REVIEW OF BASIC MATHEMATICAL CONCEPTS

Transcription

1 REVIEW OF BASIC MATHEMATICAL CONCEPTS The following document is a modified version of a laboratory exercise prepared for students taking general biology. It is provided here for individual review purposes only and may not be used for any other purpose or in any other context. J. Montvilo Review of Basic Mathematical Concepts Page 1 Version prepared 8/23/03; for individual review purposes only.

2 Page 2 Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

3 REVIEW OF BASIC MATHEMATICAL CONCEPTS Jerome A. Montvilo Copyright All rights reserved. Mathematics is the door and the key to the sciences. Roger Bacon 1. INTRODUCTION s Throughout high school, and perhaps even before and after, you may have sat in class after class of arithmetic, algebra, geometry, trigonometry and the rest asking yourself, When am I ever going to really use this stuff, anyway? The answer is: now. There was a reason for sitting through all those classes. As with this biology course, it was and is to prepare you for the future, since you can t always know or predict what and when it really would be useful to know these things 1. Since it is possible that you may not have paid attention in those classes as well as you might have, or because even if you did learn the material you may have forgotten it through lack of use, the basics of scientific mathematics will be reviewed here. s Like it or not, science is based on numbers. When data are 2 collected they must be analyzed to glean the information from them. This often means statistical analysis of some sort. Once the information is obtained, it must be presented clearly. This is often accomplished most effectively in the form of graphs. Statistical analysis of data will be covered in several labs in the future, so it will not be covered here. Because of its importance to presenting and analyzing the data you obtain in other labs in this course the basic concepts of graph preparation will be reviewed below. s It is not possible here to review all of the mathematical concepts that should have been learned previously. What will be attempted is to provide you with enough information to be able to solve the basic mathematical and statistical problems you are likely to encounter in this and other similar courses. Some, if not all, of what follows below may be familiar to you. The purpose of putting it here is to make it familiar to everyone in the course or as a review. Even if you don t think you need to go over this material, do so to 1 It is also why this author, a biologist who never did particularly well in mathematics, is now writing an introduction to basic mathematical concepts. Go figure (no pun intended). 2 The word data is plural, so technically the correct usage is data are. (The singular of data is datum.) However, in the recent past it has become more common, and acceptable, to use the word data as a singular noun. This newer convention will not be followed here. Review of Basic Mathematical Concepts Page 3 Version prepared 8/23/03; for individual review purposes only.

4 refresh your memory. If you don t remember this material, or if you have never had it before, take your time and try to learn it as best you can. s There is one important thing you can do to make sure that your mathematical calculations are correct. Simply ask yourself, Does this look right? Does it make sense? If you multiply 5 times 10 you should get 50. Therefore, if you multiply 5 times 9.9 you should get something near 50. If your answer does not seem to be in the same range as your estimate check your results again. (Assuming, of course, your estimate was correct in the first place, but that s another problem!) 2. SOME FUNDAMENTAL IDEAS AND RULES s Much of what follows should go without saying, but sometimes the fundamentals are forgotten. They are reviewed here for your convenience. l Numbers are written by using the symbols Except for 0 (zero) they all have a specified value. Zero signifies no value. For an unknown value an x is often used and for an unspecified value an n may be used. l Numbers may be manipulated (operated on) in various ways, and symbols are used to indicate what operations are to be carried out. These operations (and symbols) include the basic ones such as addition (n + n), subtraction (n n), multiplication (n n or n * n or n n or n(n)), and division (n n or n n). l Numbers also may be manipulated by squaring, by cubing, or by being raised to some other power [indicating that the number is multiplied by itself a certain number of times] (n 2, n 3, n n ). (The superscript number is called the exponent; on some calculators it may show up as En, as in 10E2 = 10 2, or in some other way.) Numbers may be manipulated by taking the square root, cube root, or some other root [essentially the reverse of squaring, cubing, and so on] (, 3, n ). l Numbers may be compared to each other and may found to be equal (=) or approximately equal (ª) to each other. One number may be greater than (>) the other number or less than (<) the other number. Other more complex manipulations and comparisons may also be carried out, but they will not be covered here. l Numbers less than zero are said to be negative ( ) numbers. The larger a negative number is the lower its value (for example, 10 < 5). Numbers greater than zero are said to be positive (+, but usually implied so not written) numbers. These numbers may extend to infinity ( ) in either direction. These relationships may be seen most easily on a number line, as shown below, where the values to the left are always less than values to the right. Page 4 Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

