U4735 Environmental Science for Policy Makers Recitation 1 Section Notes 1 If you notice any errors in this set of lecture notes, please let me know by email. Thanks. Exponents 10 2 =10 10 2 3 =2 2 2 In the last case 2 is the base, 3 is the exponent. Common bases are 2, 10, and e (= 2.71828 to five decimal places), which is a number to know and love with innumerable uses in mathematical and physical systems, particularly with rates of growth and decay (we ll talk about this in detail later). Note that exponentials such as 2 x get very large very quickly as x grows. Negative exponents: 10 2 = 1 = 1 =0.01. In units you will often see the same type of 10 2 10 10 thing: m/s = m s 1. When multiplying exponents with the same base, add the exponents (when dividing, subtract): 10 2 10 3 =10 5 10 x /10 y =10 x y Exponents with different bases cannot be multiplied directly as above. When exponents are raised to higher powers, multiply exponents: (10 2 ) 3 =10 6. 10 3.34 = 2187.76. This is completely legitimate. However, you will need to use a calculator for this sort of thing. Significant Figures Numbers, like physical equations (especially of complex systems like the earth), are only approximations. China has 1.2 billion people. Obviously this isn t exact. Likewise the USA has 260 million people, again this isn t exact. The US conducts a census every ten years, in which they get some exact figure based on the number of respondents, and from this figure the government can estimate the actual number, but given margins of error, they only trust the gross estimate to two or three (in this case two) significant figures (the rest are zeros). Here are a few other examples. To two significant figures 10 3.34 = 2200, to four 10 3.34 = 2188, to six 10 3.34 = 2187.76,etc. When multiplying or dividing, the number with the greatest uncertainty (the fewest significant figures) determines the significant figures in the answer. Thus, if the people of China consume on average 103.5 pounds of chicken per person per year, the total amount of chicken consumed
U4735 Environmental Science for Policy Makers Recitation 2 annually is 120 billion pounds, not 124.2 billion pounds. Again, the population of China was given as 1.2 billion, so the answer should only have two significant figures. Similarly, 625003 25 = 25000. When adding or subtracting, the trick is to line up the decimal points. For example, 9.4 + 3.14 = 12.5. 12.54 would be incorrect because 10.4 lacks the hundreths place. Note that the rule for multiplication/division does not apply i.e. reducing to two significant digits since both numbers have a tenths place. Similarly, 625003 25 = 624978. Rounding and Orders of Magnitude Round 0-4 down, 5-9 up, For example, 2.34 to two significant digits is 2.3, whereas, 1.86 is 1.9. Orders of magnitude are gross ballpark estimates based on powers of ten i.e., 1, 10, 100, 1000, etc. (in exponential form these equal 10 0, 10 1, 10 2, 10 3 ). The Harte textbook states that the rule of thumb is that a number between 0.3 and 3 is order of magnitude 1, between 3 and 30 is order of magnitude 10, etc. You will hear things like, There are on the order of 1 billion people in China. This just means that the exact number is not important, just its rough size. Scientists often speak in orders of magnitude because they can t recall the exact number! Scientific Notation Easy, compact way of writing and working with large and small numbers using powers of ten. For example, 1.2 10 9 is simply 1.2 1, 000, 000, 000, or 1,200,000,000, the population of China. Similarly, 260, 000, 000 = 2.6 10 8. 10 6 =0.000001 10 0 =1 10 1 =10 Note that 10 10 2 =1 10 3 =10 3 = 1000. 1 10 3 has one significant figure, 1.00 10 3 has three, 1.0000 10 3 has five. Not only is scientific notation more compact, it is generally easier to work with scientific notation. To multiply (or divide) simply multiply (or divide) the lead numbers and add (or subtract) the exponents. So, 1.2 10 9 4.2 10 5 =5.0 10 14. Note that it is not equal to 5.04 10 14. Similarly 1.2 10 9 (4.2 10 5 )=.29 10 4,or2.9 10 3. Scientific notation specifies the number of significant figures, hence the last problem is not 2.8571428571 10 3.
