Get Ready For Biology Chapter 2: Basic Math Review

Decimals Decimals are common in science, and biology is no exception. All decimals are based on 10 in a very specific way-- each place in the number represents a multiple of 10. The value increases 10 times for each space that you move to the left of the decimal, and it decreases 10 times for each space to the right. Let's consider this number: 12345.6789. You know how to read the numerals to the left of the decimal. They make 12,345. Moving from the decimal to the left, you can see how the spaces represent, in order, 1's, 10's, 100's, 1000's, and 10,000's (Figure 2.3). Our number represents 10,000 + 2,000 + 300 + 40 + 5. The spaces to the right of the decimal represent fractions: tenths, hundredths, thousandths, and so on. Thus in our number, as you go from right of the decimal, the numerals represent the fractions 6/10, 7/100, 8/1000, and 9/10,000

(Figure 2.3 shown)

GRAPHS Graphs present a more pictorial view of data. Numerical data that can be organized in a table (Figure 2.9a) can usually also be presented in graph. The main advantage is that the graph allows you to spot trends and relationships almost instantly. There are various types of graphs Let's look at three of them: line graphs, bar graphs, and pie charts. Look at Figure 2.9b. This is a line graph, and they usually are laid out in a grid. The horizontal axis at the bottom is the x-axis. It often, but not always, marks the progression of time. The vertical axis on the left is the y-axis and typically reflects some increasing value. Where these lines meet on the lower left of the graph marks the 0 position, so units go up as you move away from that point. Each axis should be labeled and should include units. When drawing a line graph, don't forget to label the axes and to include units. The most common mistake made when graphing data is to not use an appropriate scale. Be sure to size the units to maximize the space the graph fills. You don't want the graph to be cramped into one corner, making it hard to read. Spread it out both vertically and horizontally.

(See FIGURE 2.9) Typical heights for children through age 6 years, a) Table format b) Basic line graph. In our example, the x-axis tells us who the data are about- the age group of children for whom the listed height is typical. The v-axis gives height in inches. For each age group, the height (data point) is placed above the age it represents. The data points may be left unconnected, they may be directly connected (as they are here), or a line of "best fit" may be drawn that passes between the points so they are evenly distributed on each side of it. In our example, because this depicts growth with time, the line allows us to see at once the heights for each age group and to quickly comprehend the trend of a gradual increase in height with age.

PIE CHARTS Figure 2.11 is a pie chart, a type of graph that shows parts of a whole. This one shows the main molecules that make up the human body and tells the percent of the body made of each type of molecule. When looking at this, you immediately see that all the parts add up to the whole "pie," which is 100%, so these charts are very effective when showing percents. We all have a visual concept of a whole pie and a slice of pie, so even before looking at the numbers, we instantly see the differences in distribution. You know right away from this pie chart that one type of molecule makes up well over half of the human body (see Chapter 5, Chemistry, for a refresher course on molecules and related topics). Pie charts have no axes to label and the space within them is limited, so a key is often used. Labels for each "slice" may be written within the pie or placed to the outside, as in our example. Using different shading or colors for the slices makes these graphs even more readable. Because there are no axes to provide information, the graph title and key are very important

(See Figure 2.11)

Tables, Graphs, and Charts We explored how to get numbers by measuring and how to work with them. Now we will see how these numbers and other information, collectively called data, can be depicted. TABLES We have already used tables in this book, and you should be familiar with them, so let's quickly review the basics. Tables come in many forms and are a convenient way to present information so it is easy to read and compare. We will use Table 2.4 as a reference. When viewing a table, start with the table title, in this case "Fluctuations in human body temperature throughout the day." The title usually tells you what the table contains. Next, realize that tables are carefully arranged in columns and rows. All information in a single column is related, and all information in a single row is related. Look at the top of each column—these are column heads that tell you what information each column contains. Look at the beginning of each row. These are row heads, or labels that tell you what each row contains. In our example, the column heads reveal that the first column identifies each test subject, and the other columns contain the temperatures for all test subjects at specific times of the day. The row heads tell us that all temperatures in a single row belong to a single test subject, and who it is. So, by using the column and row heads, it is easy to find out, for example, what temperature Subject C had at noon.

(See Table 2.4) Time to Try On a separate piece of paper, using sentences and paragraphs, write out all the information that is included in Table 2.4. This could read as: Subject A records temperature at 4 hour interval starting at 4 AM and ending at midnight: 96°F, 96.3°F, 96.4°F, 96.8°F, 97.0°F, 96.8°F as applicable, Which version is easier to read? In which version can you more easily determine Subject D's temperature at 8 PM?

