Why do we need a measurement system? Measurement Systems Why do we need a measurement system? There’s one thing you need to know before we can begin talking about biotechnology applications. You need a quick math review! This poweroint is going to teach you how to identify significant figures, calculate in scientific notation, and convert numbers in the metric system.

Scientific Notation A way to write very large and very small numbers. A number in scientific notation is written in two parts, the coefficient and an exponent of 10. 5 x 1022 coefficient exponent of 10 In biotechnology, we often work with numbers that are very small. Our hand gets tired from writing out so many zeroes, plus it’s hard to read, so we write numbers in scientific notation. Scientific notation is a way to write very large and very small numbers. In scientific notation, all numbers are expressed by a coefficient (number between one and 9) multiplied by a power of 10. The exponent tells how many times 10s are multiplied together. For example 10 to the second means 10 times 1, or 100.

Scientific Notation Changing standard numbers to scientific notation Numbers greater than 10 Move decimal until only ONE number is to the left of the decimal. The exponent is the number of places the decimal has moved and it is POSITIVE. Ex. 125 = 15,000,000,000 = Numbers greater than 10 are written with a positive exponent. To convert a large number written in standard notation as a number in scientific notation you will need to move the decimal until there is only one number to the left of the decimal. The exponent is the number of places the decimal has moved, and it is positive. So for the number 125, you will have to move the decimal two places to the left. That means the exponent will be 2. To write 125 in correct scientific notation, you will write 1.25 times ten to the second power. To write 15 billion in scientific notation, you need to move the decimal 10 places to the left, so you will write 1.5 times ten to the 10th power. 1.25  102 1.5  1010

Scientific Notation Changing standard numbers to scientific notation Numbers less than 1 Move decimal until only one number is to the left of the decimal. The exponent is the number of places the decimal has moved and it is NEGATIVE. Ex. 0.000189 = 0.5476 = Numbers less than 10 are written with a negative exponent. To convert a small number in standard notation to scientific notation, you will again move the decimal until there is only one number to the left of the decimal. The exponent is still the number of places the decimal has moved, but this time it is negative. So for 0.000189 you will move the decimal 4 places to the right and the number written correctly in scientific notation will be 1.89 times 10 to the negative 4. The example 0.5476 only requires the decimal to move one place to the right, so this number written in correct scientific notation will be 5.476 times ten to the negative one. 1.89  10-4 5.476  10-1

Scientific Notation Changing standard numbers to scientific notation To change a number written in incorrect scientific notation: Move the decimal until only one number is to the left of the decimal. Correct the exponent. (remember: take away, add back) Ex. 504.2  106 = 0.0089  10-2 = If a number is written in incorrect scientific notation, meaning there is more than one number to the left of the decimal, you will move the decimal until there is only number one number to the left of it. You have to change the exponent to reflect the movement of the decimal. So in the first example the decimal moved two places to the left and caused the number to be smaller (from 504 to 5). Therefore, the exponent will need to increase by 2 to reflect this change and the 6 becomes an 8. So 504.2 times ten to the sixth should be written as 5.042 times ten to the eighth. In the second example, the decimal has moved 3 places to the right and has caused the number to become larger (from 0 to 8). Therefore, the exponent will need to decrease by 3 to reflect this change and the -2 becomes a -5. So this number written correctly is 8.9 times ten to the negative five. 5.042  108 The coefficient decreased by 2, so the exponent must increase by 2 8.9  10-5 The coefficient increased by 3, so the exponent must decrease by 3

Scientific Notation Changing numbers in scientific notation to standard notation If the exponent is (+) move the decimal to the right the same number of places as the exponent. 1.65  101 = 16.5 1.65  103 = 1650 If the exponent is (-) move the decimal to the left the same number of places as the exponent. 4.6  10-2 = 0.046 1.23  10-3 = 0.00123 Sometimes a number will be written in scientific notation and you will want to change it to standard notation. If the exponent is positive, you will move the decimal to the right the same number of places as the exponent. In example 1a, 1.65 times ten to the one tells you to move the decimal one place to the right and the number in standard notation is 16.5. In example 1b, the exponent is 3 so you will need to move the decimal 3 places to the right, so the number in standard notation is 1650. If the exponent is negative, you will move the decimal to the left the same number of places as the exponent. So in example 2a, 4.6 times ten to the negative two tells you to move the decimal two places to the left. This number in standard notation is 0.046. In example 2b, the exponent -3 tells you to move the decimal three places to the left and the number written in standard notation will be 0.00123.