5 l When the sign (negative or positive character) of a number is not important for a calculation but only its numerical value matters then the absolute (positive) value ( n ) is used. l When two positive numbers are added together the result is a higher positive number. When two negative numbers are added together the result is a lower negative number. Whether the result of an addition or subtraction calculation involving positive and negative numbers is positive or negative depends on the magnitude of the values of the numbers involved. l Two negative numbers multiplied together results in a positive number, as do two positive numbers multiplied together. A positive number multiplied by a negative number results in a negative number. l A negative number divided by a negative number results in a positive number. A negative number divided by a positive number results in a negative number. A positive number divided by a negative number results in a negative number. A positive number divided by a positive number results in a positive number. l Any number, negative or positive, divided by itself results in a value of 1 (positive 1). l A number divided by 1 equals itself. l Manipulations involving zero are special. Any number multiplied by zero equals zero. You may not divide by zero. l Fractions will not be covered in detail here. However, fractions can easily be converted into decimals, the preferred method of dealing with values less than 1, by dividing the numerator (on top) buy the denominator (on the bottom). (The symbol can be thought of as meaning divided by.) For example, 1 10 = 0.1 and 5 8 = This is most easily accomplished using a calculator. l Units, such as degrees (, F, C ), length units (m, mm, cm, etc.), area units (mm 2, cm 2, etc.), and so on, must be included in the reporting of numbers. (In a table where the units are clearly given in the label the units may be left out to reduce clutter.) s Other more detailed concepts will be covered in the sections below. 3. A COUNTING SYSTEM s Our decimal counting and mathematical systems are founded on the concept of base 10 arithmetic. ( Deci refers to tens.) Put simply, this means that the position of a numeral in a sequence of numerals will determine Review of Basic Mathematical Concepts Page 5 Version prepared 8/23/03; for individual review purposes only.

6 its value. This is so because the numeral indicates the number of powers of ten that are found in that position. To put it another way, that numeral is multiplied by the power of ten determined by the place it is in. Zero is used as a placeholder only and has no value of its own. The positions begin as follows and continue in both directions from the decimal point, as in the chart below: thousands hundreds tens ones tenths hundredths thousandths 1000s 100s 10s 1s.1s.01s.001s (The decimal point, of course, indicates the separation between the whole numbers on the left and the fractional numbers on the right.) This is basically a multiplication and addition scheme. For example, the number could be represented in the above chart as follows: thousands hundreds tens ones tenths hundredths thousandths 1000s 100s 10s 1s.1s.01s.001s This means that you have 2 thousands (2 1000) added to 3 hundreds (3 100) added to 0 tens (0 10) added to 1 one (1 1) plus 2 tenths ( ) added to 3 hundredths ( ) added to 5 thousandths ( ). Shown another way, it might look something like this: 2 thousands hundreds tens one 1. 2 tenths.2 3 hundredths.03 5 thousandths.005 total: There are several points of significance to this scheme, not the least of which is that it can be used to simplify mathematical manipulations through the use of so-called scientific notation which will be discussed below. 4. SCIENTIFIC NOTATION s You may have noticed that each of the numerical positions discussed above was either ten times greater than or ten times less than the position on either side of it. Therefore, for example, the position signifying tens is ten times larger than the ones position but only one-tenth as large as the hundreds position. Why is this useful? Page 6 As noted above, each position can be indicated as a power of ten. The power to which ten is raised is indicated by a superscript number (exponent) after the 10. Ten is multiplied by 1 to give you 10. In other words, 10 1 = 10. Ten is multiplied by 10 to give you 100, or 10 2 = 100. Each of the positions Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