Bar Graphs Figure 2.10 is a bar graph showing the amount of water in the human body. Looking at the axes, you see that the x-axis has three separate categories: total water, intracellular fluid (the water located inside body cells), and extracellular fluid (the water not contained in cells). Each of these categories has two bars-one for males and one for females. The y-axis tells what percent of the total body weight the water represents. This graph is drawn so that the male and female data are directly compared by being positioned side-by-side, yet easily distinguished by use of different shading. The shading is explained on the lower left in a feature called the key. This bar graph has a 3-D effect that does not change its meaning at all—it just makes it slightly more interesting. Bar graphs may be drawn vertically or horizontally. Because each bar is so distinct, these graphs are good for comparing specific bits of data, whereas line graphs may be better at showing an overall trend.

(See figure 2.10)

Moving the decimal to the right is the same as multiplying by 10 for each space moved. Moving it to the left is the same as dividing by 10 for each space moved. Quick Check Solve these problems: 1.27 + 3.6 = 14087 - 3.2= 2.4 x 1.2 = 8.4 / 0.2 =

1.27 + 3.6 = 4.87 14.87 - 3.2= 11.67 2.4 x 1.2 = 2.88 8.4 / 0.2 = 42 (move the decimal in the divisor and dividend both one space to the right, so it becomes 84 / 2 = 42.)

Time to Try Can you supply the missing information in this table? Percent 36 %, _______, _______ Decimal ______, 0.42,________ Fraction______, ______, 80/100 = 4/5

36% = 0.36 = 36/100 = 9/25; 42% = 42/100 = 21/50; 80% = 0.8 = 80/100 = 4/5.

QUICK CHECK What is the most reduced form of each of the following fractions: 60/90, 25/100, and 18/54?

60/90 has a GCF of 30 and reduces to 2/3; 25/100 has a GCF of 25 and reduces to 1/4; 18/54 has a GCF of 18 and reduces to 1/3.

To better understand this idea of normal, you need to know how to calculate the mean pulse rate. Your three trials give you the following data: Trial 1: 72 beats per minute Trial 2: 74 beats per minute Trial 3: 79 beats per minute To find the mean of a group of numbers, simply add them all together then divide the total by how many numbers you added. For your data, you would add the three pulse rates, then divide by 3.

72 + 74 + 79 = 225 225 / 3 = 75 beats per minute Did you notice that the mean is not one of the original numbers? It does not have to be. It is the average of all three numbers.

Most of the work you will do, though, will be converting units length, mass, and volume, so let's move on. TIME TO TRY You will use the metric system more easily if you have some idea of the size of the base units. Most packaged items manufactured in the United States list both metric and U.S. units. Most rulers and measuring tapes have both metric and U.S. units. Explore your home and see if you can determine the following: Which is longer, a meter or a yard? Which is heavier, a gram or a pound? PICTURE THIS If you drink carbonated soft drinks, you are likely quite familiar with their standard large plastic bottles. What is their volume in metric units? _____liters Knowing this, envision the bottle only half full. Is that amount more or less than a gallon of milk? Know the paper clip! A standard small paper clip has a mass of about 1 g (it's very light). A standard large paper clip has a side-to-side width of about 1 cm, and the wire from which it is made has a diameter of about 1 mm.

A meter is longer than a yard by a few inches. A pound contains just over 450 grams. Half of a 2-L soda is definitely less than a gallon of milk, so a liter is smaller than a gallon.

Reducing Fractions Equivalent Fractions have the same value even though they appear to be different. Consider the following fractions: 1/3, 2/6, 4/12, 7/21 All of these numbers have the same value 1/3. To see this, you need to reduce the other fractions. This is done by finding the greatest common denominator (GCF) for each fraction. The greatest common factor is the largest whole number that can be divided into both the numerator and the denominator. Consider 2/6.

Both the numerator (2) and the denominator (6) are divisible by 2, which is the greatest common factor. If you do the division, you see that 2/2 = 1 and 6/2 = 3, so 2/6 becomes 1/3

Final Stretch! Now that you have finished reading this chapter, it is time to stretch your brain a bit and check how much you learned. At the end of each chapter, be sure you have learned the language. Here are the terms introduced in this chapter with which you should be familiar. Write them in a notebook and define them in your own words, then go back through the chapter to check your meaning, correcting as needed. Also try to list examples when appropriate.

Data Product Exponent Dividend Divisor Quotient Normal Average Mean Numerator Denominator Equivalent fraction Reduce Greatest common factor (GCF) Common denominator Least common multiple (LCM) Repeating decimal Scientific notation Coefficient Ratio Proportion Cross-multiply Metric system (SI) Meter (m) Kilogram (km) Gram (g) Liter (L) Degree Celsius (°C) Centi- Milli- Kilo- Mass Volume Meniscus x-axis y-axis Key

Time to Try Look at the other fractions we listed: 4/12 and 7/21. What is the greatest common factor for 4/12? Divide the numerator by that factor: Divide the denominator by that factor: What is the reduced fraction? What is the greatest common factor for 7/21? Divide the numerator by that factor: Divide the denominator by that factor: What is the reduced fraction?

