CHAPTER 2 –MEASUREMENT NOTES
I. How to use your calculator – Individual instruction in class
HINT: When performing scientific notation, always use the EE or EXP keys! Find one of these on your calculator.
II. Scientific Notation
A. A. Used to represent either very large or very small numbers in science.
B. B. Uses a number 1 < 10 multiplied by powers of 10 (exponents)
C. C. Numbers larger than 1 are represented with positive exponents
D. D. Numbers less than one are represented with negative exponents.
E. E. Form is as follows:
a. 1) only 1 number to the left of any decimal
b. 2) use “X” as the multiply symbol. Do NOT use a DOT!
F. F. Examples:
a. 1.) 5.6 x 104
b. 2.) 9.8 x 10-5
G. G. Changing “long form” numbers to scientific notation:
a. 1.) Move the decimal until there is just one number to the left of the decimal.
b. 2.) Count the number of places you moved. That is the exponent to use for the power of 10.
c. 3.) If the “long form” number was less than 1, make the exponent negative.
d. 4.) If the long form” number was 1 or greater, make the exponent positive.
e. 5.) The exponent of “0” is equal to multiplying by 1.
H. H. Examples:
a. 1.) 1,588,293 = 1.588293 x 106
b. 2.) 0.4976 = 4.976 x 10-1p
c. 3.) 19 = 1.9 x 10 0
I. I. Changing scientific notation numbers to “long form:”
a. 1.) Move the decimal the number of places specified by the exponent.
b. 2.) Move the decimal to the right if the exponent is positive (number is 1 or greater).
c. 3.) Move the decimal to the left if the exponent is negative (number is less than 1.)
J. Examples:
a. 1. 8.4 x 10 7 = 84,000 000)
b. 2. 3.923 x 10 –4 = 0. 0003923
K. REMEMBER!! When entering problems in your calculator in scientific notation, NEVER actually type in “10 x…..” This will always cause error. Your EXP or EE key does this for you!
III. Metric (SI) System
A. In science, we always use the metric system, or as it is called, the SI system, for measurements.
1. SI is short for “Le Systeme Internationale d’Unites”
2. Used universally except in the United States and some undeveloped nations!
B. How it works
1. Uses base units with values of “1” of something (meter, second, gram, Liter, mole) and then uses powers of 10 to indicate more or less than 1. There is a base unit chart on page 34. You are responsible for learning the following base units: length, mass, time, temperature, and amount of substance.
2. Each power of ten has a name associated with it, which is used as a prefix to denote more or less than 1 of that particular item.
3. There is a chart on page 35 that contains the metric prefixes and their values. You are responsible for learning this chart! This means memorizing the prefixes and their values.
C. Examples
1. Lets say you have 1 gram of copper. Then you acquire 9 more grams, for a total of 10 grams. That is one power of ten. So that could be written as either 10 g OR 1 dekagram. Deka is the prefix that means “10 x” so, that is 1 dekagram (dag).
2. Now we decrease the amount of copper we have. Let’s say we have only one-tenth of a gram. We have decreased the amount by 1 power of ten. So we could write it 0.1 gram, OR we could say “1 decigram.” Deci (d) is the prefix that means 1/10 or 0.1 of an item. So it is correctly written as 1 dg.
3. If we have one million (1,000,000) seconds, we can say we have 1Megasecond.
4. If we have only one millionth of a second (1/1,000,000), we have a microsecond.
D. Converting in the metric system
1. To convert within the metric system, you have to keep up with the number of decimal places you must move. You can do this in your head, visualizing the metric staircase, or you can use your calculator (preferred).
2. For instance, if you have 315 grams of copper, and want to convert that to milligrams (mg), you would have to know that “milli” means 1/1000 of an item. That is 3 powers of ten. So I have to move 3 decimal places to the right. I could have just counted those in my head, or I could have multiplied 315 by 1 x 103 . Notice I did not multiply by 1 x 10 -3 . We would use the absolute value of the power of ten. We could correctly write this as “315,000 mg.”
3. Why did we multiply and not divide in the example in #2? It is because a gram is much larger than a milligram. It takes lots of little things to make a big thing! So there are many more tiny milligrams in the number, and you multiply to make the number larger.
4. Let’s take an opposite example. Using the same 315 grams of copper, let’s figure out how many megagrams (Mg) that would be. A “mega” something is one MILLION times larger than 1 of something. That is 1 x 10 6 ! So now, the opposite reasoning is true. I only need a small portion of that “big something” to represent the smaller quantity. So this time, I will either move my decimal 6 places to the LEFT or DIVIDE by 1 x 10 6 to get my smaller value. Dividing 315 by 1 x 10 6 gives me 0.000315 megagrams (0.000315 Mg).
5. What if you want to convert a quantity NOT from the “base” position? You still need to know how many powers of ten you need to move the decimal. There are a couple of ways to do this. You could look at a metric staircase and actually count the “steps” between the two quantities. Then you could move the decimal that many places. You would move the decimal to the right if you were going from a larger to a smaller prefix, and you would move it to the left if you were moving from a smaller to a larger prefix.
However, it is still preferred that you use your calculator to do this. Here are two rules for determining the powers of ten between them. 1) If you are “crossing the base” value (for instance moving between milli and kilo) you will take the absolute value of the exponents and add them together to get the power of ten. For the example I just gave, moving from milli to kilo, I would know that milli is 10 –3 and kilo is 10 3 . Taking the absolute value of both numbers, I would add 3 +3 = 6. There is a difference of 6 powers of 10. So then you could either move your decimal that many places or use the calculator to divide it out. 2) If you are NOT “crossing the base” (for instance converting from deci to micro) you use the same procedure of taking the absolute value of the two powers of ten, but instead of adding the exponents, you SUBTRACT them. For our example, deci is 10 –1 and micro is 10 –6 . Subtract 6-1=5. So there are 5 powers of ten between them. Now you can either move the decimal by that amount or use your calculator to multiply by the power of ten.
