Astronomy 113 - Using scientific calculators 0. Introduction For some of the exercises in this lab you will need to use a scientific calculator. You can bring your own, use the few calculators available in Sterling 4421, or use online calculators, such as the one on http://www.motionnet.com/calculator. Below, you will find some helpful information for using scientific calculators. These notes will be particularly useful if you have read the section on scientific notation (also called exponential notation), which describes how scientists write and handle large numbers. Your scientific calculator will look similar to the image shown below. What sets it apart from a regular business calculator is the ability to perform a number of advanced calculations, to handle exponential notation, and, in most cases, to store information for further use. Some advances calculators allow you to enter complex programs and even to plot/graph your results. You will not need these extended functions. However, if you are already familiar with your own calculator and know how to use this extended functionality, it could be very helpful in completing the lab in a shorter amount of time. This guide will not describe graphing or programming. 2.0 33 Trigonometric and special functions Basic operations Number block 1. Entering numbers and basic calculations: Your calculator has a regular number pad, in opposite order from a telephone number block (lower numbers start from the bottom). Pressing any number key will enter that number into the display. You can enter any number, digit by digit. Most calculators can only display numbers up to 10 digits, including the decimal point. You can perform simple algebraic calculations like multiplication and division by entering the first argument (or number), pressing the key for the specific computation you want to perform
(for example, the key for division) and then entering the second argument (or number). For example, pressing the sequence 123.45 0.03 will bring up the result 4115.00 on the display - you have just divided 123.45 by 0.03. Some calculators (e.g., those manufactured by Hewlett-Packard) have a more complicated pattern of entering numbers and operations, called reverse Polish notation. These notes will not present an introduction for those calculators. Please see the instructor if you are using a Hewlett-Packard scientific calculator. 2. Scientific/Exponential Notation Many things encountered in everyday life can be described by some number and a unit that translates that number into something meaningful. For example, the length of a football field is 90 yards. Societies have picked these units to make numbers most convenient for every day use. That means that they are neither excessively small nor excessively large, otherwise we would need to many zeros to describe them. For example, if we were to measure the length of a football field in units of microns, roughly the size of a typical human cell, it would measure about 90000000 microns - a number with 7 zeros. Clearly, microns are not a very practical unit to measure football related things in. In science, however, one encounters many different scales, and there is no one unique scale that would describe the entire universe in convenient numbers that are neither excessively large not excessively small. To be able to describe measurement and observables with such different scales and such large numbers, scientists use a mathematical way of writing numbers called exponential notation (also called scientific notation). The idea behind the exponential notation is simple: We can write any large number as the product of a smaller number and a power of the number ten (remember: powers of ten are simple numbers, 10 0 =1, 10 1 =10, 10 2 =100, 10 3 =1,000, 10 4 =10,000,...). This might sound complicated, but it is actually very simple. Take the number 12,345, for example: We can write this number as the product of the the number 1.2345 and the number 10,000: 12,345 = 1.2345 * 10,000 Now, 10,000 is the fourth power of the number 10 (it has 4 zeros), i.e., 10 4 = 10,000 or 10*10*10*10 = 10,000. So we can write: 12,345 = 1.2345*10 4 You might ask yourself why this should be beneficial, since the expression on the right is clearly longer than the one on the left. However, consider the number 123,450,000,000,000. This is already a rather large number - it has 15 digits. If we write this as a product between a number between 1 and 10 and a number that is a power of 10, we arrive at the following: 123,450,000,000,000 = 1.2345*10 14 We have reduced the number of digits needed to describe a large number from 14 to 10. In Astronomy, one often encounters numbers that are even much larger than this. For example, the mass of the sun is often measured in grams to be
M = 2,000,000,000,000,000,000,000,000,000,000,000.00 g This is clearly not a practical number to work with - your calculator would not nearly have a big enough display to show all of these numbers, and typing them in would easily lead to errors if one missed a zero. Written in scientific notation, on the other hand, this number becomes very manageable: M = 2.00*10 33 g Clearly, this is a better way of writing large numbers. It is very easy to convert a large regular decimal number to scientific notation: copy the number, but place the decimal point immediately behind the first digit (in the example this would be 2.0). Count the number of digits before the decimal point, subtract one, and this is going to be the power of ten you have to multiply the small number you ve created in the first step with. The mass of the sun has 34 digits (including the 2) before the decimal point, so the power of ten we have multiply 2.00 with is 33. This works well for large numbers, but it works equally well for small numbers: In many cases, scientists encounter tiny numbers. For example, the mass of an electron is m = 0.000,000,000,000,000,000,000,000,000,91 g Using the same recipe, we want to write this very small