How did you do with Time to Try? You should have found that the greatest common factor for 4/12 is 4, so 4 / 4 = 1 and 12 / 4 = 3. Thus, the fraction 4/12 reduces to 1/3. Similarly, 7/21 has a greatest common factor of 7, and 7 + 7 = 1, and 21 / 7 = 3 so, again, 7/2 reduces to 1/3. After using a number to reduce the fraction, check your result to see if it is in its simplest form or if it can be reduced further.

Time to try Calculate the mean of these body temperatures: 97.4*F 98.0*F 99.2*F 99.8*F 1. What is the mean? ______ *F 2. What does this mean mean?

If you did this correctly, you should have gotten a mean of 98.6*F, even though that was not one of the original temperatures listed, so, as stated earlier, the physiological "normal" value is a mean, and individuals' normal temperatures will vary around that mean.

Here is a tip to help you with means and with most math problems--learn to predict your results! The mean of a group of numbers will be somewhere between the highest and lowest of the numbers you are averaging. If your value does not fall in that range, check to see if you made an error. Common errors are missing a number during the addition or dividing by the wrong number.

If you estimate your result first, you can more easily recognize errors if they occur.

Adding and subtracting Fractions To add or subtract fractions, they must first be in the same format. You might think you can just add the numerators and denominators, but that won't work. By that method, 1/2 + 1/4 would equal 2/6, which reduces to 1/3. But that is smaller than 1/2, one of the numbers we added! This doesn't make sense. (See why it helps to predict your results?) Instead, you must first put the fractions into common terms. They must have the same denominator, called a common denominator. To get the common denominator, you need to know the least common multiple (LCM). This is the smallest number that can be divided by both the numerator and the denominator.

In our example of 1/2 + 1/4, 4 is the least common multiple, so we want both fractions to have 4 as their denominator. Recall that any number multiplied by 1 does not change. To convert 1/2 into fourths, we multiply it by 2/2 (=1). Thus, 1/2 x 2/2 becomes 2/4. One the fractions have a common denominator, we simply add the numerators only: 2/4 + 1/4 = 3/4 Figure 2.2 illustrates this for you.

Proportions are statements of equal ratios. A simple example would be to say 1/2 = 4/8. In science, we often use proportions to solve problems. To see how, examine a generic version: a/b = c/d Because these two ratios are equal, their cross-products are also equal, due to some basic laws of math. This means that the product of multiplying the first numerator (a) by the second denominator (d) equals the product of multiplying the second numerator (c) by the first denominator (b): a x d = c x b

Let's say we want to know how many times the heart beats in an hour. Assume that the heart beats on average 80 beats per minute. We know there are 60 minutes per hour. So, we can set up the proportion, filling in the information we know and using "x" to represent the value we are trying to determine: 80 beats / 1 minute = x beats / 60 minutes We know we can cross-multiply (Figure 2.4). When we do that, we get 4800 = x, so there are 4800 beats per 60 minutes, or per hour. In fact, there is a simpler way to write this problem, which is a shorter version of cross-multiplying. We know there are 80 beats per minute, and 60 minutes per hour, so we can calculate the beats per hour as follows: 80 beats per minute X 60 minutes per hour = 4800 beats per hour Notice that the units are shown, and in the next to last step, minutes appear on both the top and bottom. That means they cancel each other out, so we are left with beats per hour, which is the correct unit. Using the units can be an easy way to ensure that you have set up the problem correctly. This is another example of taking the time to think through the problem before you start--the units should make sense when you are done.

Time To Try Try the following problems. 1. 4 x (9 - 6) + 10 = 2. 3^3 / 9 - 4 + 5+ 3. 6^2 - 2(5 - 2) + 4 -2

Let's see how you did Problem #1: The correct answer is 22. First, do what is in the parentheses (rule 1): (9 - 6) = 3, so the problem becomes 4 x (3) + 10. Next, do the multiplication (rule 3): 4 X 3 = 12, so the problem becomes 12 + 10. Finally, do the addition (rule 4): 12 + 10 = 22. Next, Problem #2: The correct answer is 4. There are no parentheses, so start with the exponent (rule 2): 3^3 = 3 X 3 X 3 = 27, and the problem becomes 27 / 9 - 4 + 5. Next, do the division (rule 3): 27 / 9 = 3, so the problem becomes 3 - 4 + 5. Finally, do the addition and subtraction from left to right (rule 4), and you get 3 - 4 = -1, then -1 + 5 = 4. Problem #3: The correct answer is 32. Start in the parentheses (rule 1): (5 - 2) = 3, so the problem becomes 6^2 - 2(3) +4 -2. Next, take care of the exponent (rule 2): 6^2 = 6 X 6 = 36, so the problem becomes 36 - 2(3) + 4 - 2. Now, do the multiplication and division from left to right (rule 3): 2(3) = 2 X 3 = 6 and 4 - 2 = 2, so the prob lem becomes 36 - 6 + 2. Finally, do the addition and subtraction from left to right (rule 4): 36 - 6 = 30, then 30 + 2 = 32. As you see, some mathematical equations can be long and somewhat complicated, but if you keep the basic rules in mind and tackle them step by step, they become quite manageable.

