuring a physical quantity in the laboratory it is good practice to estimate the uncertainty on the value obtained. When such quantities are combined in a calculation it is straightforward, though possibly tedious, to combine these uncertainties to give the overall uncertainty. Frequently though, we are often asked to perform calculations using values where the uncertainty is not specified. In these cases we need to use some standard rules for combining the quantities based on the precision with which they are expressed,* ^{1,2}* to avoid giving values which imply a greater degree of precision than is justified. Before considering them, we need to understand what is meant by 'rounding'.

### Rounding

This is important because calculators typically give answers to greater precision than is justified by data. The basic rule is that if the digit we are discarding is above five we increase the previous digit by one, and if it is less than five we reduce the previous digit by one. Thus 4.857 rounds to 4.86 and 7.82 rounds to 7.8. An interesting situation arises when the digit to be discarded is five. Most people seem to have been taught to increase the previous digit by one. However, statistically it is better to round the previous digit to the nearest even number because then there will be an equal numbers of times when one rounds up or down in this case. This is known as Gaussian or, perhaps less fashionably, banker's rounding, and is designed to avoid an upward bias.* ^{3}* In fact, this is only likely to be an issue when dealing with large amounts of data and the most important thing is to adopt a consistent approach. Whichever method of rounding is used, it is also important to consider only the digit being discarded.

### Significant figures

Although this is the term commonly used, it is probably more meaningful to talk about significant digits. Any non-zero digit is considered to be significant. Zeros are considered to be significant if they are enclosed, as in 8.076, and may be if they appear at the end of a decimal number to the right of the decimal point, as in 76.900. When they appear at the end of an integral number it is more tricky to decide. Does 7600 represent an exact figure or is it 7643 rounded to the nearest hundred? In these cases we need to use any additional information we are given to decide. Leading zeros, as in 0.00876, are never significant because they simply indicate the magnitude of the number. Pure integers, such as those arising from stoichiometric ratios, are exact numbers and can be considered to have an infinite number of significant figures. Thus the number two which appears in the chemical equation:

H_{2}SO_{4} + 2NaOH = Na_{2}SO_{4} + 2H_{2}O

can be thought of as 2.000000000...

### Addition and subtraction

When two physical quantities are added or subtracted, the answer should be given to the least number of decimal places in the original data. Thus 1.436 + 2.3 should be quoted as 3.7, since 2.3 is given to only one decimal place. Thus, if we want to combine the electrode potentials described by the equations:

Cu^{2+}(aq) + 2^{e-} → Cu(s) *E*_{R }= 0.3419 V

and

Zn^{2+}(aq) + 2^{e-} → Zn(s) *E*_{L} = -0.76 V

in the Daniell cell, according to *E*_{cell} = *E*_{R} - *E*_{L}, we have:

*E*_{cell} = 0.3419 V - (-0.76 V)

= 0.3419 V + 0.76 V

= 1.10 V

since *E*_{L} in the original data is only given to two decimal places.

### Products and quotients

In this case we quote the final answer to the least number of significant figures given in the original data. Thus 7.82/3.164 is given as 2.47 to three significant figures. If we want to calculate the entropy of vaporisation Δ_{vap}*S* of water using the equation:

Δ_{vap}*S =* Δ_{vap}*H/T _{b}*

where Δ_{vap}*H,* the enthalpy of vaporisation, is 40.656 kJ mol^{-1} and *T*_{b}, the boiling temperature is 373 K, we would have:

Δ_{vap}*S =* 40656 J mol^{-1}/373 K

= 109 J K^{-1} mol^{-1}

which should be given to three significant figures as indicated by the precision of *T*_{b} in the original data.

### Logarithms

When considering significant figures in relation to logarithms, you must appreciate that the digits before the decimal point indicate the magnitude of the number, and those after the decimal point its actual value. Thus, if we are taking the logarithm of a number, the number of significant figures after the decimal point in the resulting value should be equal to that in the original data. For example, the log_{10} of 12.34 would be written as 1.0913 with four digits after the decimal point. Conversely, if we are taking an antilogarithm, then the number of significant figures in the answer is equal to the number of digits after the decimal point. Somewhat confusingly, leading zeros are treated as significant when we consider the logarithmic values in this case.* ^{4}* The same rule applies to natural logarithms and exponentials.

*For example, the exponential of 2.361 is written as 10.6 because there were originally three significant figures after the decimal point. A chemistry example would be the calculation of pH, defined by:*

^{5}pH = -log_{10}([H^{+}]/mol dm^{-3})

If [H^{+}] = 0.055 mol dm^{-3} then

[H^{+}]/mol dm^{-3} = 0.055

and

pH = -log_{10} 0.055 = -(-1.260) = 1.26

in which we have quoted two digits after the decimal point because this is the number of significant figures given in the original concentration.

### More complicated examples

In chemistry we are frequently dealing with equations which require us to combine these rules. Consider, for example, the calculation of heat capacity *C*_{p} using the equation:

*C*_{p} = *a* + *bT *

where *a* and *b* are constants and *T* is the absolute temperature. For oxygen *a *= 29.96 J K^{-1} mol^{-1} and *b* = 4.18 x 10^{-3} J K^{-2} mol^{-1}. At a temperature of 298.15 K, we have:

*bT* = 4.18 x 10^{-3} J K^{-2} mol^{-1} x 298.15 K = 1.25 J K^{-1} mol^{-1}

which is given to the three significant figures implied by the value of *b*. We now have:

*C*_{p} = 29.96 J K^{-1}mol^{-1} + 1.25 J K^{-1} mol^{-1}

The final answer is given to two decimal places

as 31.21 J K^{-1}mol^{-1}.

For more complicated examples with more steps it may be worth calculating the final value and then revisiting individual steps to consider the appropriate number of figures to carry through at each stage. Often one quantity will be significantly less precise than the others which will make this a much easier process.

#### Further Reading

- A method I have found works particularly well to introduce this topic is that described by Clase.
In this a decimal number is extended using a number of question marks which represent undefined digits. Students are able to appreciate that adding an unknown digit to a known digit results in another unknown digit. Unfortunately, it is less straightforward to illustrate the other rules described above. For those who would prefer to adopt a more experimental approach to the subject, a number of possibilities have been described. These include the running of a measurements lab^{6}and measurements of volume and length.^{7}^{8 } - There has been a series of studies into the validity of the rules described, and for multiplication and division
and exponentiation^{9 }alternatives have been proposed which claim to preserve a precision which is usually lost. These studies could provide a useful entry point into a discussion of how dealing with significant figures in this way is not always an exact science.^{10}

#### Related Links

Rules for Rounding Off

### Reference

- L. M. Scwartz,
*J. Chem. Educ*., 1985,**62**, 693. 2. R. C. Pinkerton and C. E. Gleit,*J. Chem. Educ*., 1967,**44**, 232. - Rounding.
- D. E. Jones,
*J. Chem. Educ*., 1972,**49**, 753. - H. L. Clever,
*J. Chem. Educ*., 1979,**56**, 824. - H. J. Clase,
*J. Chem. Educ*., 1993,**70**, 133. - B. Anderlik,
*J. Chem. Educ*., 1980,**57**, 591. - R. A. Pacer,
*J. Chem. Educ*., 2000,**77**, 1435. - W. Chen, W. Lee and C. L. Mulliss,
*Chinese J. Phys*., 2004,**42**, 335. - W. Chen, W. Lee and C.L. Mulliss,
*Chinese J. Phys*., 2005,**43**, 1017.

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