U4735 Environmental Science for Policy Makers Recitation 3 When adding and subtracting be careful that you compare like terms. 1000 110 = 890 just as 1.00 10 3 1.1 10 2 =8.9 10 2. Units Understanding and working with units is of critical importance in this course (and in life). If you are alert and careful about units, you can solve many problems with little or no thought. For example, suppose someone gives you the following problem: Blah blah blah blah 10 chickens/person/year. Blah blah blah 1.2 billion people blah blah blah. Blah blah blah chickens/year? Even without understanding it, you can guess that since the question is about chickens/year, you should multiply chickens/person/year by people like so: 10 chickens/person/year 1.2 10 9 people = 1.2 10 10 chickens/year This is also an example of dimensional analysis: applying the mathematical operations to the units as well as the numbers to ensure that the answer has the correct units. It is very important to pay close attention to units performing any calculation because sometimes conversion of units is neccessary. Show your work, write it every time. You are going to have to use a lot of crazy units don t shy away from them, just work to organize, convert and cancel. Dimensional Analysis is thorough and reliable. Distance, mass and time are the fundamental units. The SI system (today s standard for science) uses MKS units, these are the meter, the kilogram and the second. In the past, many people used the CGS system (centimeter, gram, second) and often you will see confusing units based on this system. It is best to stick to the MKS system whenever possible. Most other units (derived units) can be expressed in terms of the fundamental units. Doing so is often useful for converting units or checking the equivalence of a set of units. For example, pressure, a force per unit area, can be expressed as Newtons (MKS unit of force) per meter squared. This, in turn, can be written in fundamental units as a kilogram per meter per second squared: 1pascal(Pa)=1Nm 2 =1kgm 1 s 2 Non-MKS units cannot be handled in this way. For example: 1joule(J)=1kgm 2 s 2 =10 7 ergs = 0.2389 calories (cal) = 2.389 10 4 kcals Force, energy, power, pressure, work, heat; these are all derived units. Don t be bashful look through the appendix in Spherical Cow on units. Take the time to see how the units interelate. Sometimes you can learn a lot about what a number means by studying it s units. For example, force is measured in newtons: kg m / s 2. Energy is measured in joules: kg m 2 /s 2. The first is
U4735 Environmental Science for Policy Makers Recitation 4 the acceleration of a mass, the second is an amount of work, a force working over a distance. A joule equals a newton-meter. It is also worth noting that one way to check that an equation that you are working with is correct is to confirm that the fundamental units of each of the terms of the equation is the same. For example, 3x apples apples =9 oranges oranges is OK. But if you see, 3x apples = 9 apples + 9 oranges oranges then it is likely that there is a mistake in the equation. Now for some more practice with dimensional analysis: 30 inches is how many feet? How many meters? 30 inches 1foot =2.5 feet 12 inches 2.5 feet 1 meter =0.76 meters 3.281 feet Follow the units in combination (this is known as dimensional analysis). If there are 5 billion chickens in China, a country with 1.2 million people, how many chickens per person are there? The question asks for chickens per person. These are your goal units. We can easily calculate 5/1.2 =4 chickens. If there are 10 billion rats in China and 50 fleas per rat how many fleas per person person are there (assuming all the fleas are on the rats)? Here the goal is fleas per person: Fleas/rat rats/person = fleas/person. 50 fleas/rat 4 rats/person = 200 fleas/person. Suppose some of the fleas have bubonic plague 1 in a million. And suppose the biting rate of humans by these fleas is one in a ten thousand that is, one in ten thousand of these fleas will actually bite a human. What is the biting rate by fleas with bubonic plague? We need infectious bites per person: Infectious bite/bites, total bites of humans/flea x fleas/person. 1 infectious bite/(1 10 6 ) bites 1 bite of human/(1 10 4 ) fleas 200 fleas/person = 2 10 8 infectious bites (of a human)/person. A small number. Suppose the average flea bites ten times a day. How many bites a year is that? 3650 bites/flea-year.
U4735 Environmental Science for Policy Makers Recitation 5 How many infectious bites/person year? Assume one in ten thousand bites is of a human. 3650 bites/flea-year 1 bite of human/(1 10 4 ) bites 1 infectious bite/(1 10 6 ) bites 200 fleas/person = 7 10 5 infectious bites of a human/person-year. Finally, how many infectious bites per year in China? 7 10 5 infectious bites of a human/person-year 1.2 10 9 people = 8 10 4 infectious bites of a human/year. Not a small number! A final but very important note about units: remember that the conversion cm 3 (= cc, cubic centimeter) to m 3 requires the conversion of centimeters to meters three times: 1cm 3 The same caution applies to converting areas. Mathematical functions 1m 100 cm 1m 100 cm 1m 100 cm =1 10 6 m 3 A function is a mathematical expression that takes an input (called the arguement) and produces an output (called the result). Several examples are: a(x) =10x +5 b(x) =2.2x 3 c(y) =e 5y Each of these takes an arguement (x or y) and produces a result (a, b or c). f(x) is a linear function of x because x is not raised to a power. b(x) and c(y) are non-linear functions. You will hear scientists use the words non-linear frequently. Mathematically, the meaning of this is very simple. The connotation is that a quantity that is non-linear may experience disproportionately large or sudden changes for small changes in something that it depends on. Linear vs. non-linear functions An attribute of a linear function is that a plot of the output against the input forms a straight line; thus if you know what happens when you double in input, you also know by extrapolation what happens if you quadruple in input. For example, here is a linear function where x is the input and f(x) is the output: f(x) =2x
U4735 Environmental Science for Policy Makers Recitation 6 You can see that x =1gives f(x) =2and x =2 1 gives f(x) =4. If you plot these two points on a graph, you will find that the line they form contains all the other points produced by the function f(x). Thus it is easy to extrpolate away from x =2, simply extend the line and read the answer off the graph. Another way to state that a function is linear is to say that its slope is a constant. In mathematical terms this means that its derivative is constant. A non-linear function has a slope that is not a constant. The world of non-linear functions, however, is vast. Take, for example, these three non-linear functions: g(x) =x 3, h(x) =cosx, k(x) =e x. Of these three, the first and the third are increasing functions of x. Asx gets bigger, the function gets bigger too, and it does so much faster than the rate of increase of x! The second is oscillatory. That means that as x gets bigger, h(x) bounces back and forth between a maximum value, 1 and a minimum value, 1.