MULTIPLYING AND DIVIDING FRACTIONS When doing mathematical operations with fractions, the rules are different for multiplication and division than they are for addition and subtraction. For multiplication, you simply multiply the numerators in one step, then multiply the denominators. Consider 2/3 X 3/4. The numerators are 2 and 3. Multiply them to get 6, and that goes on top. Next, multiply the two denominators, 3 X 4, to get 12. So the product is 6/12, which reduces to 1/2: 2/3 x 3/4 = 6/12 = 1/2

Let's try another: 1/3 X 2/5 X 3/4 = First, multiply all the numerators (1 x 2 x 3 = 6), then multiply all the denominators (3 x 5 x 4 = 60) and you get 6/60, which reduces to 1/10. To multiply fractions, first multiply all the numerators, then multiply all the denominators. Reduce the result as needed.

At what Fahrenheit temperature does water freeze? Boil?__ _ In the Celsius scale, normal body temperature (98.6°F) is 37°C. To convert between the two temperature scales, there are two specific equations-one to convert from degrees Celsius to degrees Fahrenheit, and another to convert in the opposite direction. Each of these equations is listed below, first in its original form, which includes a fraction, and then with the fraction converted to a decimal. You will likely use a calculator to do any conversions, and it will be easier to multiply using the decimal. °Celsius = (°Fahrenheit - 32) X 5/9 = (°Fahrenheit - 32) X 0.556 °Fahrenheit = "Celsius X 9/5 + 32 = 'Celsius X 1.8 + 32

Let's try one of each type of conversion To convert 37°C to °F we use the second equation: F = 37°C X 1.8 + 32 = 66.6 + 32 = 98.6°F Now let's convert 75°F to °C. We use the first equation as follows: °C = (75°F - 32) X 0.556 = 43 X 0.556 = 23.9°C

How do you divide when both numbers are decimals? Let's divide 2.1 by 0.7. All you have to do is move the decimal of the divisor until you have the whole number, then move the decimal of the dividend by the same number of spaces in the same direction. If you try 1.68 (dividend) / 0.3 (divisor), you move the decimal in 0.3 one spot to the right to get 3, then you must also move the decimal in 1.68 one spot to the right, getting 16.8, so the problem becomes 16.8 / 3. Do long division to get the result:

Moving the decimal may seem confusing, but there are some easy shortcuts to remember. Moving the decimal to the right one space is the same as multiplying by 10; two spaces multiplies by 100; and so on, so the numbers get bigger. Moving the decimal to the left means you are dividing by 10 for each space moved, and the number always gets smaller.

WHAT DID YOU LEARN? Try these exercises from memory first, then go back and check your answers, looking up any items that you want to review. Answers to these questions are at the end of the book. PART A: SOLVE THESE PROBLEMS. 1. (4 + 2) +6 - 5 X 2^3 2. 2 x 10^4 = 3. 27/36 reduced is 4. 3/8 X 2/3 = 5. 5/6 -7/12 = 6. 0.5 X 0.4 = 7. If the respiratory rate is 12 breaths per minute, how many breaths are taken in 1 hour? 8. 4 meters = ______centimeters 9. 1 inch = 2.54 cm, so 1 foot = _______cm 10. If a man weighs 200 pounds and 60% of his body weight is water, how many pounds of water does he have?

PART B: ANSWER THESE QUESTIONS. 1 What is the mean of 8, 9, 12, 18, and 23? 2. What is the numerator in 4/5? 3. Express 3/10 as a decimal as a percent 4. From Figure 2.11, how much of the human body is made of protein? 5. What would you be measuring if you are looking at a meniscus? 6. In the metric system, list the base unit for each of the following: mass: length: volume: 7. What is meant by "normal" blood pressure? 8. At what Celsius temperature does water boil? 9. The amount of space something occupies is called____? 10. How many milligrams are there in 1 gram?

