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Taking measurements, significant digits and scientific notation

Science 10. Taking measurements, significant digits and scientific notation. Uncertainties of Measurements.

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Taking measurements, significant digits and scientific notation

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  1. Science 10 Taking measurements, significant digits and scientific notation

  2. Uncertainties of Measurements All measurements are subject to uncertainties. All instruments used are influenced by external circumstances, and the accuracy of a measurement may be affected by the person making the reading. In everyday language, the words precise and accurate mean roughly the same thing . . . not in science.

  3. Accuracy A measurement is accurate if it is close to the accepted value. The accuracy of an individual measurement depends on the quality and performance of the instrument used to make the measurement.

  4. Precision The precision of a measurement is the degree of exactness to which it can be reproduced. After taking a lot of measurements, you will find that they are close to each other. The precision of an instrument is limited by the smallest division on the measurement scale.

  5. For example, A measurement of the mass of the clock made with this scale would be more precise than a measurement made with this one.

  6. Recording Measurements When you read any measuring device, you always record the measurement by . . . . reading the smallest division on the scale and then “guessing at” or estimating, the tenth of the smallest division. These estimates create uncertainty!

  7. Record the correct readings on the ruler. e) a) 13.50 cm c) 12.63 cm 14.35 cm b) 11.10 cm d) 12.26 cm = estimated digit

  8. Record the correct readings from the thermometers. 32.6°C 8.95°C 8.2°C - 8.3°C = estimated digit

  9. Significant Digits and Scientific Notation The value that an answer is expressed in is determined by significant digits SIGNIFICANT DIGITS Significant digits are the specific number of digits used to communicate the degree of uncertainty in a measurement

  10. Significant Digits I can “sig dig” it. Grooovy baby! Significant digits are all those digits obtained from a properly taken measurement: all of the certain digits plus the one estimated digit. The number of significant digits indicates the precision of the measurement. More sig digs means a more precise measurement. For example, 10.0 cm 10.0365 cm less precise more precise

  11. Rules for counting significant digits • Digits 1-9 are significant For example, 321, 0.321, 0.00321, 3.21 and 3.21 x 103 all have 3 significant digits • The position of any zeros determines whether or not the are significant • LEADING ZEROS ARE NOT SIGNIFICANT • For example, 0.321 and 0.0321 both consist of only 3 significant digits • TRAILING ZEROS ARE SIGNIFICANT • For example, 4.00, 400 and 4.00 x 105 all consist of 3 significant digits • ZEROS IN BETWEEN DIGTS 1-9 ARE SIGNIFICANT • For example, 204.6 has 4 significant digits, 10.204 has 5 significant digits

  12. Exact numbers have an infinite number of significant digits, because they do not involve an estimated measurement. Exact numbers are: eg. 1000 m = 1 km 1 dozen = 12 a) numbers that are definitions b) numbers that result from counting objects eg. 40 students 150 books

  13. For each of the following measured values, indicate the number of significant digits. these zeros are placeholders 1) 12.7 m 3 2) 10.03 kg 4 3) 0.0034 s 2 4) 200 min 3 5) 200 students infinite 6) 2.746 x 1012 4

  14. What happens when we perform mathematical operations with these estimated, measured values? The result contains some significant digits and some that are not because the arithmetic was performed with uncertain numbers. Our answer can NEVER be more precise than the least precise value we used in our calculations. Measurements are subject to uncertainty! Always remember:

  15. Rules for performing mathematical operations ADDING OR SUBTRACTING The answer MUST have the same number of decimal places as the number with the least number of decimal places in the original data Example: 2.9 + 8.276 13.45 24.626  Since 2.9 has 1 decimal place, the final answer should also have 1 decimal place 24.6

  16. Try these.... 1) 42.3 g + 16.452 g = 58.8 g 2) 924 + 63.2 + 27.54 = 1015 3) 205.65 kg – 60 kg = 146 kg

  17. MULTIPLYING OR DIVIDING The answer should be rounded to the least number of significant digits Example: 2.3 X 0.05 X 2.0 = 0.23  0.2 Since 0.05 has only one significant digit, the final answer should have only one significant digit

  18. IMPORTANT!! When doing a series of calculations to arrive at an answer, never round off your sub-steps. Only round your final answer. If possible, keep the significant digits in your calculator. A good rule of thumb is to write three more significant digits than you’ll need to round to at the end.

  19. Try these... Use scientific notation when the number of sig digs in your calculator is greater than the number you need to round to! 1) 1.32 m x 0.4 m = 0.5 m2 2) 8.33 x 10-5 3) 25.7 x 2403 = 6.18 x 104 4) 0.0028 x 983 ÷ 6.5 x 10-7 = 4.2 x 106

  20. Scientific Notation In science, you often deal with extremely large or small numbers. To express these without using a lot of zeros, scientists use scientific notation Scientific notation is always written in the form a x 10 b where a is the coefficient and b is the exponent • The coefficient tells you how significant ( # of sig digs) the number is • a is ALWAYS a number between 1-9!!!!!! • The exponent tells you where the decimal is

  21. A zero or positive exponent tells you that the number is larger than one. • Decimal is to the RIGHT of the first number • Example: 1.23 x 104 = 12 300 (decimal moved 4 spaces to the right) • A negative exponent tells you that the number is smaller than one. • Decimal is to the LEFT of the first number • Example: 1.23 x 10-6 = 0.00000123 (decimal moved 6 spaces to the left)

  22. Try these... • Express 5904m in scientific notation • Express 0.000 072mm in scientific notation • Express 478356 cm in scientific notation • Express 0.00000000034532 dm in scientific notation

  23. Answers • 5.904 x 103 m • 7.2 x 10-6 mm • 4.78356 x 105 cm • 3.4532 x 10-10 dm

  24. Homework: • read student reference #6- Significant digits and scientific notation page 467-468 • Skills practice #1-3 page 132 • Worksheet –scientific notation and significant digits

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