7 can be indicated by a particular power of ten (how many tens are multiplied together), as shown below: thousands hundreds tens ones tenths hundredths thousandths 1000s 100s 10s 1s.1s.01s.001s Notice that 10 0 equals 1. Notice that as you move to the left of the decimal point the power of ten is raised by an additional +1 for each position. Notice that as you move to the right of the decimal point the power of ten is raised by an additional 1 for each position. Notice that the numeral in the exponent indicates how may tens away from the decimal point the number is and the sign indicates whether it is to the left of the decimal point (+, larger value) or to the right of the decimal point (, lower value). Why is this important? For one thing, it provides an easy way to do complicated problems. Multiplication problems are converted into addition problems and division problems are converted into subtraction problems. How? Notice that if you multiply 10 times 10 you get 100. Or, = times 10 equals 1000 (or = 10 3 ). To multiply these two numbers, then, you just add the exponents. What is 1000 divided by 10? 100. [ = ( = 100).] Just subtract the exponents. What if you want to add or subtract numbers using exponents? Then it becomes a little tricky. You can only do this if the exponents are the same. You cannot directly add 10 3 plus 10 1, for example. You first need to convert one of the numbers so that the exponents are the same. How do you do this? Usually, by moving the decimal point which, in effect, changes the exponent. Note that in the examples above only the base number and the exponent were given. It was assumed that you only had one of them. To be more correct, these numbers should have been written as something like or, more correctly, Notice that if you move the decimal point to the right three spaces you will have a value of That s what the exponent means. Every time you move a decimal point to the left one place the exponent loses a value of one. Every time you move a decimal point to the right the exponent gains a value of one. So, the number 1000 could be written as (1 1000) or as (10 100) or as (100 10) or as (1000 1). The number.01 could be written as (1.01) or as (10.001) and so on. (While it could be done, it is not an accepted method of writing numbers in scientific notation. It is used here just as an example.) Review of Basic Mathematical Concepts Page 7 Version prepared 8/23/03; for individual review purposes only.

8 What about a number such as 106? Can it be written using an exponent? The answer is yes. 106 simply means that you have one 100 and six 1s. 106, then, means 100 ( ) plus 6 ( ). But remember, you cannot add or subtract these kinds of numbers unless the exponents are the same. Therefore, either the 10 2 or the 10 0 must be changed. Which one? Technically, it doesn t matter, but in this case it is easier to change the The decimal point needs to be shifted two spaces to the left. Since is the same as which is the same as , it can now be seen that 106 can be written as the sum of plus , or (Note that for consistency additional zeros were not added to the decimal numbers above. Technically, and from here on, so there is no ambiguity, a leading zero should and will be added in front of the decimal place. Therefore,.1 will be written as 0.1 and so on. Adding a zero after the decimal point, as in changing 1 to 1.0 should not and will not be done because to do so changes the accuracy and precision of the number, a matter taken up in the discussion of significant figures.) These same rules apply to all numbers and form the basis of what is known as scientific notation. The primary usefulness of scientific notation lies in the fact that both very large and very small numbers can be written in a compact manner. The number can be written as (Notice that the positive exponent here indicates the number of zeros representing powers of 10 in the number.) The number can be written as (Notice that the negative exponent here indicates how many places there are to the right of the decimal point.) The main difference between what was discussed at the beginning of this section and scientific notation is that it is most common to express a number in scientific notation as a single digit from 1 to 9 followed by a decimal point and the appropriate rest of the decimal followed by the exponential power of ten to which that decimal is raised. Therefore, the number is expressed in scientific notation as (Moving the decimal point two spaces to the right returns the decimal equivalent.) Briefly, then, to convert a decimal number to its scientific notation equivalent you move the decimal so that a single digit appears before the decimal point. If you moved the decimal point two places to the left then the exponent is a positive 2. If you moved the decimal point two spaces to the right then the exponent is a negative 2. To convert a number from scientific notation to its decimal equivalent you would just reverse the process. A positive exponent indicates to move the decimal point from where it is to the right the number of places designated by the exponent. A negative exponent indicates to move the decimal point from where it is to the left the number of places designated by the exponent. Page 8 Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

