Numbers and Measurement
- understand the difference between precision and accuracy
- determine the number of significant figures in a measurement or estimate
- convert numbers between standard form and scientific notation
The scientific approach to understanding our universe involves, to various degrees, the following activities:
- making observations; being surprised; wondering
- analyzing and classifying observational data; recognizing laws
- formulating theories to explain relationships in data sets
- deducing hypotheses from theories and observations; making predictions
- designing experiments to explore consequences of hypotheses
- revising theories on the basis of new observations
- using theories to guide expansion of understanding
Science is a human endeavor; this is not a recipe to be followed in sequence, but a complex set of logical relationships. Science, as a pathway to understanding, is defined by several important qualities:
- evidence is valued based on experience, reproducibility, and mathematical reasoning
- existing theories are open to revision on the basis of new evidence and reasoning
Observations, whether they be of the natural world or of controlled experiments, provide the basis for understanding. Observations can be qualitative (What is it like?) or quantitative (How much is there?).
Quantitative observations (measurements and estimates) are made with a degree of precision that is represented by the value of the measurement and its uncertainty. For example, suppose we want to measure the length of an object using a ruler marked with tenth of an inch divisions. We could record the measurement using digits for the whole inches, a digit for the last tenth mark that the object extended beyond, and one more digit representing our estimate, to the nearest tenth of a division (i.e. hundredth of an inch), of where our object ends. We might measure a credit card as having a width of 3.35 inches, for example, where the '5' represents our best guess of where the credit card ends between the 3.3 and 3.4 inch marks. Other measurers using the same ruler might record 3.34 or 3.36 inches as the width.
Precision is represented by the number of significant figures in the value, and uncertainty is assumed to be ±1 in the last significant figure, the digit that represents the estimate to the nearest tenth of a division between the graduations on our measuring device.
Many measuring devices in current use have digital displays instead of scales and pointers. The only difference is that the estimate of the last digit is done by the electronic circuitry rather than the operator. We can still assume that a digital readout is uncertain by ±1 in the last digit.
Accuracy in a scientific measurement is not the same as precision. Accuracy refers to how close the measurement is to the 'true' value. This will depend on how well our measuring device is calibrated, how well it is used, and how appropriate it is for the particular measurement being made.
A more sophisticated treatment of reproducibility and estimation of errors in measurements will be encountered in Quantitative Analysis, CHEM 3211/3201.
A particular chemistry instructor weighs 154 lb and has a waist measurement of 32.5 inches. The weight was determined on a digital bathroom scale and has an uncertainty of ±1 lb, and the waist was measured with a tape and has an uncertainty of ±0.1 inch. Both measurements have three significant figures (3 sf.).
It is important to know the number of significant figures in a measurement so that when we use it in a calculation we can properly estimate the precision of our answer.
Rules for counting significant figures in a value (the examples given have three significant figures):
- all non-zero digits are significant : 32.5
- zeroes between non-zero digits are significant : 109
- trailing zeroes to the right of the decimal point are significant: 8.70
- leading zeroes used to hold the decimal point are not significant: 0.00246
Values of measurements are rational numbers expressed to an appropriate number of significant figures except in one special case: integers (counting numbers) are infinitely precise.
Estimates of large numbers may look like integers, but some of the trailing zeroes may be included only as placeholders. For example, if the crowd at Neyland Stadium for the Vols-Gators football game was reported to be 105,000, we can't tell how many significant figures there are in this number unless we know it was estimated to the nearest thousand (3 sf.), nearest hundred (4 sf.), nearest ten (5 sf.), or counted exactly.
Ambiguities of this kind are removed by using scientific notation.
Scientific notation is a specific form of exponential notation in which a measurement is expressed as a
number between 1 and 10 (the coefficient) multiplied by 10 (the
base) to a power (the
exponent). Examples include numbers such as 1.2 x 105,
the average number of hair follicles on a human head, and
3 x 10-3, the average diameter of a human hair measured in inches.
Pay particular attention to the exponent when a number is presented in scientific notation. This tells you the order of magnitude of the value you are looking at. A change of 1 in the exponent is a factor of 10 in the value! Note that large numbers have positive exponents, and numbers between 0 and 1 (fractions) have negative exponents.
It's important to remember that ten to a positive integer power, n, is equivalent to 10 multiplied by itself n times: e.g. 104 = 10 x 10 x 10 x 10 = 10000. Other important meanings of exponents include:
- 100 = 1
- 10-1 = 1/10 = 0.1
- 10-n = 1/10n For example, 10-2 = 1/102 = 1/100 = 0.01
So 1.2 x 105 = 1.2 x 100000 = 120000 (one hundred and twenty thousand) and 3 x 10-3 = 3/1000 (three thousandths). Given that the exponents in these examples were relatively small, the numbers written in standard form are relatively easy to recognize and comprehend, but consider the number 5 x 1013, the average number of cells in the human body. In standard form, 50000000000000 is not easy to comprehend without carefully counting the zeroes, and it is easy to make a mistake when copying the number. In science we frequently encounter numbers with exponents up to ±25, and sometimes even larger, so it is important to be able to use scientific notation, and to comprehend the magnitude of a number presented in this form. I repeat, a change of 1 in the exponent is a factor of 10 in the value! Numbers between 0.1 and 1000 are often not presented in scientific notation, although they can be.
Not only is scientific notation useful for rapidly judging the magnitude of very large and very small numbers, it also removes the ambiguity associated with estimates of large numbers written with trailing zeroes. For example, if we knew that the estimate of 105,000 for the crowd at Neyland stadium was made to the nearest hundred, we would represent it as 1.050 x 105 people (4 sf. - the trailing zero is significant).
To convert a number to scientific notation, move the decimal point to the right of the first significant digit. The number of places moved will be the exponent, with a negative sign for movement to the right. When converting a number to scientific notation, neither the value nor the precision change. For example:
- 1024.0 is 1.0240 x 103 (there are 5 sf. in both representations)
- 0.000000832 is 8.32 x 10-7 (there are 3 sf. in both representations)
- 1,590,000,000 is assumed to be 1.59 x 109 (3 sf. in both representations) in the absence of further information that the value was measured more precisely.
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