Time to try Now that you see the simple secret to this process, complete the following conversions: 5 kg = ____ g 8 mL = _____ L 6 cm = _____ m If you did these conversions correctly, you should see that 5 kg - 5000 g; 8 mL = 0.008 L; 6 cm = 0.06 m. See, the metric system is easy! Now let's convert from metric to U.S. units, and vice versa. To do these you need the correct conversion factor (see List 2.1). Let's convert 18 inches into centimeters. From the list we see that there are 2.54 cm per inch, so we merely multiply 18 inches x 2.54 cm/inch. The answer is 45.72 cm. Let's convert 30 cm into inches. We can try a proportion to solve this. 2.54 cm / inch = 30 cm / x inches Cross-multiplying gives us 2.54 x = 30, so to get x we divide both sides by 2.54: 30 = 2.54 = 11.81 inches. You now have all the tools you need to do conversions between the two systems. The easiest solution however, is to only work in metrics, like the rest of the world!

Quick Check Complete these conversions: 1. 30 miles = _______ km 2. 8 L = ________ mL 3. 110 pounds = _______ kg 1. 30 mi x 1.6 km/mi = 48 km 2. 8 L x 1000 ml/L = 8000 mL 3. 110 pounds / 2.2 pounds/kg = 50 kg

Why should I care? All science is based on data and experimental trials. There is a certain amount of error possible with each trial. Consider the pulse values we used as examples. Three trials gave us three results. Using the mean helps to minimize the error from individual trials.

Quick Check What is meant by saying that normal human heart rate is 80 beats per minute? It means the average , or mean, heart rate for humans is 80 beats.

Fractions Fractions are written as a/b, in which a and b are both whole numbers and b is not 0. The first (top) number is called the numerator and the one on the bottom is the denominator. A fraction represents parts of some whole group (Figure 2.1). For example, 3/4 represents 3 equal parts out of 4 equal parts, where the 4 equal parts make up the whole (Figure 2.1a). Whole numbers can be represented as fractions as well. The whole number simply becomes the numerator, and the denominator is 1, so 3 = 3/1.

Quick Check: For 5/8, What is the Numerator? The denominator? Express 6 as a fraction:

Quick Check An average person takes about 12 breaths per minute. How many breaths do they take in an hour?

Set up the proportion: 12 breaths/minute = x breaths/60 minutes. Cross-multiply : 12 x 60 = 720 = x, so x = 720 breaths per hour.

You will become more familiar with the metric system as you use it. You may occasionally need to convert from U.S. units to metric units, although this is done more as an exercise than out of need--in class almost all measuring and discussion will use metric units. Still, it is useful to know how to convert between the two systems. List 2.1 shows some of the basic conversion factors. Let's try some conversions. First, let's do the easy stuff: converting: between metric units. Remember that the difference between the units will always be some multiple of 10. Let's convert 13 meters into centimeters. 13 m = ____cm A centimeter is 1/100 of a meter, so there are 100 centimeters per meter. Thus: |13 m X 100 cm/m = 1300 cm

Special relationships 1 milliliter (mL) = 1 cubic centimeter (cc), a unit often used in administering liquid medications. The mass of 1 milliliter of water = about 1 gram. the mass of 1 liter of water = about 1 kilogram. Approximate Conversion Factors Multiply inches x 2.54 cm/inch to get centimeters. Multiply feet x 0.305 m/foot to get meters Multiply miles x 1.6 km/mile to get kilometers Multiply Pounds by 2.2 pounds/kg to get kilograms. Multiply Gallons x 3.8 L/gallon to get Liters. °Celsius: (°Fahrenheit - 32) x 5/9 = (°Fahrenheit - 32) x 0.556 °Fahrenheit: °Celsius x 9/5 + 32 = °Celsius x 1.8 + 32

Measuring Volume Volume refers to the amount of space a substance occupies. In Biology labs, you will most often measure liquid volumes by using beakers or graduated cylinders. Always use the smallest container in which the substance will fit--the smaller it is, the more accurate your reading will be. Always read the scale, usually in milliliters, at eye level for accuracy. You should also know how to read the meniscus (Figure 2.8)-this is especially critical in a graduated cylinder, Liquid in a container tends to climb slightly up the sides, so the center is lower than the edges, where the liquid contacts the container. This dip is called the meniscus. When reading the scale, always read it at the low point of the meniscus for the best accuracy.

TIME TO TRY Find the narrowest clear container you can and fill it halfway with water. Look at it at eye level. Use a ruler on the outside of the container to measure the highest point of the water.________, measure the lowest point _______ The dip that you see is the meniscus. Whenever you measure a liquid volume, always measure at the lowest point of the meniscus.

