Aug 27, 2012 - ... 1 sig fig. 2. Zeros at the end of a number containing NO decimal point ... must write 5000. or 5.000 x 103 ... when determining the...
Homework Answers…p.148 #6 and #7 Express as decimal: 6a) 4.83 x 102 = 483 b) 7.221 x 10-‐4 = 0.0007221 c) 6.1x 100 = 6.1 Put in standard scien?fic nota?on: 7a) 142.3 x 103 = 1.423 x 105 b) 0.0007741 x 10-9 = 7.741 x 10-‐13 c) 22.7 x 103 = 2.27 x 104
Scientific notation 1.27 x 102 9.07 x 10 – 2 5.06 x 10 – 4 2.3 x 1012
What time is it? • Someone might say 1:30 or 1:28 or 1:27:55
How big? • Each is appropriate for a different situation
• In science we describe a value as having a certain number of significant digits How small?
• The # of significant digits in a value includes all digits that are certain and one that is uncertain • 1:30 likely has 2, 1:28 has 3, 1:27:55 has 5
How accurate?
So…. 45.50 has 4 signifcant figures while 45.5000 has 6 sig figs and .0005 has only 1 sig fig
Look at the difference adding a 1 makes! .0045 has 2 significant figures but 1.0045 has 5 significant figures
Reminder: bring a calculator to class
Here is a one sentence rule for coun/ng sig figs: All digits ARE significant except 1. Zeros preceding a decimal frac/on Example: .0025m contains 2 sig figs .0008g contains 1 sig fig 2. Zeros at the end of a number containing NO decimal point Example: 5000 contains 1 sig fig
Numbers with no decimal are ambiguous... • Does 5000 ml mean exactly 5000? Maybe.... Maybe Not! • So 5000, 500, 50, and 5 are all assumed to have 1 significant figure • If a writer means exactly 5000, he/she must write 5000. or 5.000 x 103
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A Significant Figures Rules Summary To Determine the Significant Figures in a Number:
A Significant Figures Rules Summary
• Non-zero digits are ALWAYS significant • Ex: 1234.56 à 6 sig figs • Counted numbers or conversion factors are considered to have infinite significance and will NEVER be considered when determining the number of digits to report in an answer.
• “Leading” (placeholders) and are NEVER significant • Ex: 0.00031 à 2 sig figs • “Captive” (between nonzero numbers) and are ALWAYS significant • Ex: 1013 à 4 sig figs
Answer: a. b. c. d.
1 2 3 4
2. How many significant figures in 0.005? 1 2 3 4
And More Prac?ce: a. b. c. d.
1 2 3 4
a. b. c. d.
1 2 3 4
More Prac?ce: a. b. c. d.
3. How many significant figures in 1.005?
1. How many significant figures in 4.50?
• “Trailing” (at the end of a number) • IF there is a decimal and a non zero number before the zero à ALWAYS significant • 131.400 à 6 sig figs • IF no decimal à NEVER significant • 131,400 à 4 sig figs
ZEROS have to be examined carefully….
1. How many significant figures in 4.50?
Now you try:
• ZEROs may be:
Answer: 2. How many significant figures in 0.005? a. b. c. d.
1 2 3 4
Answer: 3. How many significant figures in 1.005? a. b. c. d.
• It is better to represent 100 as 1.00 x 102 • Alternatively you can underline the position of the last significant digit. E.g. 100. • This is especially useful when doing a long calculation or for recording experimental results • Don t round your answer until the last step in a calculation.
Math and Significant Figures
• Addition and subtraction • The number with the least significant place value determines the sig figs for the answer. • Sometimes your answer will have more counted sig figs than one of the numbers used in the calculation.
13.64 + 0.075 + 67.