Scientific Notation Multiplication and Division in Scientific Notation To multiply numbers in scientific notation Multiply the coefficients. Add the exponents. Convert the answer to correct scientific notation. Ex: (2  109) x (4  103) = 8 x 1012 To multiply numbers in scientific notation you will multiply the coefficients and add the exponents. So in the example, 2 multiplied by 4 is 8 and the exponent will be 9 + 3 which is 12. The final answer is eight times ten to the twelfth.

Scientific Notation Multiplication and Division in Scientific Notation To divide numbers in scientific notation Divide the coefficients. Subtract the exponents. Convert the answer to correct scientific notation. Ex: (8.4  106)  (2.1  102) = 4 x 104 To divide numbers in scientific notation you will divide the coefficients and subtract the exponents. So in the example, 8.4 divided by 2.1 is 4 and the exponent will be 6 – 2 which is 4. The final answer is four times ten to the fourth.

Scientific Notation Addition and Subtraction in Scientific Notation Before numbers can be added or subtracted, the exponents must be equal. Ex. (5.4  103) + (6.0  102) = (5.4  103) + (0.6  103) = 6.0  103 Adding and subtracting numbers in scientific notation is a little different. The exponents have to be the same before you can add or subtract the coefficients. So in the example, let’s change the exponent of 2 to be a 3 so that it matches the exponent in the first number. Remember the take away/add back rule. If we change the exponent to be one more, we must move the decimal so that the coefficient is one less. Therefore the 6 becomes 0.6. So now, to add 5.4 times ten to the third with 0.6 times ten to the third, we can just add the coefficients together and keep the exponent the same. 5.4 + 0.6 is 6 and the exponent remains 3. The final answer is 6 times ten to the third.

Significant Figures Are all the numbers for which actual measurements are made plus one estimated number. 1 2 You would estimate this measurement as 1.5 First of all, let’s talk about how we take measurements. It is important to make accurate measurements and to record them correctly. No physical measurement is exact, there is always some degree of uncertainty. The number of significant figures in a quantity is an indication of this uncertainty. When recording data, the number of significant figures is the number of known digits plus one estimated digit. For example, look at the top ruler. Let’s say this is a meter stick. What number would you record for this measurement? You know for sure it is between 1 and 2, so the 1 is the accurate digit. But you will have to guess where it is between the one and the two. I think it looks half-way, so I will record 1.5 m. The 0.5 is estimated. See? Now you try on the second ruler. This time you know for sure it is between 1.4 and 1.5, but you will have to guess how far between. My guess is 1.48, but you may have guessed 1.47. Do you see how that last number is an estimate? Good! 1 2 You would estimate this measurement as 1.48

Significant Figures Tells the person interpreting your data about the accuracy of the measuring instrument used to obtain the data. It is important to record your data with the correct number of significant figures because it tells the person interpreting your data about the accuracy of the measuring instrument. For example, on the electronic balance in this picture you would record your data with 2 decimal places, even if the last number is a zero. This brings up an important question. How do you know when zeroes are significant? I’m so glad you asked! Let’s talk about rules for counting significant figures.

Significant Figures Rules for counting sig figs 1. Digits other than zero are always significant. 96 = 2 sig figs 61.4 = 3 sig figs 2. Zeroes between 2 other sig figs are always significant. 5.029 = 4 sig figs 306 = 3 sig figs If a number is not zero, it is significant. Period. So in example one, 96 has 2 significant figures and 61.1 has three. That’s easy, but what about zeroes? Well, if zeroes are sandwiched between 2 non-zero numbers they are significant. So in example 2, 5.029 has 4 signifcant figures and 306 has three.

Significant Figures Rules for counting sig figs Leading zeroes are never significant when they are to the left of non-zero numbers. 0.0025 = 2 sig figs 0.0821 = 3 sig figs Trailing zeroes are only significant if there is a decimal present and they are to the right of nonzero numbers. 100 = 1 sig fig 100.0 = 4 sig figs 0.0820 = 3 sig figs It gets a little trickier when zeroes are in front or at the end of a number. Leading zeroes are never significant when they are to the left of non-zero numbers. So in example 3, only the non-zero numbers are significant. That means that 0.0025 has only two significant figures (the 2 and the 5) and 0.0821 will have three significant figures. The trailing zeroes are a little trickier. The trailing zeroes are only significant if there is a decimal present AND they are to the right of non-zero numbers. So, in example 4a, there is no decimal so those zeroes are not significant. The number 100 has only one significant figure. However, in example 4b the addition of the decimal causes all of the zeroes to be significant. So 100.0 will have 4 significant figures. In example 4c, the leading zeroes are only place holders and are not considered significant. The zeroes trailing after the decimal point are significant because, for example, they tell you that the measurement was taken on a balance that goes to 4 decimal places. So 0.0820 has three significant figures.