Percents As you learn biology, you will encounter many values that are stated as percents. Percents are based on 100, where 100% is total. For that reason, when working with percents, always be sure they add up to 100% and no more than that. Percents are easy to work with. They are essentially fractions expressed as hundredths. For example, 25% is the same as 25/100, which can be further reduced to 1/4 And because fractions can be expressed as decimals, so can percents. You simply put the decimal two places to the left of the percent. So, 25% becomes 0.25 and 7% becomes 0.07. Percents, decimals, and fractions are all interchangeable, but when doing math operations, percents should be converted into either decimals or they must all be whole numbers, or fractions, or decimals. Table 2.1 explains the relationship between these expressions.

Table 2.1 The relationship between percents, decimals, and fractions.

Can you Feel the Power? Understanding exponents We briefly discussed exponents earlier, when we stated that a number written with an exponent is basically a multiplication problem: 2^3 = 2 x 2 x 2 = 8. In science, very large and very small numbers are often written in a special format that uses exponents based on powers of 10. This format is called scientific notation. Let's consider the number 200 to see how this format is used. To write a number in scientific notation, first place the decimal immediately after the first digit, and then drop the zeroes. This number is called the coefficient. In this example, the coefficient is 2. Next, count how many spaces you moved the decimal--two places to the left. Each of those places means 10 x 10 = 100. In scientific notation, we would write 100 as 2 x 10^2. As you can see, this is 2 x 10 x 10 = 200. Remember, the first number in scientific notation must be greater than 1 but less than 10. If a number is less than 1, the exponent is a negative power of 10. For example, 0.0004 would be 4 x 10^-4 because the decimal was moved four spaces to the right. You will rarely need to do math operations with scientific notation, so we will skip those. You should, however, understand scientific notation so that you can understand some of the measurements you will read about--such as cell sizes, which are often measured in micrometers (10^-6 meter).

Table 2.2 lists some common exponents. Time to try Express the two numbers below in scientific notation. 24,000,000 0.003 If you did this correctly, you got 2.4 x 10^7 and 3 x 10^-3.

The metric system is amazingly simple because it is all based on the number 10, which means decimals are easier to use than the U.S. system. For our purposes, we'll learn four main units used in science: those that deal with length or distance, mass, volume, and temperature. Each of these has a standard or base unit: * The basic unit of length (or distance) is the meter (m). * The basic unit of mass is, technically, the kilogram (kg), but many sources use the gram (g) as the base unit instead. The basic unit of volume is the liter (L). • The basic unit of temperature is the degree Celsius (°C). These are the base units, but more convenient units are derived from these. For example, a meter is just a bit longer than 3 feet (39.34 inches), so it is not a convenient unit for measuring the size of, say, your finger or a cell. Smaller units of the meter, based on the powers of 10, are used instead. These units are named by adding the appropriate prefix to the term meter (Table 2.3). Centi- means 1/100, and there are 2.54 centimeters (cm) in an inch, so centimeters work well for measuring fingers. Cells are microscopic, so they are best measured in even smaller units, such as micrometers—one micrometer = 1 millionth of a meter Driving between cities, you can best measure the long distances in kilometers, each of which equals 1000 meters. Again, all metric units are based on 10. Think about that-you first learn to count from 1 to 10, then you can count by tens to 100, then by hundreds to 1000, and so on. It is an easy system.

Table 2.3 provides many of the prefixes and their base-10 equivalent. In biology, you will use some units more often than others. For length or distance, which is a straight linear measurement meters, you will mostly work in meters, centimeters, millimeters (1/1000 m), and micrometers. For mass, which is the actual physical amount of something, you will most often refer refer to kilograms (1000 grams), grams, and milligrams (1/1000g). For volume, which refers to the amount of space something most common units will be liters and milliliters.

When reading a graph, always read the title first, then the axes, key, and labels. Finally, just let your eyes take in the relationships depicted. Quick Check 1. In figure 2.9, which age group is shortest? What is the typical height for children aged 1 to 3 years? 2. In figure 2.10, what percent of the total body weight is water in an average female? 3. In Figure 2.11, which molecule makes up most of the human body?

The Shortest group is aged 0-6 months, and children aged 1-3 are typically 35 inches tall. 50% Water

Quick Check Solve these problems: 3/5 + 1/4 + 1/10 = 9/16 - 3/8 =

The smallest common denominator is 20, so 3/5 x 4/4 = 12/20, 1/4 x 5/5 = 5/20, and 1/10 x 2/2 = 2/20. Then add the numerators: 12 + 5 + 2 = 19/20. The smallest denominator is 16, 20 3/8 x 2/2 = 6/16. Then 9/16 - 6/16 = 3/16.