9 These manipulations are summarized in the table below. FROM DECIMAL NOTATION TO SCIENTIFIC NOTATION Æ (The decimal point is moved 3 places to the left to give a single digit before the decimal point, so the exponent is +3. As a check, remember that 10 3 = 1000, so this looks right.) Æ (The decimal point is moved 4 places to the right to give a single digit before the decimal point, so the exponent is 4. As a check, remember that 10 4 represents , so this looks right.) FROM SCIENTIFIC NOTATION TO DECIMAL NOTATION Æ (The +2 exponent indicates that the decimal point should be moved 2 places to the right. As a check, remember that 10 2 represents 100 and therefore 3.4 times 100 would be about 340.) Æ (The 3 exponent indicates that the decimal point should be moved 3 places to the left; zeros are added as necessary. As a check, remember that 10 3 represents and 6 times would be ) 4.1 Convert each of the following numbers either from decimal notation to scientific notation or from scientific notation to decimal notation. Place your answers on the worksheet. 4.1a 4.1b 4.1c 1,359 Æ? Æ? Æ? 4.1d 4.1e 4.1f Æ? Æ? Æ? 5. ORDER OF OPERATIONS s Most non-arithmetic errors in calculations are the result of not following the correct order of operations in the calculations. While this can become problematical if an equation is not set up properly, in a well laid out equation what the operations are should be clear and the results unambiguous. By way of review, the operations to be dealt with here are the following: raising to a power, multiplication, division, addition, and subtraction. These should all be familiar at this point. In order to force a calculation between two numbers they may be included in parentheses, ( ), also sometimes called brackets. Liberal use of parentheses simplifies the problem of what calculations to do first in an equation. s Although there may be minor exceptions, the general rule is that operations are carried out in the following order from left to right 1 : [parentheses] [exponents] [multiplication and/or division] [addition and/or subtraction] 1 How do you remember the order? One old mnemonic (memory aid) is the following sentence: Please excuse my dear Aunt Sally. The first letter of each word corresponds to the first letter of an operation. The operations are carried out in the same order as they occur in the sentence. Review of Basic Mathematical Concepts Page 9 Version prepared 8/23/03; for individual review purposes only.

10 (Actually, what is really meant is that any calculation or number that occurs within a grouping symbol such as the parentheses, ( ), but also straight or curly brackets, [ ] or { }, the absolute symbol (representing only the positive value of the number is used),, or the radical/root sign,, should be carried out or evaluated first. Then exponents should be evaluated. Then, working from left to right, multiplications and divisions are done you do the divisions first if they appear first from left to right. Finally, additions and subtractions are done you do the subtractions first if they appear first from left to right.) NOTE: Do not confuse the slash symbol used to represent a division operation (/) with the line separating the numerator from the denominator in a fraction ( ). While they look similar (but look carefully and you will see that are not exactly the same) and mean basically the same thing, the interpretation in terms of order of operation may be different. In a properly written equation this should not be a problem. In the examples that follow the / symbol represents a division, as in 10 / 5 = 2. Fractions will be written as 5 10 or s What is the result of the following calculation? =? Is it then multiply by 7 to give you 77? Or is it 6 x 7 then add 5 to give you 47? Following the rules for the correct order of operations the answer is 47. The multiplication is done first (6 x 7) and then the addition (5 + 42). s What is the result of the following calculation? (5 + 6) 7 =? Now the answer is 77 because you do the operation within the parentheses first and then do the multiplication. s One final example follows: (75 + (70 + 5)) å The expression in the parentheses is evaluated first. Where there are nested parentheses, as here, always work from the inner parentheses outward. Therefore, = = 150. This leaves the equation as: ç The exponent is now calculated. 2 5 = 32. This leaves the equation as: Page 10 Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

11 é Next, the multiplication and division are carried out from left to right = 2. 3 x 32 = 96. This leaves the equation as: è Finally, the additions and subtractions are done. These are normally done from left to right, giving: = s It is very important that these rules be followed carefully in order to get the correct answers in your calculations. 5.1 What are the results of the following calculations? 5.1a 5.1b 5.1c (( ) 12) 3 = = (( 3 3) 4) 5 2 = 5.1d 5.1e 5.1f = = (24 (3 + 5)) 3 = 6. RATIOS AND PROPORTIONS s A ratio is a comparison made between two or more numbers. In a situation in which there is one of one type of thing and three of another type the ration is said to be a three to one ratio. This is usually written in one of three ways: 1 to 3 1 : or 1 3 What this means, simply, is that for every one of the first thing there are three of the second things. Depending on what is being represented, it is also possible to indicate that for every three of the first thing there is only one of the second things, as seen below. 3 to 1 3 : or 3 1 In biology, especially in genetics, you can have a situation where you are comparing the amounts of more than two things. For the traits of offspring, for example, you may have nine of one trait (called a phenotype), three of another Review of Basic Mathematical Concepts Page 11 Version prepared 8/23/03; for individual review purposes only.

