Objectives

To review the essentials of mathematics: algebra scientific notation exponentiation (powers of 10)

To review the essentials of math and the scientific process as they relate to astronomy: scaling uncertainties in measurements statistics scientific method

Materials

• 15-cm Ruler (we measure using the metric system)

• Scientific Calculator (borrow one if necessary)

Background and Theory

Mathematics is the language of the Universe. Imagine learning English and not being able to use any vowels:

or given a coded message without a key:

Similarly, astronomy cannot be taught adequately without using the proper language. Some concepts can be explained best and most completely by the use of a formula. You may be surprised to see how much information is contained in a simple algebraic equation; for example, Einstein's famous equation: E = mc².The math involved is at the level needed to be accepted into this great university. Please do not worry about the math used in this course; you will always have your fellow students and your instructor to help.

Procedure

Each of the following steps of your review starts with a sample problem that is similar to math you will encounter this quarter. Work the problem given. If you cannot work the problem quickly, without hesitation, then take the review link offered. After your review, come back to this page and try to work the next problem. Check your answer. Continue this procedure for the complete math section. Then, move onto the introduction to astronomical and scientific methods.

1. Review of Mathematics

   1. Algebra
      • Solve for y:          
        Answer y = sqrt(x/4)

        Review of Algebra

      • Solve for m (mass):     
        Answer m = E/c^2
        Understanding: In the equation above, E represents the energy; m, the mass; and c the speed of light -- approximately 300,000 km/sec. Why can a whole lot of energy be obtained from a tiny bit of mass?
        
        Answer Because the energy is the product of the mass and a very large number: the speed of light squared is 90,000,000,000!
   2. Scientific Notation
      • Write the following in scientific notation:
        3,042; 231.4; 0.00012; 0.0000000000667
        
        Answer 3.042 x 10^3; 2.314 x 10^2; 1.2 x 10^(-4); 6.67 x 10^(-11)

        Review of Scientific Notation (Note the sections on how to use your calculator!)

      • Convert the following numbers from scientific notation:
        4.2 x 10⁽¹⁴⁾; 4 x 10^(-11)
        
        Answer 420,000,000,000,000 (the mass, in kilograms, of Pan, a moon of Saturn) 0.00000000004 (the approximate wavelength, in meters, of blue light)
      
      
      
   3. Exponentiation (powers of 10)

      Work the following problems using a scientific calculator. If you do not own a calculator that uses scientific notation, then find someone who does and borrow it.

      Multiply: 3.1 x 10⁷ by 3 x 10⁵
      Divide: 1.496 x 10¹¹ by 5.2 x 10^-3
      Simplify: (3 x 10⁸)²
      Simplify:
      
      Answers 9.3 x 10^(12);   2.9 x 10^(13) -- two significant digits;   9 x 10^(16);   4000^2 or 16,000,000

      Exponentiation (powers of 10)

   4. Scaling and Scale Factors
      

      
      If you've ever read a map, used the scale to determine how many miles were represented by an inch, and then used a measured number of inches to determine how far you were going to travel, then you have all the skills needed to start your career as an astronomer! Think about how the cartographers calculated the map scale in this case: someone had to measure the actual number of meters or yards on the ground and then, perhaps using a ruler, translate that to inches on the map.

      How far, in kilometers, is it from Roosevelt Way NE to 25th Avenue NE along NE 45th Street?

      Answer Approximately 1.27 km

      You live just north of NE 45th Street on 17th Avenue NE. You leave from your house to go to sections in the A-wing of the Physics and Astronomy Building. You are late and so you are walking briskly, covering approximately 1 meter per second. How far is it to the A-wing (approximately at the corner of NE Pacific Street and 15th Avenue NE) in meters?

      Answer 980 meters

   5. Measurement Errors and Uncertainties

      The term "error" signifies a deviation of the result from some "true" value. Often we cannot know what the true value is, and we can determine only estimates of the errors inherent in the experiment.

      If we repeat the experiment, the results may differ from those of the first attempt. We can express this difference as a discrepancy between the two results. The fact that a discrepancy arises is due to the fact that we can determine our results only within a given uncertainty or error. The more precise our measuring tool, the more precise our answer; the more accurate our input data, the more accurate our results. But, we will always have some uncertainty.

      In science, we can never hope to have absolute precision, absolute accuracy. (Thought question: What is the difference between a "precise" measurement and an "accurate" measurement?) It is perfectly acceptable to have large uncertainties, as long as we let the reader know our estimate of those uncertainties so that an intelligent evaluation of the results can be made. As scientists, we strive to perfect our experiments and observations in order to improve the precision and accuracy of our results.