MEASURING MASS Mass is the actual amount of something, and it is closely related to weight, but weight takes into account the force of gravity acting on the mass. Mass will be constant, but weight will vary with gravity-just ask the astronauts, who have no weight in outer space because there is no gravity. Here on Earth, where the gravitational pull is rather steady, the terms mass and weight are often used interchangeably, but you should realize that they are different. Mass can be measured with a digital scale that measures in grams, but often in labs we use the triple beam balance (Figure 2.7a). Before using the balance, be sure all the attached standard masses are pushed as far to the left as possible. Next, you must "zero" the balance before adding anything to the pan. Notice how the two lines on the far right line up. One line is on the arm and moves with it; the other is on the end piece of the scale. If they are not perfectly aligned, slowly turn the zero knob located on the left of the scale, usually under the pan. Rotate the knob in either direction until the two lines are aligned. The arm will move up and down a bit--wait until it has stopped and the lines are aligned. The scale is now zeroed. NOW you are ready to measure the mass of your object. (refer to figure 2.7 for details about your object as you weigh it) Always zero the balance before starting, with all standard masses at the far left.

There are three beams on the arm of the balance, each suspending a different standard mass. The largest mass is 100 g. Each spot that you move that mass to the right equals 100 g. Another beam has a 10-g mass, and the front beam has a 1-g mass. Starting with the 100 g mass, slide it across until it causes the arm to swing too far to the right. That was too heavy, so back it up to the left by one spot. Note the number--that is how many hundred grams are in your object. Next, slide the 10-g mass until it is too much, back up one notch, then note how many 10's of grams there are. Finally, slide the l-g mass carefully until the two lines on the right side again align, as they did at the beginning. At that point, the scale is balanced. Read all of the whole numbers on the scale; each line beyond the last whole number is 1/10 g. If the 100 g mass is at 200, the 10-g mass is at 80, and the 1-g mass is halfway between the 3 and 4, what is the mass of the object? (It would be 283,5 g, as shown in Figure 2.7b.)

Ratios and Proportions Sodium and potassium move across cell membranes in a 3:2 relationship. In the United states, the ratio of males to females at birth is about 105:100. The ratio of males to females declines steadily until, after 85, it is only 40.7:100 HUH? Welcome to the comparatively interesting realm of ratios. A ratio expresses a relationship between two or more numbers--it is a way to compare them. Ratios can be expressed using a colon between the numbers (as above), as a fraction, or by using the word "to." For example, carbohydrates contain hydrogen and oxygen in a 2 to 1 ratio, meaning that there are twice as many hydrogen atoms in carbohydrates as there are oxygen atoms. Ratios are used for comparison, and they can also be expressed as fractions. For example, a ratio of 1:2 means the same as 1/2. Look at that carefully, though. Let's say the ratio of men to women in your biology class is 1:2. We're not saying that half of the class are men, we are saying there are half as many men as women. If ratios can be expressed as fractions, they can also be expressed as decimals and percents. Because they can be written as fractions, they can also be reduced like fractions. For example. a ratio of 4:6 is the same as 4/6, which is the same as 2/3. When working with ratios it is critical to write them in the correct order. If a biology class has 10 males and 20 females, the ratio of males to females is 10:20. If we write it as 20:10, it means there are twice as many males as females, which is not true.

Time to Try Empty your pocket or purse of change. Separate the coins by denomination. Count all of the coins in each category. Now express those numbers in a ratio: Why is it important to indicate the order in which you are listing the coins? If you did this correctly, you should have indicated the order of the coins, because without that reference, we have no idea what number corresponds with which coin. Perhaps you had 5 pennies, 4 nickels, 3 dimes, and 2 quarters. If you wrote your ratio in that order, it would be 5:4:3:2.

Subtracting fractions also requires a common denominator. To get the common denominator, you need to know the least common multiple (LCM). This is the smallest number that can be divided by both the numerator and the denominator. Once you have the denominator, simply subtract the second numerator from the first denominator. Let's try one: 1/3 - 1/4. The smallest common denominator for these two fractions is 12, so they both need to be converted, as follows: 1/3 x 4/4 = 4/12 1/4 x 3/3 = 3/12 Now line up your fractions in the correct order from left to right, then subtract the second numerator from the first: 4/12 - 3/12 = 1/12

To add or subtract fractions, use a common denominator to put the fractions into a common form, then add or subtract the numerators only. Remember to always subtract from left to right.

Dividing Fractions may seem difficult at first, but a simple trick actually makes it easy! These problems may be written two different ways: (see attached) Solving them is easy. First, invert (flip) the second fraction, which is the divisor: 2/3 becomes 3/2. Then you simply multiply the two fractions: 4/5 / 2/3 = 4/5 x 3/2 = 12/10 Now, 12/10 can be reduced to 6/5. Here the numerator is larger than the denominator, which means this fraction is greater than 1. Usually when this happens, it is best to express the answer as a mixed number--one combining both whole numbers and fractions. To do this, first reduce the fraction: 12/10 = 6/5. Then realize that 6/5 = 5/5 + 1/5. Since 5/5 equals 1, the mixed number would be 1-1/5 (read as 1 and 1/5).