12 Page 12 phenotype, three of another phenotype, and one of another phenotype giving what is known as a 9 : 3 : 3 : 1 phenotypic ratio. s A proportion is a comparison between two ratios. The comparison is usually made between two ratios that have the same numerical relationships so that they are usually separated from each other by an equal sign. An example follows: 1 2 = 5 10 This means that each of the expressions on either side of the equal sign is equivalent. This is convenient if you need to find out an unknown amount, as in the example below. (As usual, an x stands for the unknown quantity.) In a very simple case, let s say that one horse can eat two bales of hay in one day. How many bales of hay would five horses eat? Set up as a proportion the relationship would look like this: 1 2 = 5 x Notice that the same type of objects is either above or below the line in each ratio: 1 horse 2 bales of hay = 5 horses x bales of hay This may be read as: 1 horse is to 2 bales of hay as 5 horses is to how many bales of hay? The answer is obtained through cross-multiplication and simple algebra. The two numbers that are diagonal to each other in the relationship are multiplied and the equivalence is kept. This gives you: 1O N5 2N O x 1 x = 2 5 x = = 10 1 =10 It would therefore take ten bales of hay to feed five horses. 6.1 Thirty-three red maple trees were found in a section of forest covering an area two square kilometers in size. What is the probable number of red maple trees present in the whole forest if the entire forest covers 98 square kilometers? Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

13 s As can be seen from the above examples, in a typical proportional relationship as one value goes up so does the other. In an inversely proportional relationship the reverse is true as one value goes up the other goes down or as one value goes down the other goes up. Using the example above, as you increase the number of horses the number of bales of hay needed also increases. This is a proportional relationship. If it takes one person one day to do a job, how long will it take two people? The answer is one-half day. This is an inversely proportional relationship, since as the number of people goes up the time it takes to complete the job goes down. 7. PERCENTAGES s A percentage is merely a ratio in which one of the numbers is always 100. ( Cent refers to 100; per means out of or for each ; percent therefore means a number out of 100.) The symbol % represents percent. If you say that something makes up 33 % of the total this means that for each 100 things that there are 33 of them will be of that certain type. This can actually be represented in three different ways, all of which are equivalent: 33% = = 0.33 (To convert the decimal to a percentage just multiply by 100 and add the percent sign.) Since it is often easiest to compare two quantities when they are both represented using equivalent measurements, in this case percentages, it is usually a simple matter to convert all data compared to percentages, as shown in the example below. s You receive a 67 out of 80 on one exam and a 47 out of 65 on another exam. In terms of percentages, which grade is higher? Clearly, the answer would be easy if both exams were scored out of 100. What can be done is to use the techniques learned above to convert each of the grades to a percentage, as if they actually had been graded out of 100. For the first exam set up the following proportion: = x 100 Do the math as shown in the previous section (it does not matter which of the cross-multiplied numbers goes first): Review of Basic Mathematical Concepts Page 13 Version prepared 8/23/03; for individual review purposes only.

14 80 x = x = = = The results of the first exam, then, are equivalent to a grade of out of 100. For the second exam a similar process is gone through: = x x = x = = = The results of the second exam are equivalent to a grade of out of 100. In terms of percentage, then, you did better on the first exam than on the second exam. (A quick way to convert any fraction to a percentage, especially on a calculator, is to do the division indicated by the fraction and then multiply the result by 100.) s There are other ways of doing these types of problems, but they all should yield comparable results. 7.1 A biologist collect 940 containers of pond water in order to study the single celled organism called Paramecium. Of those containers, only 47 contained Paramecium. What percentage of all the containers is represented by Paramecium-containing containers? 8. SIGNIFICANT FIGURES AND DIGITS 1 s Some of the trickiest basic mathematical concepts to grasp are the concepts of significant figures (sometimes called significant digits or significant numerals) and rounding. Although the idea behind these concepts is simple, carrying them out is not. This is because there is not complete agreement on how they are carried out. 1 There are a number of good tutorials on the World Wide Web that deal with mathematical concepts. The one from which some of the following material comes can be found at the following URL: Page 14 Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