number as the product of a number between 1 and 10 and another number that is a power of 10. The first number is obviously 9.1, the second number has to be much smaller than on: 0.000,000,000,000,000,000,000,000,000,000,1. If we want to write this second number as a power of ten, the power has to be negative, because the number is smaller than one (remember, negative powers of ten are 10-1 =0.1, 10-2 =0.01, 10-3 =0.001, 10-4 =0.0001...). In the example of the electron mass, the negative power of ten is -28 (the electron mass has 28 zeros before the first non-zero number appears). So, in scientific notation we can write the electron mass as m = 9.1*10-28 g Again, this is a much more economical way of writing the same number. So, the recipe for converting a small number to scientific notation is this: copy the original number, but place the decimal point directly behind the first number that is non-zero (in the example, this is the number 9). Then count the number of zeros before the first non-zero number appears in the original small number (in the example, there are 28 zeros) and take the negative of that number (in the example, this is -28). This is the power by which you have to raise 10 to get the second part of the scientific notation. Sometimes scientific notation is even further abbreviated: Instead of writing 9.1*10-28 g, a different convetion is to write 9.1e-28 (for example, this is often how numbers in scientific notation are entered into computer programs). On your calculator, you can enter numbers in scientific notation as well. The button to do this is either called EE or EXP. To enter the mass of the sun into your calculator, first enter 2.0,
then press EE (or EXP) and then enter 33. You will see a large 2.0 and a small, raised 33 right next to it (see the picture of the scientific calculator above for how this might appear on your calculator). To enter the mass of the electron into your calculator, enter 9.1, then press EE (or EXP) and then enter -28 (on some calculators, you cannot enter the minus sign first, in which case you enter 28 and then press the ± or +/- button). You will see a large 9.1 on the main display and a small, raised -28 right next to it. 3. Trigonometric functions on your calculator Another important use of your calculator in astronomy is the calculation of angles and of trigonometric functions. This introduction is not meant to explain basic trigonometry (please refer to your high school or introductory college mathematics text for that), but we will describe briefly how to calculate trigonometric functions on your calculator. Before explaining how to do this, take a look at the sketch below to orient yourself and to remind yourself what the different trigonometric functions are: 90 α Looking at the triangle, you will immediately remember from high school that pythagoras theorem tells you that a 2 + b 2 = c 2. The angle between a and c is called α. The basic trigonometric functions relate α and different ratios of a,b, and c: sin α = b/c, cos α = a/c, tan α = b/a, cot α = a/b Let s say α =1.23. To calculate the sin of α, enter the sequence 1. 2 3 sin To calculate the trigonometric functions of α, enter the value of α into your calculator and press any of the trig-function keys. If your calculator does not have a function for tan or cot, you can still calculate them: cot α = 1/tan α, so enter the sequence 1. 2 3 tan 1/x This also introduces you to the 1/x function, which is just that: it divides 1 by the number you entered (in this case tan α).
To calculate the inverse of a trigonometric function, use the -1 function (usually on the same key as the trigonometric function, printed above the key). In order to access this function, you usally have to press the INV or SHIFT key (it might be called something else on your calculator, but it usually has the same color as the small print above the trigonometric function you are calculating the inverse of). The inverse of a trigonometric function allows you to calculate the angle if you enter the sin, cos, tan, or cot of that angle. Say you know that sin α = 0.5. Entering 0.5 and pressing sin -1 will give the result 30 or 0.524 (depending on whether you are using radians or degrees on your calculator). Radians and degrees: Angles can be measured in different units (just like length scales can be measured in yards, meters, micron,...). There are two wide spread ways to measure angles: One is in degrees, where a full turn equals 360 degrees (you are familiar with degree measurements from high school and most every aspect of everyday life where angles are involved). The other is radians (this concept is most often used in scientific and mathematic applications, because it is a more natural way to measure angles). A full turn equals 2π=6.28319... radians. The fact that a full circle is should remind you of the fact that the circumference of a full circle is 2π times the radius of that circle (i.e., 2π radians). That makes the definition of angles in radians quite simple: Divide the length of arc between the two sides of an angle by the radius of that arc, and you have the angle, measured in radians. So, a 180 angle corresponds to a half circle, which has a length of arc of half of a full circle, which is just π times r. Thus, 180 = π = 3.14159... If you have trouble visualizing angles in radians, talk to your instructor about an explanation during the open lab / office hour period. Scientific calculators allow you to switch between angle measurements in radians and degrees. You will get wrong answers if you enter an angle measured in degrees into a calculation where the trigonometric functions are expressed in radians. How to switch between radians and degree varies from calculator to calculator. Talk to your instructor if you need help in figuring out how to switch between the two modes. There are many other special functions, like log, ln, exp, x y, and more. They operate in much the same way the trigonometric functions work on your calculator.