Error in Measurement Some error or uncertainty always exists in any measurement. Therefore, the last digit is always the estimation of the value of a questionable digit. You know that it is above 39. So 39 is a certain number. You also know It is above 39.2. What you don’t know Is where exactly the meniscus is Between 39.2 and 39.3. It looks like it Is just below the half way mark, so 39.24 would be an acceptable guess, With the 4 being the questionable digit.
Significant Figures Significant Figures in a measurement consist of all the digits known with certainty plus one final digit, which is somewhat uncertain or is estimated. The term significant doesn’t mean certain-like the example with the meniscus, the last digit was significant, but not certain. There are rules for determining if numbers are significant or insignficant
Rules for determining Significant Zeros RuleExamples 1. Zeros appearing between nonzero digits are significant a.40.7 L has three significant figures b.87,009 km has five significant figures 2. Zeros appearing in front of all nonzero digits are not significant a.0.095897 has five significant figures b.0.0009 kg has one significant figure 3. Zeros at the end of a number and to the right of a decimal point are significant a.85.00 g has four significant figures b.9.000000000 mm has 10 significant figures 4. Zeros at the end of a number but to the left of a decimal point may or may not be significant. If a zero has not been measured or estimated but is just a place holder, it is not signficant. A decimal point placed after zeros indicates that they are significant. a.2000m may contain from one to four significant figures, depending on how many zeros are place holders. In this class we will assume it only has 1 significant figure b.2000. m contains four significant figures, as indicated by the decimal point.
Practice with significant figures How many significant figures are in the following measurements? 1.28.6 g 2.3440. cm 3.910 m 4.0.04604 L 5.0.0067000 kg
Practice with signficiant figures How many significant figures are in the following measurements? 1.28.6 g 3 sig figs 2.3440. cm4 sig figs 3.910 m2 sig figs 4.0.04604 L4 sig figs 5.0.0067000 kg5 sig figs
Rounding If the digit following the last digit retained is: Then the last digit should be: Example (rounded to three significant figures) Greater than 5Be increased by 142.68 g 42.7 g Less than 5Stay the same17.32 m 17.3 m 5, followed by nonzero digit(s) Be increased by 12.7851 cm 2.79cm 5, not followed by nonzero digit(s), and preceded by an odd digit Be increased by 14.635 kg 4.64 kg (because 3 is odd) 5, not followed by nonzero digit(s), and the preceding significant figure is even Stay the same78.65 mL 78.6 mL (because 6 is even)
Addition or Subtraction with Significant Figures When adding or subtracting decimals, the answer must have the same number of digits to the right of the decimal point as there are in the measurement having the fewest digits to the right of the decimal point. Example: What is the sum of 2.099 g and 0.05681 g?
Addition and subtraction 2.099 g + 0.5681 g = 2.156 g Practice: 1.87.3 cm – 1.655 cm 2.5.44 m – 2.6103 m 3 sig figs to the Right of the decimal 3 sig figs to the Right of the decimal
Multiplication and Division with Significant Figures For multiplication and division, the answer can have no more significant figures than are in the measurement with the fewest number of significant figures. Practice: Calculate the area of a rectangular crystal surface that measures 1.34 meters and 0.7488 meters (area=length x width)
Multiplication and Division Area= 1.34 m x 0.7488 m = 1.00 m 2 Practice: 1.What is the volume, in cubic meters, of a rectangular solid that is 0.25m long, 6.1 m wide, and 4.9 m high? (volume= length x width x height) 2.12 m x 6.41 m 3 sig figs
Scientific Notation Numbers are written in the form of M x 10 n where M is a number between 1 and 9 and n is whole number Example: 1.0.00012 mm = 1.2 x 10 -4 2.65,000 km = 6.5 x 10 4 km (2 sig figs) If you wanted to have 3 sig figs you would write 6.50 x 10 4
Practice Calculate the volume of a sample of aluminum that has a mass of 3.057 kg. The density of aluminum is 2.70 g/cm 3. Pay attention to your units!!
Direct Proportions Two quantaties are directly proportional to each other if dividing one by the other gives a constant value Example: If you make $20.00 an hour, the more hours you work, the more money you make. hoursmoney
Inverse Proportions Two quantities are inversely proportional to each other if their product is constant. Example: As the distance from the Earth increases, the gravitational pull decreases. distancegravity