15 The concept of significant figures is related to measurement, precision, and accuracy, which will be covered in more detail in another lab exercise. (For now it is important to know that accuracy refers to how close a number is to the actual value and precision refers to the exactness of the number.) Simply put, there are exact numbers and measured numbers. s Exact numbers are exact because they are defined that way. By definition, for example, there are 2.54 centimeters in 1 inch and there are 1.8 Fahrenheit degrees in 1 Celsius degree. A number may also be exact because the numbers are whole integer numbers not usually measured in fractions. For example, the number of wings an insect may have is usually 0, 2, or 4. s Measured numbers are just that they have been generated by some sort of measuring device such as a thermometer or a balance. As such they are inexact because a certain amount of estimation is made concerning the true value of the measurement. Measured numbers are also obtained as a result of a mathematical operation, such as finding the mean (or average ). The mean value of the numbers 3, 4, 5, 5 is 4.25; the numbers are exact (assuming they are not measurements) because they are integers but the mean is measured because it was obtained from a calculation. When you deal with a measured number, how do you know how accurate it is? Let s say you measure the length of some object and it is 5.2 centimeters long. What does this mean? It means that it is greater than 5.1 centimeters long but less than 5.3 centimeters long. Could it be 5.19 centimeters in length? Yes, because of the inherent problems with measuring something accurately. What if we say the object is 5.23 centimeters long? This means that it is greater than 5.22 centimeters in length but less than 5.24 centimeters in length. By adding the last digit we significantly add to the precision of the reporting of the measurement. And that is the point behind significant figures. They tell you how precise a measurement is (but not necessarily how accurate). If you report something as having a mass of 6 grams, that is different from saying that something has a mass of 6.0 grams. The second number is more precise. The last digit gives you a clue to the precision. s In all measurements the last digit is always an estimate. Someone reading the number should realize that. If you multiply two numbers, such as 3.4 and 7.319, how do you know the significance of the result? How do you know which of the digits in a number really have value which are significant? The answer is in a set of rules that have been developed. These rules are as follows: u All non-zero digits (1 through 9) are significant. [In the number there are five significant digits.] Review of Basic Mathematical Concepts Page 15 Version prepared 8/23/03; for individual review purposes only.

16 u All zeros between non-zero digits are significant. [In the number there are six significant digits.] u Zeros at the end of a number are significant if they are to the right of a decimal point. [In the number the last zero is significant; there are four significant digits.] (Note that in some numbers the zeros at the end of a number can be misleading. In the number 800, for example, are the zeros significant? If the number is accurate, the zeros are significant. If the number is an estimate they are not significant. This ambiguity can often be dealt with by using scientific notation because only significant figures should be used in scientific notation.) u Zeros at the beginning of a number, usually used as a placeholder, are not significant. [Therefore, the zeros in 003 and 0.85 are not significant.] Likewise, any zero to the left of the first significant digit is not significant. [In there are only two significant digits, the 4 and 5; in scientific notation this would be written as ] s Knowing the number of significant digits is important for being able to report the answers of calculations. The number of significant digits reported for any calculation cannot be any more than the fewest number of significant digits in any of the original measured numbers used in the calculation. For example, if you add the measured numbers 3.6 plus 0.61 plus the result is However, since the fewest significant digits in any of the original numbers was two the calculated number should be reported as 5.2. (Rounding may be necessary, and this will be taken up in the next section.) The same is true for subtraction. [It should be stated that not everyone agrees with this practice. For some, the answer is valid. To a certain extent this may be a difference between how they are handled in science or in mathematics. Check with your instructor as to the preferred manner of dealing with this.] If you multiply times 2.53 the calculated answer is but since the fewest significant digits in the original numbers is only three the number should be reported as This rule is generally accepted. Division is handled in the same way. 8.1 How many significant digits are in each of the following numbers? 8.1a 8.1b 8.1c d 8.1e 8.1f Page 16 Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

