1. Scientific Measurements Measurements are always approximate. They depend on the precision of the measuring instrument – that is the amount of information that the instrument can provide. Look at the graduations on a beaker versus the graduations on a graduated cylinder. Which instrument will provide better information?

4. During the investigations this year, you will use equipment such as rulers, balances, scales, graduated cylinders, beakers, and thermometers to measure matter and use that data to perform calculations. Can you just plug all sorts of numbers into your calculator and get an answer. Is the answer with the most digits is the best? How exact can your measurements and calculations be? How exact should they be?

5. Significant Digits (also known as significant figures) All measurements involve uncertainty. One source of this uncertainty is the measuring device itself. The other source of uncertainty is your ability to perceive and interpret a reading. The fact is, you cannot measure anything with complete certainty. The last (farthest right) digit in any measurement is always an estimate.

6. Estimating If you look carefully on most measuring scales, you can estimate readings between lines. Scientists agree to add only one additional figure to a measurement. Significant digits include the digits that are certain and a final, uncertain digit that you estimate.

9. Measuring Correctly When measuring with a mechanical instrument, record all the digits marked on the instrument ’ s scale and then estimate one more digit. When measuring with an electronic instrument, record all digits on the readout and consider the last digit to the be approximate.

11. The Rules to tell you which digits in a measurement are significant? 1. All non-zero digits are significant. 7.886 has – four sig digs 19.4 has – three sig digs 527 266.992 has – nine sig digs

12. 2. All zeros that are located between two non-zero digits are significant. 408 has – three sig digs 25 074 has – five sig digs 98 000 003 has – eight sig digs

13. 3. For values less than one, the zeros located to the left of a non-zero digit are not significant. 0.000 764 has – three sig digs 0.45 has – two sig digs 0.000 000 000 000 7 has – one sig dig

14. 4. For values greater than one, that do not have a decimal, the zeros that are located to the right are not significant. 22 770 000 has – four sig digs 343 000 has – three sig digs 4 000 000 has – one sig dig.

15. 5. For values greater than one, that do have a decimal, the zeros that are located to the right are significant. 1230.0 has – five sig digs 3 827 000.0 has – eight sig digs

16. Rules four and five When you run into a situation where you have zeros at the end of a value, write the value in scientific notation. » 96 0000 If the zeros are significant, then include them after the decimal. » 9.600 0 x 10 4 If the zeros are not significant, then drop them after the decimal. » 9.6 x 10 4

17. 6. All counted objects have an infinite number of significant digits. There are 15 students in this class – the fifteen can be written as 15, 15.0, or 15.00, or 15.00000. It has an infinite number of sig digs. There are 37 pens in my drawer – the thirty seven has an ∞ number of sig digs.

18. Accuracy and Precision The terms “ accuracy ” and “ precision ” mean the same thing to most people, but to a scientist, their meanings are quite distinct. Accuracy refers to how close a measurement is to an accepted value. Precision refers to how close a measurement is to another measurement. Scientists often repeat their experiments to reduce uncertainty.

19. High Accuracy High Precision Low Accuracy High Precision High Accuracy Low Precision Low Accuracy Low Precision

20. If measurements are approximate, then calculations based on them must be approximate as well. Scientists agree that calculated answers should be rounded so that they do not give a misleading idea of how precise the original measurements were (more numbers does not make your answer more precise).

21. Calculating with Significant Digits You will often take measurements and use them to calculate other quantities. You must be careful to keep track of which digits in your calculations and results are significant. Your results should never imply more certainty than your measured quantities. Remember, calculators do not make decisions about certainty – you do!

22. Multiplying and Dividing with Significant Digits The value of significant digits, going into the calculation, determines the number of significant digits that you should have in your answer. 39.786 x 3.45 » 137.2617 39.786 has five sig digs 3.45 has three sig digs Answer can only have three sig digs Answer is 137

23. Adding and Subtracting with Significant Digits The value with the fewest number of digits after the decimal going into the calculation, determines the number of decimal places that you should report in your answer. 145.684 + 34.7293 180.4133 145.684 has three digs after the decimal 34.7293 has four digs after the decimal The answer can only have three after the decimal 180.413

24. Rounding To get the appropriate number of sig digs or places after the decimal, you may need to round your answer. If you answer ends in a number that is greater than five, increase the preceding digit by one. If you answer ends in a number that is less than five, leave the preceding digit.