IV. Significant Figures/ Accuracy and Precision
A. Accuracy vs. Precision
1. accuracy=how close a measurement is to an accepted value.
Examples: water boils at 1000 Celsius. If you boil water and get a measurement of 99.90 Celsius, you are accurate, because you are extremely close to the accepted value. If you get a measurement of 75.00 you are NOT accurate, because it is far away from the accepted value.
2. Precision= the ability to repeat measurements that are very close to one another, but not necessarily close to an accepted value. For instance, if I am measuring a quantity of iron oxide, and I take 3 readings of 5.6 g, 5.59 g, and 5.61 g, I am precise, because those 3 measurements are very close to one another. However, if I took 3 readings that were 5.6, 4.2, and 6.0, those are NOT precise, because they are widely scattered.
3. Generally, good precision usually means the experimenter has good technique. However, it can also indicate that the measuring instrument is in good working order. Poor precision can result either from experimenter’s error or bad equipment. However, bad equipment usually affects accuracy rather than precision.
4. The illustration below shows a graphic representation of measurements with accuracy and/or precision.
B. Significant Figures (Sig Figs) or Significant digits
1. In science, we acknowledge that measuring instruments can have inaccuracies. If we use inaccurate equipment, we might get inaccurate readings. To minimize the effects of this, science uses significant figures.
2. In any measurement, science allows all absolutely known numbers plus one “uncertain” number. This “uncertain” number is the last number in the measurement. It is considered an estimation.
3. Depending on the measuring instrument you use, you may have more significant figures or less. The more precise your measuring instrument, the more significant figures you can place in your measurement.
4. For example, beakers are not used for measuring in chemistry because there are very few “measurement lines” – called graduations -- on them. However, a graduated cylinder has many “measurement lines” – graduations – and is therefore more precise. We can get many more significant figures from a graduated cylinder than a beaker. Look at the comparisons below:
The beaker in this picture only shows a graduation every 25 mL. So we cannot be very precise in our measurements. However, the graduated cylinder has a graduation for every 1 mL. We can be much more precise with our measurements in the graduated cylinder.
And even among graduated cylinders (or other devices such as rulers) there are differing levels of precision, and therefore, significant figures. Each of these rulers below has a different number of graduations. The one on top has the least, and is the least precise. The one on bottom has the most graduations and is therefore most precise. We can get more significant figures from the bottom reading.
C. Rules for Significant Figures -- General
1. All non-zero numbers in a measurement are significant. Always. In a measurement of 59.4576 there are 6 significant figures (SF). In a measurement of 0.067 there are 2 SF.
2. For zeroes, there are 3 rules:
a. Leading zeroes are NEVER significant! Example: 0.0056 = 2 SF. All the zeroes are leading zeroes. They are place holders and attention getters. They are NOT actually measured, but tell us the place value of the first measured number.
b. Captive, or Trapped zeroes are ALWAYS significant. We assume that we measured those zeroes on the way from one non-zero number to the next one. Example: 1,001 = 4 SF 10.055 = 5 SF
c. Trailing zeroes are SOMETIMES significant. The ARE significant if they follow a decimal OR have a decimal right after them. Example: 23,000. =5 SF 15.00 = 4 SF
Trailing zeroes are NOT significant if there is no decimal involved. Examples: 1,000 = 1 SF 700 = 1 SF
D. Rules for Significant Figures – math operations
1. For multiplication/division – your answer cannot have more SFs than the measurement in the problem with the least number of SF. It must be rounded to that number. Example: 126/3 – the pure math answer is 42. However, looking at the two numbers in the problem, we see that 126=3 SF and 3=1 SF. The number 3 is therefore the measurement with the least number of SF, so we MUST round our math answer to only show 1 SF. This is done by rounding 42 down to 40, showing NO decimal after the zero. Example: 7,345 x 0.25 – the pure math answer is 1,836.25. However, examining the two measurements involved, 7,345 = 4 SF and 0.25 = 2 SF. Therefore, our answer must only have TWO SF in it. This is done by rounding 1,836.25 to 2 SF 1,800 with no decimal after the zeroes. You could also write it in scientific notation 1.8 x 103 .
2.For addition and subtraction – the answer cannot have more numbers in it than the measurement whose last SF is furthest to the LEFT. This is actually based on place value rather than number of SF. I usually find it is easier to figure out this value by always adding or subtracting in a vertical column. In the examples below, I’ll do that and underline/bold the LAST SF.
Example:
125.56
50.1
+ 1.25
______
176.91
In this example, the pure math answer is 176.91. However, looking at the LAST SF of each measurement, you can see that the 1 in 50.1 is the last SF that is furthest to the left. It is sitting in the 10ths place. Therefore, our answer cannot go past the 10ths place! We would round it to 176.9.
Example:
77.35
- 11
______
66.35
In this example, the pure math answer is 66.35. However, you can see that the 2nd 1 in 11 is the last SF furthest to the left. It is in the one’s place. Therefore, our answer must be rounded to the ones place, or simply 66.
E. General hints for significant figures – 1) Numbers that are universally accepted as correct values (such as the value of pi) or conversion factors (such as 12 inches to 1 foot) are considered to have an infinite number of significant figures. They are IGNORED in measurement calculations. 2) When you have a problem to solve involving measurements, always use your initial given quantity to figure out the limit on your significant figures. 3) When doing a problem with mixed math (addition and then multiplication, for instance) we generally use the rules for multiplication/division.