17 9. ROUNDING s In doing mathematical operations, particularly using a calculator or a computer spreadsheet, it is easy to assume that the long numbers that may result are very precise. As seen above, this is not necessarily the case. Only a certain amount of precision, or significance, can be given to the digits of a calculated result. Since the last digit of most measured and calculated numbers dealt with in a laboratory is an approximation only, it sometimes becomes necessary to round off or to remove the additional digits the number to ensure accuracy and consistency. As with most things, not everyone agrees on a process for rounding. Some techniques have been shown to be more accurate while others tend to be more convenient. Which is used depends on a number of factors. For this laboratory simply use the technique preferred by your lab instructor. s Why do you round off a number? There are several answers. As seen above, you round off a number when dealing with significant digits. You may also round off a number for a variety of other reasons, not the least of which is convenience or when you don t need an exact answer. s When do you round off a number? If you are going to round off in a calculation always round off a number at the end of a calculation. Do not round off any numbers within a calculation. s How do you round off a number? Again, there are different ways of accomplishing this. First, you need to know how precise the rounded number needs to be. Do you round to the nearest whole number? To the nearest million? To the nearest thousand, hundred, ten? To the nearest tenth (one decimal place), hundredth (two decimal places), thousandth (three decimal places)? To one significant digit? To two or three significant digits? The answer is usually up to you unless your lab instructor gives you explicit directions. ( Always round to the nearest. ) Be consistent. Second, you need to know how to round. The technique will vary depending upon which precision level you choose, but, in general, the technique you were probably taught in previous classes is generally acceptable. If the first digit beyond the one you are rounding to is 4 or less (0, 1, 2, 3, 4) round down to the next lowest value. If the first digit beyond the one you are rounding to is 5 or above (5, 6, 7, 8, 9) round up to the next highest value. A more accurate way of rounding uses the following rules: (It is assumed in the examples that you are rounding to the nearest tenth.) Review of Basic Mathematical Concepts Page 17 Version prepared 8/23/03; for individual review purposes only.

18 u If the digit beyond the one to be retained is less than 5 (0, 1, 2, 3, 4) don t change the digit to be retained. Therefore, 6.43 becomes 6.4. u If the digit beyond the one to be retained is greater than 5 (6, 7, 8, 9) change the digit to be retained to the next higher digit. Therefore, 6.47 becomes 6.5. u If the digit beyond the one to be retained is exactly 5 then look to the digit just to the left of it. If that digit is even don t change the digit to be retained. If the digit is odd then change the retained digit to the next higher digit. Therefore, 6.45 becomes 6.4 and 6.75 becomes 6.8. u If there are two or more digits to the right of the digit to be retained then treat them as a group. Therefore, in the 321 is treated as a group, is considered to be less than 5 (or 500), and so the number is rounded to 6.4. In , the 652 group is evaluated as being greater than 5 (or 500) so the rounded number is Round the following numbers to the nearest whole number. 9.1a 9.1b 9.1c Round the following numbers to the nearest tenth. 9.2a 9.2b 9.2c Round the following numbers to the nearest hundredth. 9.3a 9.3b 9.3c GRAPHING s It might be assumed that with spreadsheets and other graphing applications available that you would not need to know the fundamentals of graphing. This is not true for several reasons, not the least of which is that you need to tell the application what to graph and how to label the graph and that, in lab, no graphing application may be available so any graphs will have to be drawn by hand. Page 18 s The numbers you are plotting are called variables. In science there are two types of variables, independent variables and dependent variables. Independent variables are not changed by the experimental conditions during Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

19 the experiment and they do not depend on anything else. The dependent variable (also known as the responding variable) reacts to, or depends upon, the independent variable. Therefore, at a certain time or in a certain grouping (independent variables) you may have a certain number of results (dependent variable) depending upon the conditions present. Independent variables are usually graphed along the horizontal, or X, axis of the graph with lower values towards the left. Dependent variables are graphed along the vertical, or Y, axis of the graph with lower values towards the bottom. (A third, or Z, axis may be used in some cases as for plotting threedimensional data, but that will not be covered here.) This is shown below: It is necessary that the axes themselves be marked off in appropriate intervals. What the intervals are may vary depending on the data, but the intervals should be evenly spaced and labeled. Make sure that both axes are labeled with both what the variable is as well as what the units are, as in the following example: s The three types of graphs you are most likely to be using in this course are scatter graphs, line graphs, and bar graphs. In a scatter graph only the points are plotted. In a line graph, the plotted points are connected by lines. (Technically, all the plotted points should be connected by a curved line, or a curved line of best fit, but the same basic idea may be obtained by connecting the points with straight lines. Use a straight edge to do so.) In a bar graph (an example of which is called a histogram), grouped data are presented in the form of a column or a rectangle. Examples of these three graph types are shown below: Review of Basic Mathematical Concepts Page 19 Version prepared 8/23/03; for individual review purposes only.