To divide one fraction by another, first invert the second fraction to turn it into a multiplication problem. Next, multiply the numerators, then multiply the denominators. Finally, reduce the result.

Multiplying and dividing decimals Multiplication and division of decimals is a bit trickier because you must keep track of how many decimal places you should have at the end. Let's try an easy one: 0.5 x 0.3. First, multiply the numbers as if they are whole numbers: 5 x 3 = 15. Now, add the numbers as if they are whole numbers: 5 x 3 = 15. Now, add the number of decimal places you started with. Both numbers you multiplied originally had one decimal place, so that adds up to two. Realize that your answer of 15 is 15.0, so you know where the decimal begins. Now you have to move the decimal. The numbers that you started with had a total of two digits after the decimal, so you must move the decimal point left by two places, giving you 0.15. Here is a way to double check that. If the original numbers were fractions, they would be 3/10 and 5/10. Recall how to multiply fractions--you multiply the numerators, then multiply the denominators: 3/10 X 5/10 = 15/100 = 0.15

What if the numbers had been 0.03 x 0.5? Although you still get 15, now you need to move the decimal three places to the left., but there are only two. You simply add zeros to the left until you have the correct number of decimal places, in this case giving you 0.015.

Now we will convert 27 millimeters into centimeters, but let's try another method. All we really have to do to convert between metric units is move the decimal, but by how many spaces and in which direction? How many spaces you move is determined by the difference in the power of 10. We know that millimeters are thousandths of a meter, and centimeters are hundredths of a meter. millimeters = 10^-3 centimeters = 10^-2 So, if we look at the exponents, they are different by one. We will move the decimal in our number (27) by one spot. But in which direction? When converting from smaller to larger units, the decimal moves left. When converting from larger to smaller units, the decimal moves right. Back to our example: Converting 27 mm to cm gives us 2.7 cm.

When Converting within metric units: 1. Put the units in scientific notation and subtract the smaller exponent from the larger one. The difference is how many spaces the decimal will move in your coefficient. 2. If you are converting from small units to larger ones, the number gets smaller, so the decimal moves to the left. If you are converting from larger units to smaller ones, the number gets bigger, so the decimal moves to the right.

Picture this Assume that you have worked three extra jobs for pocket cash this week. From them you earned $33.70, $45.28, and $21.02. How much extra money did you earn? With this money you buy a pizza for $8.99, soft drinks for $1.49, gas for $20, and a new CD for $19.95. How much do you have left? Congratulations, you just added and subtracted decimals, as you do on a regular basis in daily life. You should see that you earned $100 and have $49.57 left.

When adding or subtracting decimals, always align the decimal point in the two numbers before doing the operation.

The metric system In the United States, we all grew up learning there are 12 inches to a foot, 3 feet to a yard, and 100 yards to a football field, and we measure driving distance in miles, which contain 5280 feet. In the kitchen, we use cups, half cups, quarter cups, table spoons, teaspoons, eights of teaspoons, pints, quarts, and gallons. There are so many units in our system it is amazing we can keep them straight. But of course there is a simpler way to measure. It is called the metric system, or System Internationale (SI). It is universally used in science and by almost every country in the world except the United States. You have undoubtedly had brushes with learning the metric system, and you may have found it delightfully simple. Instead, the problem is with out complicated U.S. (also called English) system and the need to convert between the two systems. This requires -- you guessed it -- math.

Why should I care? Science uses metric measurement almost exclusively, so you will need a basic understanding of metric units for all your future coursework. In addition, almost everyone on our planet, except the United States-uses the metric system.

Math in Science You probably remember doing story problems when learning math in your younger years. Those problems helped you see how math can be used. Many students are surprised to learn that they have to use math in biology. But you must remember that science—all science-deals will that which is testable. A scientific test, as you know, is called an experiment.. Results collected from experiments are called data and, more often than not, the data are numbers. When you try to make sense of the data, you are working with numbers, and that means math. Many students entering this class may only need a brief reminder of what they learned before, whereas others may need to learn it again. Regardless of your math history a quick refresher will help you better understand the numbers

You will likely do some experiments in lab and then analyze the data. These experiments may involve the study of physiology, the internal processes of living organisms, as a part of biology. For example, you may investigate one of the many aspects of body function that have "normal" conditions, these are often expressed in numerical values. For example, normal human body temperature is 98.6*F, normal blood pressure is 120/80, and normal pulse is around 70 to 80 beats per minute. In addition, you will work with chemical solutions and you will need to understand their concentrations. You will also measure in metric units, refer to percentages and ratios, and interpret graphs and charts