Scientific Notation. A measurement written in scientific notation consists of: 1. a between 1 and 9.99 followed by 2. a of 10.

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1 Scientific Notation A measurement written in scientific notation consists of: 1. a between 1 and 9.99 followed by 2. a of 10. The number in front of the power of 10 is called the. When creating coefficients, the decimal point in the original number will be moved in a direction that will provide in front of the. Example: "2300" would be written as "2.3 x 10 3 " in scientific notation " " would be written as "7.2 x 10-4 " in scientific notation. "2.3" and "7.2" are the coefficients. Explination: In order to change 2300 into its corresponding number in scientific notation, the (invisible) decimal point at the end of the number had to be moved 3 places to the left. Each place represents a power of 10. Since "2.3" (the coefficient) is much smaller than "2300", these places are being stored. So, we represent this in the number with "x 10 3 ". In order to change " " to the coefficient "7.2", the decimal point was moved four places in the opposite direction. The coefficient "7.2" is much larger than the original number of " ". In this case, those 4 places were "borrowed". A loan was made from the bank. This is expressed as "x 10-4 ". *Note: Observe the way " " has been written with a space between the "0" and the "7". This is not a typographical error. Our use of the comma in measurements can be confusing in some parts of the world. In certain countries, a comma is used where we would use a decimal point. For instance, where we might write "1.00", other countries would write "1,00". Clear communication is extremely important in science. In an effort to avoid confusion, it has been agreed that a decimal point will be used as we use it; however, the use of commas, as in "460,000,000" will be dropped in favor of a blank space If everyone is consistent, confusion will be avoided. For addition, subtraction, multiplication and division we will use our calculators.

2 Scientific Notation Practice: Format--Write the following numbers in proper notation x x x Multiplication-- 1. (2 x 10 9 ) x (4 x 10 3 ) 2. (6.2 x 10-3 ) x (1.5 x 10 1 ) 3. (3.4 x 10-3 ) x (2.5 x 10-5 ) Division: x 10 6 / 2.2 x x x x x 10-9 Addition & Subtraction: 1. (5.4 x 10 4 ) + (2.7 x 10 4 ) 2. (9.4 x 10-2 ) - (2.1 x 10-2 ) 3. (6.6 x 10-8 ) - (4.0 x 10-9 ) 4. (6.7 x 10-2 ) - (3.0 x 10-3 ) 5. (2.34 x 10 6 ) + (9.5 x 10 4 ) 6. (5.7 x 10-3 ) + (2 x 10-5 ) Significant Digits When making, we may not always be able to use the same equipment each time. Perhaps we are recording values in a laboratory as well as out in a swamp. The very expensive laboratory equipment may allow us to make measurements to decimal places. The less expensive field equipment will only permit decimal places. Even though we will record measurements with as many decimal places as the equipment will allow, we will report answers in the number of decimal places of which we are sure in all measurements. In this example, that is 2 decimal places. This is called expressing the measurement in " " or "significant figures".

3 There are rules concerning which numbers are considered to be significant in a measurement: 1. Every nonzero digit in a recorded measurement is significant has 3 significant digits. 2. Zeros appearing between nonzero digits are significant has 4 significant digits. 3. Zeros appearing in front of nonzero digits are NOT significant. They are acting as place holders has 2 significant digits. 4. Zeros at the end of a number and to the right of a decimal point are significant has 4 significant digits. (The measured zero, in this case, is significant, because it tells us that the equipment had the capability to measure to that point. Had a nonzero measurement been there, it would have shown up.) 5. Decimal points will be used with zeros at the end of a measurement to show that the zeros were measured and to distinguish the measurement from an estimation. "300"--shows that this value is an estimate. It has 1 significant figure. "300.00"--shows that this value was actually obtained when measuring to the second decimal place. This number has 5 significant figures. 6. In scientific notation, count the significant figures from the coefficient x 10 2 has 3 significant figures. BUT A handy way to count significant digits without memorizing all of these rules is the Rule: Pacific Atlantic If the number in question has a decimal point: (1), moving from left to right, begin counting with the first number encountered and continue counting to the end of the number. Zeroes to the left of the first nonzero number will not count; zeroes to the right of the first nonzero number will count. (2) T, moving from right to left, begin counting with the first number encountered and continue counting to the end of the number. Zeroes to the right of the first nonzero number will not count; zeroes to the left of the first nonzero number will count.

4 Rounding: A final answer be reported with accuracy than the measurement in a group of numbers which shows the accuracy. Final answers will need to be rounded. Rules of Rounding: 1. If the digit immediately following the last significant digit is less than, all the digits after the last significant place are dropped. 2. If the digit following the last significant digit is 5 or, the value of the digit in the last significant place is increased by. Addition & Subtraction: The final answer can have no more than the measurement with the number of decimal places. e.g., a) has 2 decimal places has 1 decimal place has 2 decimal places. The final reported answer can only have decimal place because of the measurement of Calculate the answer (369.76) and then round it to the appropriate number of decimal places. The reported answer will be "369.8". b) = Because has only decimal places, the final answer must be rounded to decimal places. This results in a reported answer of Multiplication & Division: The final answer can have no more than the measurement with the total significant digits. e.g., a) 7.55 x 0.34 Calculate the answer (2.567) and then round to the least total number of significant digits. The measurement 7.55 contains total significant digits; 0.34 contains total significant digits. The final answer then can have only total significant digits and must be reported as 2.6. b) / 8.4 = Since 8.4 contains only significant digits, the final answer must be rounded to significant digits, Rounding: *CAUTION-- a) 8792 m (round to 2 significant digits) Since "9" is greater than 5, the "7" in the number must be rounded up to "8". Unfortunately, due to carelessness, the magnitude of the number is often lost in the reporting and an answer of "88" results. The correct response should be. Insert zeroes as place holders in these situations.

5 b) 89.8 (round to 2 significant digits) Since "8" is greater than 5, the "9" in the number must be rounded up. "89" rounds to "90". Unfortunately, "90" contains only one significant digit. If you try to fix this by placing a decimal point behind the zero, you must also place another zero behind the decimal point giving your answer 3 significant digits (90.0). So, are we lost and without hope? Not a chance. It's Scientific Notation to the rescue!! Change "90" into scientific notation and you can create a number with 2 significant digits, 9.0 x 10 1 or you can place line over the last significant digit or you can place a decimal behind the zero, 90. c) (round to 2 significant digits) Resist the temptation to round the "4" to a "5" first and then perform the task of rounding to 2 significant digits. The answer will correctly be Significant Digits Practice: Counting: How many significant figures are in each of the following? m cm mm x 10 4 m m L m m Round each measurement to 3 significant digits and then round each measurement to 1 significant digit m x 10 8 m m m x 10-3 m m Addition & Subtraction: m m m m m m m m m g g + 19 g L L Multiplication & Division: m x 2.22 m 2. (1.8 x 10-3 m) (2.9 x 10-2 m) m / x 10-2 / / m x 3.21 m x 1.871

6 Precision and Accuracy refers to the agreement of a particular value with the true value. refers to the degree of agreement among several measurements made in the same manner. Differences Accuracy can be true of an measurement or the of several Precision requires measurements before anything can be said about it See examples