20 s Neatness counts. This means, among other things, that: every line should be drawn using a straightedge; symbols should be clearly drawn and easily recognizable; proper spacing should be used so that the graph is both easily readable and accurate (in general, use the space available); do not go outside the boundaries of the graph (plan your graph accordingly). s In most cases data are plotted starting at the origin (0, 0) of the graph unless there is a particular reason for not doing so. s It is important, especially when the graphs contain data from multiple data sets, that the data be identifiable. An identification key or legend, usually placed on the upper right side of the graph, should be included. The legend may include colors, different point types, or different hatching or fill patterns to identify the data. These points and other identifiers should be easily distinguishable from each other. s Each graph should have a title describing what the graph is. That title is usually centered above the top of the graph. s The axes of each graph should be labeled to indicate both what the axis represents and what units are being used On the graph paper provided on the worksheet plot the following data as two superimposed line graphs. Mean Heart Rate (beats per minute) After Exercise Time after exercise (min) Males Females Page 20 Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

21 11. SOME DEFINITIONS AND FORMULAS s For reference, some of the more common terms and formulas that you will need to know for this course are given below. More will be provided as needed. Area. The amount of surface covered by a geometric figure. It is reported in square units or units 2. (For example, two square meters = 2 m 2.) Surface area refers to the amount of surface covering the outside of a three-dimensional object. u The area of a rectangle equals its length times its width. u The area of a circle equals pi times the radius squared. (A circle = ϖr 2 ) Average. (See Mean.) Circle. A geometric figure in which the outer boundary is always the same distance from a central point. The diameter of a circle is the distance across the center of a circle. The radius of a circle is the distance from the center of a circle to the outside of the circle; it is equal to one-half the diameter. Circumference. The total length of the outside of a geometric object. u The circumference of a rectangle or triangle is the sum of the length of all of its sides. u The circumference of a circle is equal to pi times the diameter of the circle (ϖd). Diagonal. A line drawn which connects the opposite corners of a rectangle. The diagonals of a rectangle are always equal in length. Diameter. (See Circle.) Mean ( X ). A statistical calculation in which the sum ( S) of the values in a data set are divided by the number of values (N) in the data set. Pi (ϖ). The ratio of a circle s circumference to its diameter. It is a number with an infinite number of decimal places, but it is usually shortened to or or even just Radius. (See Circle.) Rectangle. A geometric figure with four sides and each angle equal to 90. A square is a rectangle with four equal sides. Square. (See Rectangle.) Review of Basic Mathematical Concepts Page 21 Version prepared 8/23/03; for individual review purposes only.

22 Volume. The amount of space taken up by a three-dimensional geometric figure. It is reported in cubic units or units 3. (For example, nine cubic centimeters = 9 cm 3.) For regular solids, such as a cube, it represents a length times a width times a height. s Give these final problems a try To the nearest tenth, what is the area of a circle with a diameter of 4.5 cm? 11.2 In the diagram below is shown one quarter of a circle with a rectangle inscribed within it. The radius of the circle is 10 centimeters. What is the length of line AB, a diagonal of the rectangle? (It s not hard, and it looks more complicated than it really is. No advanced math is necessary, just an understanding of the terms discussed above.) Page 22 Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only

23 Name: Section: Date: BASIC MATHEMATICAL CONCEPTS WORKSHEET a 4.1b 4.1c 4.1d 4.1e 4.1f a 5.1b 5.1c 5.1d 5.1e 5.1f a 8.1b 8.1c 8.1d 8.1e 8.1f a 9.1b 9.1c a 9.2b 9.2c a 9.3b 9.3c Review of Basic Mathematical Concepts Page 23 Version prepared 8/23/03; for individual review purposes only.

24 Page 24 Review of Basic Mathematical Concepts Version prepared 8/23/03; for individual review purposes only