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29.03.2016

More precision by more digits?

Precision is always good, isn´t it? Especially in science, accurate and precise data and calculations are absolutely essential. But most of us might also have a feeling of “too much precision” when our pocket calculator presents us numbers with seven digits. Indeed, most of them usually are without meaning, as we will discuss in this short summary about precision of numbers.

To understand which level of precision makes sense, we first have to understand what written numbers really mean and which digits are relevant when presenting numbers. Second, we are going to summarise some basic rules for communicating meaningful numbers to others.

Presenting numbers
The precision of numbers is determined by their number of digits. The last digit of any number indicates the intended precision, e.g. of a measurement. Stating that a person weighs 82 kg means the true weight lies between 81.50 and 82.49 kg. In this case, the weight is rounded to the last digit before the decimal point. If we want to point out that the true weight was exactly 82 kg on a gram scale, we would have to write 82.000 kg or 82,000 g. Scaling the level of precision up or down this way is easy for digits after the decimal point: just add or omit digits. But how can we express different levels of accuracy in digits before the decimal point, e.g. that an apple weighs “about 150 grams”, i.e. 145-154 g? The first option is easy: we can change the unit from g to kg and write 0.15 kg. If that is not an option, the order of magnitude can be indicated by powers of ten, e.g. 0.15x102 g. This is also very useful to make very small or large numbers easier to read, e.g. 0.00000015 = 1.5x10-7.

This last example also shows us that the number of leading zeros does not seem to add important information as the leading zeroes may be omitted. In fact, the number of digits after the decimal point is only indicating the magnitude of an effect on the scale of the chosen unit, but not its precision. For example, 1 gram can be also expressed as 0.001 kg, 0.000001 tons, or 0.5x10-33 solar mass. However, “1 gram” does not become more “precise” by adding several digits before the first non-zero digit; it still remains a gram.

Thus, the relevant digits are the significant digits. The basic rules for identification of significant digits are:

1. Any non-zero digit is significant.
2. Leading zeroes prior to the first non-zero digit are never significant, even after the decimal point.
3. Trailing zeroes are significant if they are written down. The last digit indicates the rounding place of the number. For example, 82.00 kg means 81.995-82.004, but not 81.5-82.4 kg. Any non-significant trailing zeroes are omitted.

Thus, 1.00 has three significant digits but 0.01 has only one and 0.10 has two.

After having learned about accuracy in presenting numbers and measurements, we take a closer look at handling of calculated numbers that have been derived from measurements with different levels of accuracy.

Calculations with different levels of accuracy

For calculations with numbers of different levels of accuracy, there are two basic rules:

1. For addition and subtraction, the result of the calculation has as many digits following the decimal point as the input with the lowest number of digits after the decimal point had. For example, a human weighing 81.5 kg carrying an apple weighing 0.15 kg has a calculated total weight of 81.5+0.15 = 81.65. But we would have to write 81.7 kg because the result is rounded to the lowest number of digits any of these numbers had.

2. For multiplications and divisions, the rule is similar but refers to the significant digits: the result of a calculation has the lowest number of significant digits involved. Natural numbers are considered having an infinite number of digits, e.g. 2 apples are exactly 2.00000… . Thus, if calculating the average price per kg of apples from the price of a single apple, we would calculate: 0.49 € per 1 apple weighing 152 g => 0.49/(1x152) €/g = 0.32 €/kg. For the result, we provided the lowest number of significant digits involved, which were the two digits of the price.

Accuracy needed vs. accuracy possible
For presentations of numbers, the level of digits theoretically possible may not be useful or even meaningless. Thus, it also has to be adapted according to common sense and realistic needs. For example, human body weight can be determined exactly on a gram scale, e.g. 87,654 g. But whenever a human being eats, drinks, sweats, or visits a restroom, this weight is going to change significantly. Thus, no matter what precision the measurement of weight might have: providing the (average) weight of a human on a kilogram scale is usually the most reasonable option.

Thus, we have to take into account two main criteria when deciding about precision of numbers: The correct mathematical level of accuracy and the reasonable level of accuracy according to the suggested “use” of a number.

Summary
The accuracy of numbers needs to be properly thought through. The number of digits does not always make sense to the reader but may imply a misleading message, if too many digits are provided, indicating accuracy which is not really there. In conclusion, accuracy should be always presented …accurately.



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