Measurement & significant Figures

1. Measurement & significant Figures

2. The Purpose of Significant Digits • When we count we use exact numbers • If we count people we have exactly 4 people • There is no uncertainty about this number of people. • Measurements using an interval scale (like a ruler) has some uncertainty • Is it 29.68cm or is it 29.67cm? • This involves some estimation on the last digit

3. Systems of measurement • Invented in order to compare quantities • The metre was once defined as the distance between two scratches on a bar of platinum-iridium alloy at 0oC • All other measurements of distance are estimated • Not all measurements, however, are known to the same accuracy • Accuracy is how close the measurement is to the real value. (Sometimes the real value is not known) • A micrometer, for example, will yield a more accurate measurement of a hair's diameter than will a metre stick

4. Sig-Figs • Significant digits tell us something about how the measurement was made • A better instrument allows us to make better measurements and we record a greater number of significant digits in reporting this value • A metre stick would only allow us to record the thickness of a hair as 0.1mm • A micrometer allows us to record the thickness as 0.137 mm • This measurement is closer to the true value and is a more accurate measurement

5. Too Accurate? • Is it possible to be too accurate? • When is it not suitable to use a more precise measuring tool? • For each measurement would you use a micrometer or a metre stick? • Length of table • Width of fruit fly • Thickness of a penny • Width of hand • Distance to the sun • Having the length of a table recorded at 1.2345678909876m is not needed for most cases • A recording of 1.23m is sufficient most of the time

6. Suppose the length of a table is measured with a ruler calibrated to 10 centimetres • The table is definitely less than 2.8 metres but greater than 2.7 metres 2m 20cm 40 60 80 3m

7. Is the extra length 0.04 or is it 0.05? • We make the best possible estimate • A proper measurement would be recorded as 2.74 metres. This has 3 Sig-Figs • Where did the 0.04 come from? I thought the ruler could only do 10 centimeters at a time? • This indicates the table has a length of 2 metres plus 70 centimetres plusa little bit more • A measurement of the same table with a ruler calibrated to centimetres could yield 2.742 metres. This has 4 Sig-Figs 2m 20cm 40 60 80 3m

8. How to determine the # of Sig-Figs in a measurement copy • All non-zero digits are significant • Eg. 374 (3 sig-figs) • 8.1 (2 sig-figs) • 8.365 X 104 (4 sig-figs) • All zeroes betweennon-zero digits are significant (Captured Zeros) • Eg. 50407 (5) • 8.001 (4) • 9.05 X 104 (3)

9. Leading zeroes in a decimal are not significant • Eg. 0.54 (2) • 0.0098 (2) • 0.05 X 10-7 (1) (Not proper Scientific notation) • Trailing zeroes are significant only if they are to the right of a decimal point • Eg. 2370 (3) • 16000 (2) • 16.000 (5) • In numbers greater than 1, trailing zeroes are not significant unless stated so • Eg. 37000 (2)

10. The last three zeroes may or may not be part of the measurement. • To show that they are, we use scientific notation. All the zeroes written in the number in scientific notation are significant. • 37000 with 3 sig. digits would be 3.70 x 104 • 37000 with 4 sig. digits would be 3.700 x 104 • 37000 with 5 sig. digits would be 3.7000 x 104 • 37000.0 has 6 sig. digits • Exact Numbers do not affect Sig-Figs • If I count 4 people, I have exactly 4, not 4.01 or 3.99 • If I take the average of 7 tests, that 7 is an exact number, will not affect sig-figs in any way.

11. rounding • Round each value to 3 sig figs • 1.234 • 1.23 • 9.865 • 9.87 • 0.07888 • 0.0789 • 0.5399 • 0.540 • 12990 • 1.30 x 104

12. Calculations: Addition and Subtraction • General Rules: • Add or subtract as normal. • Count the number of digits to the right of the decimal. • The answer must be rounded to contain the same number of decimal places as the value with the LEAST number of decimal places

13. Calculations: Addition and Subtraction • Example • 12.0 + 131.59 + 0.2798 = ? • Add as normal: 12.0 + 131.59 + 0.2798 = 143.8698 • The least number of decimal places is 12.0, with one decimal place • Round to the least number of decimal places • Final Answer = 143.9 • 0.0998 – 1.0 • = -0.9002 • = -0.9 • 5.4 x 102 + 2.8 x 101 • = 568 • = 570

14. Calculations: Multiplication and Division • General Rules: • Multiply or divide as normal. • Count the number of significant figures in each number. • The answer must be rounded to contain the same number of significant figures as the number with the LEAST number of significant figures

15. Calculations: Multiplication and Division • Example • 51.3 × 13.75 = ? • Multiply as normal: 51.3 × 13.75 = 705.375 • The least number of significant figures is 3 in 51.3 • Round to the least number of significant figures • Final Answer= 705

16. Example • 2.00 x 7.00 • = 14 • = 14.0 • (3.0×1012 ) / (6.02×1023)= ? • Put the numbers into your calculator as usual: 3.0×1012 ÷ 6.02×1023 = 4.98338×10−12 • The least number of significant figures belongs to 3.0×1012 with 2 sig figs • Final Answer = 5.0×10−12 • 250 x 4.0 • = 1000 • = 1.0 x 103

17. Using proper sig-figs, find how many seconds are in 1 year (365.25 days) • = 31557600 • = 31558000