Measurement and Significant Figures

The Trailing Zeros
Suppose you measure length of an object using a metre scale with least count of \(0.1 \, cm\). The length comes out to be \(35 \, cm\). Which of the following, in your opinion, is the best way to state this data?

1. \(35.000 \, cm\)
2. \(35 \, cm\)
3. \(35.0 \, cm\)
4. \(35.00 \, cm\)

Are these numbers equivalent? Mathematically, there is no difference between these numbers. However, to an experimentalist, each of the numbers above tell a different story. The number \(35.0 \, cm\) conveys that we are certain about the first two digits \(3 \, \mbox{and} \, 5\) of this measurement and we can say with a 'good degree of confidence' that the number in the tenth place is 'close' to zero, if not exactly zero. However, we are not able to say anything about numbers in hundredth position and onward.

The number \(35.00 \, cm\), on the other hand, conveys that we are certain about the first three digits — \(3, \, 5\) and the first \(0\) after the decimal place — of this measurement and we can say with a 'good degree of confidence' that the number in the hundredth place is 'close' to zero, if not exactly zero. However, we are not able to say anything about numbers in the thousandth position and onward.

Similarly. in case of the number \(35.000 \, cm\), we are certain about the first four digits — \(3, \, 5\) and the two zeros in the tenth and hundredth place — of this measurement and we can say with a 'good degree of confidence' that the number in the thousandth place is 'close' to zero, if not exactly zero. But, we are not certain at all about digits in the ten-thousandth position and onward.

Finally. in case of the number \(35 \, cm\), we are certain about the first digit, 3, of the measurement, we have a 'good degree of confidence' the number in ones place. But, we are not certain at all about digits in the tenth place and onward.

Thus, numbers \(35 \, cm, \, 35.0 \, cm, \, 35.00 \, cm\) and \(35.000 \, cm\) suggest that scales of differing precision were employed in making these measurements &mdash \(35.0 \, cm\) is measured using a scale that can discriminate lengths in the tenth position, \(35.000 \, cm\) is measured using a scale that can discriminate lengths in the thousandth position. This is an important piece of information that enables us to determines the range of validity of any inference we might make on the basis of any of these numbers. For instance, suppose these numbers are results of measurement of the length of a uniform rectangular block of wood using scales with differing precision. Then, volume of the block calculated using the length \(35.0 \, cm\) will be less precise and reliable than the volume of the block calculated with the number \(35.000 \, cm\).

In the context of experimental measurements, trailing digits convey important information about precision of measurements. We must practice caution when stating experimental data; overstating or understating the precision of measurements can easily lead anyone to make erroneous conclusions. Therefore, it is important that we formulate a method to state experimental data in a way that accounts for the precision of the measuring device used. Such a method would require that only the digits that are meaningful be included when stating data. Concept of significant figures serves this express purpose.

For instance, suppose a mass measured by a weighing scale is stated as \(9.30 \, gm\) but the precision with which the weighing scale can measure is only \(0.1 \, gm\), i.e. upto a tenth of a gram, then zero in the hundredth place in \(9.30 \, gm\) is meaningless as hundredth of a gram is well beyond the sensitivity of the scale. Stating zero in the hundredth place here is also misleading as it can lead anyone to believe that the mass is measured with precision of hundredth of a gram. Hence, this particular measurement should be reported as \(9.3 \ gm\) — \(9\) and \(3\) are the only meaningful digits here. Similarly, \(12.3 \, gm\), \(5.7 \, gm\), \(32.3 \, gm\) are examples of measurements made using this scale. Note that all digits included in these measurements are meaningful and of significance as regards the precision / sensitivity of the scale. From this discussion, we may conclude that in the context of data obtained from measurement:

Trailing zeros are significant.

The Leading Zeros (not so important)
The leading zeros, in stark contrast with trialing zeros, convey no new or meaningful information about a measurement. Consider the following measurements:

1. \(0005.3 \, cm\)
2. \(5.3 \, cm\)
3. \(05.3 \, cm\)
4. \(005.3 \, cm\)

Unlike trailing zeros, presence or absence of zeros to the left of digit \(5\) does not affect, in any way, the information carried by the remaining two digits — \(5\) and \(3\). These zeros are, therefore, insignificant. In the context of data obtained from measurement, we may say that:

All leading zeros are insignificant.

What about non-zero digits
Of course they are significant! If a non-zero number is obtained from reading the scale of a measuring device, it is just as important a part of the measurement as any other number read off of the scale regardless of the position of the number. Say number \(05003045.678\) is obtained from reading the scale of a measuring device. Here, numbers \(5, \, 4, \, 6\) and \(8\) are just as important as \(3\) in the thousandth place and \(7\) in the hundredth place; leaving out any digit here would compromise the integrity of the measurement. (Note that the leading zero is insignificant.) It is always a good idea to give all digits obtained directly from the measuring scale of a device their due regard. Therefore,

All non-zero digits in a number are significant.

We missed those zeros trapped between non-zero digits
They are all significant. The only way zeros can get trapped between non-zero numbers is when they are part of the measurement and are read off of the scale just as non-zero are. Therefore,

Zeros between non-zero digits are significant.

Hold on! What about that decimal point?
Suppose a person measures a distance and reports it as \(3000 \, km\). Now, based on the rules for significant figures developed above, we would say that there are four significant figures in this value. But can we make this statement with certainty? Let's take a step back and take a closer look at what this number exactly conveys.

Using this value, is it possible to make any statement about the sensitivity of the scale used to make this measurement? Is it obvious from this figure that scale determines digits in thousands, hundreds and tens places with certainty but digit in ones place may have a degree of uncertainty? In this case, the number of significant figures would indeed be four. What if the scale had sensitivity of \(10 \, m\) only, i.e., could yield reliable measurements in tens place only and the digit in ones place was completely uncertain, hence insignificant. There would be three significant figures in this case (zero in ones place being insignificant). What if the sensitivity of the scale was, at best, \(100 \, m\) implying digits in ones and tens place are completely uncertain. In this case, there would be two significant figures (zeros in ones and tens places being uncertain). Using the number \(3000 \, km\), is it possible to tell which of these scales was used to measure this distance? It is not.

The number \(3000 \,km\) is, thus, ambiguous as regards the sensitivity of the device used to measure it. In this form, it is not possible to make any statement about the precision and the degree of reliability with which this measurement is made. This can be avoided by including a decimal point in the data. We will see how this is done in the next post. For now, we may summarise this discussion into fifth and final rule for identifying significant figures in a given number.

In cases where decimal point is not stated explicitly, the trailing zeros may or may not be significant.

SUMMARY
The four rules to identify significant figures in a measured number:

1. Trailing zeros are significant.
2. All leading zeros are insignificant.
3. All non-zero digits in a number are significant.
4. Zeros between non-zero digits are significant.
5. In cases where decimal point is not stated explicitly in a given number, the trailing zeros may or may not be significant.