Significant Figures Rules for calculating with sig figs In addition and subtraction, the answer should be rounded off so that it has the same number of decimal places as the quantity having the least number of decimal places. 1.1 + 225 = 226.1 = 226 (rounded to no decimal places) 2.65 – 1.4 = 1.25 = 1.3 (rounded to 1 decimal place) In multiplication and division, the answer should have the same number of significant figures as the given data value with the least number of significant figures. 4.60  45 = 207 = 210 (rounded to 2 sig figs) 1.956  3.3 = 0.5927 = 0.59 (rounded to 2 sig figs) Sometimes you will need to perform calculations on your data. The number of significant figures in your calculation are going to be determined by the data you recorded. When you are adding and subtracting, the answer will be based on the data with the least number of decimal places. Therefore, in example 1a, when you add those two numbers together the sum will have to be rounded to 0 decimal places because the data point 225 does not have any decimal places. So 226.1 will have to be rounded to 226. In example 1b, the data point 1.4 has only one decimal place, which is the least. That means the answer 1.25 will be rounded to 1.3 so that there is only one decimal place. When you are multiplying and dividing numbers with significant figures, the answer will be based on the data with the least number of significant figures. Therefore, in example 2a, you need to determine the number of significant figures for each data point. 4.60 has three because the zero follows a decimal, it is significant. 45 only has two significant figures. Two is less than three, so the answer 207 will have to be rounded to two significant figures. The answer will be 210. Wait a minute. 210….doesn’t that have 3 significant figures? No! There is no decimal, so the zero is a placeholder, but not significant. The number 210 only has 2 significant figures. In example 2b, the answer will also have to rounded to 2 significant figures because the data point 3.3 has only 2 significant figures, which is the least number of significant figures. So we round 0.5927 to 0.59.

Metric System Unit of length…..meter (m) Unit of mass ……gram (g) Unit of volume …liter (L) Unit of time …….second (s) Unit of temperature…degrees Celsius (°C) Whew! That’s a lot of math! Now that you understand about significant figures and scientific notation, let’s move on to talk about the international system of measurement. Did you know that prior to the 1800’s the system of measurements varied from country to country and depended on the body parts of the ruling monarch. For example, a foot was the length of the ruling monarch’s foot. As Napoleon conquered countries in Europe he wanted to use the laborers in these countries to produce the artillery he would use to conquer the next country. Since every country had different measuring systems, he found it impossible to have the conquered countries manufacture his needed supplies. In 1793 he ordered a consortium of French scientists to come up with an international system of measurements. This is what we call the SI system of measuring, or the metric system. The SI unit of length is the meter, the SI unit of mass is the gram, the SI unit of volume is the liter, the SI unit of temperature is degrees Celsius and the SI unit of time is the second.

Metric System The metric system is based on units of 10. Prefix symbol Prefix name Prefix value Fraction or Multiple Power G giga one billion 1,000,000,000 109 M mega one million 1,000,000 106 k kilo one thousand 1000 103 BASIC UNIT: m, g, L, 1 10 d deci 1/10 0.1 10-1 c centi 1/100 0.01 10-2 m milli 1/1000 0.001 10-3 µ micro 1/1,000,000 0.000 001 10-6 n nano 1/1,000,000,000 0.000 000 001 10-9 The metric system is very useful because it is based on units of 10. In the chart above you can see that units larger than the base include the kilo, which is 1000 times bigger than the base unit, Mega which is a million times larger and giga which is one billion times larger. Conversely, the units smaller than the base include the deci, which is 10 times smaller, or one tenth the size of a base unit. Centi is 100 times smaller, milli is 1000 times smaller and micro is a million times smaller. We most frequently work on the micro scale in the biotechnology lab. You will get very comfortable converting between milli and micro units. So, how do you do this conversion?

