### Measuring Underlies Significant Figures Human-built Instruments Always have Error

At the most basic level, chemistry (indeed all of science) depends upon experimentation; experimentation in turn requires numerical measurements. And measurements are always taken from instruments made by other human beings.

1) Examples we will study include the metric ruler, the thermometer, the graduated cylinder, and the triple-beam balance.

2) Because of the involvement of human beings, NO measurement is exact; some error is always involved. This means that every answer in science has some uncertainty associated with it. We might be fairly confident we have the correct answer, but we can never be 100% certain we have the EXACT correct answer.

3) Measurements always have two parts - a numerical part (sometimes called a factor) and a dimension (sometimes called a label or a unit). The reason for this is that we are measuring quantities - length, elapsed time, temperature, mass, etc. Not only do we have to tell how much there is, but we have to tell how much of what.

In a mathematics class, units are inconsistently used. This is because much of mathematics discusses the relationships between pure numbers, not the use of a number which describes an amount of something. Many ChemTeam students have the unfortunate tendancy to see units are unnecessary. THEY ARE NOT.

Measuring gives significance (or meaning) to each digit in the number produced. This concept of significance, of what is and what is not significant is VERY IMPORTANT. Especially the "what is not" portion. Pay close attention to the examples presented. The concept of significant figures (or significant digits) is important and will play a role in almost every unit studied by the ChemTeam.

A measurement can be defined as the comparison of the dimensions of an object to some standard.

The dimensions of an object refer to some property the object possesses. Examples include mass, length, area, density, and electrical charge. Dimensions are often called units.

For example, the meter is the standard unit of length in science. It was first defined as one ten-millionth of the distance from the equator to the pole. Then, a standard meter was made out of a platnium-iridium alloy and kept in a carefully controlled environment in Paris. The third definition was the distance of a certain number of wave crests of a certain wavelength in the emission spectrum of krypton. The most recent definition of the meter is the length of the path travelled by light in a vacuum during a time interval of 1/299,792,458 of a second.

We will just use a ruler, thank you very much!

The metric ruler will be the first example of a measuring device. The whole numbers in the images represent centimeters. The divisions are tenths of a centimeter, otherwise called millimeters. An arrow will represent the end of an object being measured. Zero is always to the left.

Example #1: 1) We know for sure the object is more than 2 cm, but less than 3 cm.

2) We know for sure the object is more than 0.8 cm, but less than 0.9 cm.

How do we know these two things?

Look where the arrow is, it is to the right of 2, but short of 3. It is to the right of 0.8, but short of 0.9. So, I can say the object is more than 2.8 cm, but less than 2.9 cm. We can say this with complete confidence because of the markings on the ruler.

Can I say anything more about the length?

1) Look at the gap between the 0.8 and 0.9 cm, where the arrow is and, mentally, divide that gap into 10 equal divisions.

2) Estimate how many tenths to the right the arrow is from the 0.8 cm.

Let us say your answer is two-tenths. We then say the object's length as 2.82 cm. The first two digits are 100% certain, but the last, since it was estimated, has some error in it. But all three digits are significant.

This issue of estimation is important. Experience tells us that the human mind is capable of dividing a short distance into tenths with acceptable reliability. However, there is error built in and it cannot be escaped. Since the reliability is acceptable, we say the digit is significant, even with the built-in error.

However, the process stops there. ONLY ONE estimated digit is allowed to be significant.

Make sure to always include the unit with the number. In the ChemTeam classroom, the coach says "2.82 what? Apples?" when the units are dropped in an oral answer.

Example #2: What length is indicated by the arrow?

1) More than 4 cm, but less than 5 cm.

2) More than 0.5 cm, but less than 0.6 cm.

Correct answer = 4.50 cm. The arrow is pointing directly at the mark and is neither to the left nor to the right of it. At least, according to the ChemTeam.

Notice that whatever the smallest division in your scale is, you can always estimate to the next decimal place after. In this case, the smallest division is in the tenth place, so we can estimate to the 0.01 place.

Be aware that there is some error, some uncertainity in the last digit of 4.50. While one should make an effort to estimate as carefully as possible, there is still some room for error.

The rule about uncertain digits is that there can be one and only one estimated or uncertain digit in a measurement. It is always the last digit in the measurement.

Here are two more examples of centimeter rulers. Decide what length is being shown, then click the link for the answer.

Example #3: Check Answer to Example #3

Example #4: Check Answer to Example #4

Celsius thermometers used in high school typically have a scale with only whole numbers marked on the scale. The gap between each whole number does not have any register marks for tenths.

Example #1: Answer = 15.0 °C. Since the line stops exactly on the 15 line and DOES NOT go any farther, we estimate that it has gone zero-tenths of the way from 15 to 16. We are allowed to include the tenth degree value and have it be considered significant. Remember that when you use thermometers in chemistry experiments. Many ChemTeam students have left off the `point zero' and get deducted for it.

Example #2: The indicated temperature is 28.5 °C.

Remember to mentally divide up the gap between 28 and 29, then make your best estimate of how many tenths are covered by the mark.

The ChemTeam's best estimate was 0.5, but yours might have been a bit different. 0.4 and 0.6 are both acceptable.

You might be interested to learn that sometimes magnifying glasses are used to make the scale bigger. This aids in the estimation process.  Here are two more examples of Celsius thermometers. Example #3 is to the left and example #4 is to the right. Decide what temperature is being shown, then click the link for the answer.

The graduated cylinder is another common component found in the chemistry laboratory. When reading cylinders made of glass, the water forms a meniscus. This is the downward-curved surface of the water. (Mercury forms an upward-curved meniscus.) Readings should always be made to the bottom point of the meniscus.  What are the volumes indicated by these two graduated cylinders, first the left one (example #1) and then the right one (example #2)? Keep in mind that you are allowed to estimate ONE decimal place PAST the smallest scale division.

The left graduate (example #1)indicates a volume of 30.0 mL, since you can estimate one digit BEYOND the smallest division. The smallest division is in the one's place, so that means you can estimate the tenth's place.

The right graduate (example #2)indicates a volume of 4.28 mL (according to the ChemTeam). The smallest division is the tenth place, so we are allowed to estimate to the hundredth place. Here are two other graduated cylinder examples: Answer to example #3 Answer to example #4

A short bridge between this measurement lesson and the next major lesson (significant figure rules) is called "From Measurement to Significance." You may wish to examine it before continuing on.