      For example: How accurate do you feel your scale factor was (that is, how many meters or kilometers are represented by each millimeter or centimeter on your ruler) when working with the UW map above? Working from the computer screen, we got 1 mm = 17.86 m. Could we eliminate the possibility that 1 mm = 18.3 m? 1 mm = 17.1 m? No. Our methods were not that precise; we had some uncertainty.

      • Measure the following white lines:

        Line A	Line B	Line C

      • Which measurement will be the most precise? Least? Which measurement will warrant the greatest number of significant figures? Least? For which line could we justify the use of a micrometer capable of measuring to 4 decimal places? What are your estimates of your uncertainties (the possible errors in your measurements) for each line?
      • Answer We will be able to measure Line A with the highest precision and should use the greatest number of significant figures for this line. My measurements and uncertainties: A) 23.5 +/- 0.2 mm; B) 24.0 +/- 0.5 mm; C) 23 +/- 1 mm
      • The measurement of the height of line C would be even more uncertain. Do you see why?

   6. Statistics

      We use an absolute minimum of statistics in this class. You are probably very familiar with the terms "mean" and "standard deviation" from your high school math as well as from large lecture classes here where the grades are based on a curve. The mean is the average, while the standard deviation gives a measure as to how spread out the data are.

      If we have enough data, we may wish to graph the observations and see if there is a relationship between the variables. Here's a trivial example. Assume that a Ferrari can go from 0 to 60 mph (about 100 km/hr) in 6 seconds, and that the acceleration is constant. Here is what the graph would look like:

      .

      Let's assume another Ferrari is having air intake problems. When we plot the data, we notice that the data do not fall on a straight line. We do not want to simply "connect-the-dots" as we do not care about the second-by-second acceleration, just the overall relationship. In this case, one draws the best-fit straight line, attempting to get a balance between the number of data points above and below the line. This is very important!

      .

      What is the slope of the line? How fast would the car be traveling after 3.5 seconds? What is the uncertainty in the slope of the line?

      Answer The slope is (63 mph)/(6 sec), or 10.5 mph/sec. The car would be traveling (10.5 mph/sec) x (3.5 sec) or 36.8 mph. The slope could be as high as 12 mph/sec or as low as 9 mph/sec, making the uncertainty 1.5 mph/sec. HOW WAS THIS FIGURED OUT????

   7. The Scientific Process

      Recall the scientific process:

      1. Observe

      2. Hypothesize (make an "educated guess", predict)

      3. Test

      4. Evaluate: reject, modify, or retain hypothesis

      5. Form a new hypothesis if necessary - Predict

      6. Retest

      7. Loop until hypothesis is retained and a new scientific theory or model is formed.

      The process must be modified a bit in astronomy as we cannot bring the objects into our laboratory for direct examination and testing. We use all of the physics we know here on Earth and assume they apply to the whole universe (and, so far, there is little or no evidence that this assumption is invaled). Piece-by-piece, we have come to understand our universe.

   Scaling and Scale Factors: Here is a simple example for practice. You know the blue line is 50 pixels and the yellow line 200 pixels from the left edge of the following figure. You measure the blue line at 1.77 +/- 0.05 cm and the yellow line at 7.05 +/- 0.05 cm from the left edge on a printed copy. (You have a precise measuring tool.) How many pixels/cm? You know the red line is at 350 pixels in the figure. How many cm is this line from the left edge?

   
   Answer The scale is approximately 28.3 pixels/cm. The red line is approximately 12.4 cm from the left edge: (350 pixels)/(28.3 pixels/cm).

   Working with Units and Ratios:

   Units are important to note. For example: The distance to Sirius is 8.6. The radius of the Universe is 15,000,000,000. Kilometers? Astronomical Units? Parsecs? Light years? Megaparsecs?

   You will be asked in labs and homeworks to make a quantative comparison between two quantities. That is, you will be asked to give numbers that compare two different things. If you are asked to compare things quantitatively, the most enlightening comparison is found by taking a ratio. That is, divide the larger number by the smaller (or vice versa) to find how many times larger (or smaller) it really is.

   For example: The mass of Jupiter is approximately 1.9 x 10^27 kg. The mass of the planet orbiting the star 51 Pegasi is approximately 8 x 10^26 kg. How do these two masses compare quantitatively? Isn t it easier to understand the comparison if one says that the planet orbiting 51 Pegasi is about 45% of the mass of Jupiter?

    

   
   
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