Metric System To convert measurements within the metric system is a simple matter of multiplying or dividing by 10, 100, 1000, etc. Even simpler, it is a matter of moving the decimal point to the left or right. Well, because the metric system is based on units of 10 it is mathematically as simple as multiplying and dividing by factors of 10. Practically it’s as simple as just moving the decimal the correct number of places to the left or the right.

Metric System Ex. 3 mg = 3000 µg Ex. 3 L = 0.003 kL One way to know where to place the decimal is to draw a "metric line" with the basic unit in the center, marking off six units to the left and six units to the right. To convert from one unit to another simply count the number of places to the left or right, and move the decimal in that direction that many places. One way to know how far to move the decimal is to draw a metric line as the one shown here with the base unit in the center and marking off 6 units to the left and right of the base unit. To convert between units, you simply count the number of lines between units and move the decimal that number of places. For example, to convert 3 milligrams to micrograms you will have to move three lines to the right, that means you move the decimal over three places to the right and 3 mg becomes 3000 ug. The example on the left shows how to convert 3 liters to kiloliters. You will need to move 3 lines to the left which means you need to move your decimal 3 places to the left and 3 L becomes 0.003 kL. Ex. 3 mg = 3000 µg Ex. 3 L = 0.003 kL

Two Systems M e t r i c E n g l i s h yard, mile, feet M e t e r Gram Liter Celsius E n g l i s h yard, mile, feet pound, ounce quart, gallon Fahrenheit Why is the US not on the metric system? Napoleon never conquered England. The United States was established as English colonies, so it ended up using the English measurement system. Ironically, the British and its Commonwealth Countries now use the metric system, but the US still holds out. However, most of our technical measurements and all scientific measurements are in metric.

F a c t o r - L a b e l T h e m o s t i m p o r t a n t m a t h e m a t i c a l p r o c e s s f o r s c i e n t i s t s . T r e a t s n u m b e r s a n d u n i t s e q u a l l y . M u l t i p l y w h a t i s g i v e n b y f r a c t i o n s e q u a l t o o n e t o c o n v e r t u n i t s . One way to convert between the metric and the English system is to use the factor label method.

F a c t o r - L a b e l A f r a c t i o n W h a t i s g i v e n e q u a l t o o n e W h a t i s g i v e n

F a c t o r - L a b e l H o w m a n y b a s k e t b a l l s c a n b e c a r r i e d b y 8 b u s e s ? 1 bus = 12 cars 3 cars = 1 truck 1000 basketballs = 1 truck

F a c t o r - L a b e l H o w m a n y b a s k e t b a l l s c a n b e c a r r i e d b y 8 b u s e s ? 1 bus = 12 cars 3 cars = 1 truck 1000 basketballs = 1 truck 8 buses

F a c t o r - L a b e l 12 cars 8 buses 1 bus H o w m a n y b a s k e t b a l l s c a n b e c a r r i e d b y 8 b u s e s ? 1 bus = 12 cars 3 cars = 1 truck 1000 basketballs = 1 truck 8 buses 12 cars 1 bus

F a c t o r - L a b e l 8 buses 12 cars 1 truck 1 bus 3 cars H o w m a n y b a s k e t b a l l s c a n b e c a r r i e d b y 8 b u s e s ? 8 buses 12 cars 1 truck 1 bus 3 cars

F a c t o r - L a b e l 12 cars 1 truck 1000 bballs 8 buses 1 truck H o w m a n y b a s k e t b a l l s c a n b e c a r r i e d b y 8 b u s e s ? 12 cars 1 truck 1000 bballs 8 buses 1 truck 1 bus 3 cars

F a c t o r - L a b e l 3 2 0 0 0 b a s k e t b a l l s c a n b e c a r r i e d b y 8 b u s e s .

F a c t o r - L a b e l C o n v e r t 5 p o u n d s t o k i l o g r a m s .

F a c t o r - L a b e l C o n v e r t 5 p o u n d s t o k i l o g r a m s 5 lb 1 k g = 2 . 2 7 k g 2 . 2 0 lb

F a c t o r - L a b e l C o n v e r t 8 . 3 c e n t i m e t e r s t o m i l l i m e t e r s .

F a c t o r - L a b e l C o n v e r t 8 . 3 c e n t i m e t e r s t o m i l l i m e t e r s 8.3 cm 1 m 100 cm 1000 mm 1 m = 83 mm

F a c t o r - L a b e l M e t h o d