267.8 – 9.36
81 80.715
258.44
Adding Prac?ce:
• Rounding • The measured values are used to determine the number of digits in your final answer. • If the number cannot be rounded to the correct sig figs in expanded form, use scientific notation. • Ex: 400 to 2 sig fig à 4.0 x 102
Adding with Significant Digits
• Note accuracy of measurements (nearest .1? .01? .001?) • Answer can be no more accurate than the LEAST accurate number that was used to calculate it (line up the decimals). • E.g. a) 13.64 + 0.075 + 67 b) 267.8 – 9.36
1. How many significant figures in the answer to: 4.50 + 0.5? a. b. c. d.
1 2 3 4
Answer: 1. How many significant figures in the answer to: 4.50 + 0.5? a. b. c. d.
1 2 (5.0) 3 4
• Ex: 200 + 4.02 = 200 (200 is sig to 100s place) • Ex: 200.02 + 4.021 = 204.04 (sig is limited to 1/100s place, giving 5 sig figs for final answer even though 4.021 has only 4 sig figs)
Subtrac?ng Prac?ce: 2. How many significant figures in the answer to 1.0 -‐ 0.005? a. b. c. d.
1 2 3 4
Answer: 2. How many significant figures in the answer to 1.0 -‐ 0.005? a. b. c. d.
• Now Try B
1 2 (1.0) 3 4
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B) Answers
i)
83.25 – 0.1075
ii)
83.14
4.02 + 0.001 4.02
iii)
0.2983 + 1.52 1.82
Multiplication and Division • Determining sig. digits for questions involving multiplication and division is slightly different • You must COUNT significant figures • The answer can have only AS MANY significant figures as the LEAST of the numbers used to get it • E.g. a) 608.3 x 3.45 b) 4.8 ÷ 392 a) 3.45 has 3 sig. digits, so the answer will as well 608.3 x 3.45 = 2.10 x 103 b) 4.8 has 2 sig. digits, so the answer will as well 4.8 ÷ 392 = 0.012 or 1.2 x 10 – 2
Mul?plying Prac?ce: 1. How many significant figures in the answer to: 4.50 x 0.5? a. b. c. d.
1 2 3 4
Answer:
a. b. c. d.
1 2 (2.0 x 102) 3 4
• Count the sig figs in all numbers; report answer to least number of sig figs counted. • No answer will ever have more sig figs than a number used in the calculation. • Ex: 200 x 4.02 = 800 (1 sig fig in 200, 1 sig fig in 800) • Ex: 200.02 x 4.021 = 804.3 (4 sig fig in 4.021, 4 sig fig in 804.3)
Answer: 1. How many significant figures in the answer to: 4.50 x 0.5? a. a. b. c.
2. How many significant figures in the answer to 1.00 / 0.0050?
Math and Significant Figures • Multiplication and Division
1 (2.25 rounds to 2) 2 3 4
Here s a tougher one..... 3.00 m/s x 60 s/min x 60 min/hr = Note: standard conversion factors (such as 60 s/min) never limit significant figures-‐-‐ instruments and equipment do.
Dividing Prac?ce: 2. How many significant figures in the answer to 1.00 / 0.0050? a. b. c. d.
1 2 3 4
Did you get it? 3.00 m/s x 60 s/min x 60 min/hr = 10800 m/hr
The measurement of 3.00 m/s has 3 sigfigs. The
other numbers are conversion factors and are considered to be infinitely significant (they never limit your answer).
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Unit conversions • Sometimes it is more convenient to express a value in different units. • When units change, basically the number of significant digits does not. E.g. 1.23 m = 123 cm = 1230 mm = 0.00123 km • Notice that these all have 3 significant digits • This should make sense mathematically since you are multiplying or dividing by a term that has an infinite number of significant digits E.g. 123 cm x 10 mm / cm = 1230 mm • Try question E on the handout
E) Answers
THAT S ALL THERE IS TO IT!
i) 1.0 cm = 0.010 m ii) 0.0390 kg = 39.0 g iii) 1.7 m = 1.7 x 103 mm
• Use least accurate measurement (line up the decimals) when adding and subtrac/ng • Count sig figs when mul/plying and dividing • Conversion factors and counted numbers have infinite significance and will not limit your answer.
