# Introduction to Chemical Engineering Processes/Significant figures

## Importance of Significant Figures

Significant figures (also called significant digits) are an important part of scientific and mathematical calculations, and deals with the accuracy and precision of numbers. It is important to estimate uncertainty in the final result, and this is where significant figures become very important.

### Precision and Accuracy

Before discussing how to deal with significant figures one should discuss what precision and accuracy in relation to chemical experiments and engineering. Precision refers to the reproducibility of results and measurements in an experiment, while accuracy refers to how close the value is to the actual or true value. Results can be both precise and accurate, neither precise nor accurate, precise and not accurate, or vice versa. The validity of the results increases as they are more accurate and precise.

An useful analogy that helps distinguish the difference between accuracy and precision is the use of a target. The bullseye of the target represents the true value, while the holes made by each shot (each trial) represents the validity.

As the above images show, the first has a lot of holes (black spots) covering a small area. The small area represents a precise experiment, yet it seems that there is a faultiness within the experiment, most likely due to systematic error, rather than random error. The second image represents an accurate though imprecise experiment. The holes are near the bullseye, even "touching" or within, though the problem is that they are spread out. This could be due to random error, systematic error, or not being careful in measuring.

### Counting Significant Figures

There are three preliminary rules to counting significant. They deal with non-zero numbers, zeros, and exact numbers.

1) Non-zero numbers - all non-zero numbers are considered significant figures

2) Zeros - there are three different types of zeros

• leading zeros - zeros that precede digits - do not count as significant figures (example: .0002 has one significant figure)
• captive zeros - zeros that are "caught" between two digits - do count as significant figures (example: 101.205 has six significant figures)
• trailing zeros - zeros that are at the end of a string of numbers and zeros - only count if there is a decimal place (example: 100 has one significant figure, while 1.00 has three, as well as 100.)

3) Exact numbers - these are numbers not obtained by measurements, and are determined by counting. An example of this is if one counted the number of millimetres in a centimetre (10 - it is the definition of a millimetre), but another example would would be if you have 3 apples.

Example:
How many significant figures do the following numbers have? Assume none of them are exact numbers.


a) 4.2362 - all numbers, so five

b) 2.0 - zeros after a decimal point count, so two

c) 9900 - only two in this case, because there is no decimal point

d) .44205 - there is a "captive zero," which means it counts, so five

e) .05 - only the five counts, so one

f) 3.9400E9 - tricky one, but scientific notation helps make the zeros at the end noticeable; there are five

### The Parable of the Cement Block

People new to the field often question the importance of significant figures, but they have great practical importance, for they are a quick way to tell how accurate a number is. Including too many can not only make your numbers harder to read, it can also have serious negative consequences.

As an anecdote, consider two engineers who work for a construction company. They need to order cement bricks for a certain project. They have to build a wall that is 10 feet wide, and plan to lay the base with 30 bricks. The first engineer does not consider the importance of significant figures and calculates that the bricks need to be 0.3333 feet wide and the second does and reports the number as 0.33, figuring that a precision of $\pm 0.01ft$ (0.1 inches) would be precise enough for the work she was doing.

Now, when the cement company received the orders from the first engineer, they had a great deal of trouble. Their machines were precise but not so precise that they could consistently cut to within 0.0001 feet. However, after a good deal of trial and error and testing, and some waste from products that did not meet the specification, they finally machined all of the bricks that were needed. The other engineer's orders were much easier, and generated minimal waste.

When the engineers received the bills, they compared the bill for the services, and the first one was shocked at how expensive hers was. When they consulted with the company, the company explained the situation: they needed such a high precision for the first order that they required significant extra labor to meet the specification, as well as some extra material. Therefore it was much more costly to produce.

What is the point of this story? Significant figures matter. It is important to have a reasonable gauge of how accurate a number is so that you know not only what the number is but how much you can trust it and how limited it is. The engineer will have to make decisions about how precisely he or she needs to specify design specifications, and how precise measurement instruments (and control systems!) have to be. If you do not need 99.9999% purity then you probably don't need an expensive assay to detect generic impurities at a 0.0001% level (though the lab technicians will probably have to still test for heavy metals and such), and likewise you will not have to design nearly as large of a distillation column to achieve the separations necessary for such a high purity.

### Mathematical Operations and Significant Figures

Most likely at one point, the numbers obtained in one's measurements will be used within mathematical operations. What does one do if each number has a different amount of significant figures? If one adds 2.0 litres of liquid with 1.000252 litres, how much does one have afterwards? What would 2.45 times 223.5 get?

For addition and subtraction, the result has the same number of decimal places as the least precise measurement use in the calculation. This means that 112.420020 + 5.2105231 + 1.4 would have have a single decimal place but there can be any amount of numbers to the left of the decimal point (in this case the answer is 119.0).

For multiplication and division, the number that is the least precise measurement, or the number of digits. This means that 2.499 is more precise than 2.7, since the former has four digits while the latter has two. This means that 5.000 divided by 2.5 (both being measurements of some kind) would lead to an answer of 2.0.

### Rounding

So now you know how to pick which numbers to drop if there is a question about significant figures, but one also has to take into account rounding. Once one has decided which digit should be the last digit kept, one must decide whether to round up or down.

• If the number is greater than five (6 to 9), one rounds up - 1.36 becomes 1.4
• If the number is less than five (1 to 4), one rounds down - 1.34 becomes 1.3

What does one do when there is a five? There is a special case that deals with the number five, since, if you have not noticed, it is in the middle (between 1 and 9). Often in primary school one learns to just round up, but engineers tend to do something different, called unbiased rounding.

• If the number before the five is even, then one rounds down - 1.45 becomes 1.4
• If the number before the five is odd, then one rounds up - 1.55 becomes 1.6
• Another case is this: 1.4501, where the numbers after five are greater than zero, so one would round to 1.5

Note: Remember that rounding is generally done at the end of calculations, not before the calculations are made.

Why is this done? Engineers make many calculations that often matter, since time, money, etc. are being taken into account, it is best to make sure that the final results are not synthetic or untrue to what the actual value should be. This relates back to accuracy and precision.