Measurement and Certainty Or, “I weigh 152.3746382 pounds”

SI (Système international d’unités) International system of units (metric system) based on multiples of 10. Examples: kilograms, liters, meters

standard Exact, agreed-upon quantity used for comparison

SI Base Units Learn NOW Learn LATER Distance: meters (m) Seven measurement units from which all other units can be derived: Distance: meters (m) Time: seconds (s) Mass: kilograms (kg) Temperature: Kelvin (K) Number of particles: moles (mol) Electric current: Amperes (A) Brightness of light: candellas (cd) Learn NOW Learn LATER

Derived units Units that come from (derived from) a mathematical operation of two or more standard units Example: (speed) m/s, (density) g/L 1 Pa = 1 N/m2 = 1 kg/ms2

Error in measurement A measurement is an assignment of a value to a characteristic of something. When a measurement is made, uncertainty is always involved There are two limitations of a measuring system: Accuracy Precision

accuracy degree of closeness of measurements of a quantity to that quantity's true value Measurement instruments need to be calibrated properly to achieve best accuracy

precision How similar/close repeated measurements are to each other. (Reproducibility or variation of measurements)

(For example, 5 mm exactly) (avg measured value 5.12 mm) (Likelihood of getting a certain value as your measurement)

Illustration: accuracy vs. precision

Which is more precise?

Significant figures The digits in a measurement about which we are confident, plus one estimated digit. Confidence of our measurement (the number of sig figs) is determined by the precision of our instrument

Sig Figs rules (assumptions) The following rules assume that a measurement was taken and recorded properly…

Sig Figs rules (assumptions) Any non-zero digit is considered to be significant. Any zero between significant digits is considered significant. Only final zeros after a decimal are considered significant. All other zeros, unless indicated are not significant and are considered placeholders.

3b. A line placed over a zero may be used to indicate that it is significant: 0

Practice 305 m 3050 s 30500 km 3.50°C 0.00305 m 0.003050 kg 30.00500 mm

Multiplying/Dividing The product or quotient of two numbers is rounded to the same number of sig. figs. as the number with the least number of sig. figs. 46.2 X 11 = 508.2 510

Adding/Subtracting 34,400 s 0.0641 kg The precision of the answer is only as precise as the least precise number. 1. perform the calculation. 2. round at the place value of the last significant digit of the least precise number… 34500 s 0.00789 kg - 99 s + 0.0562 kg -------- --------- 34401 s 0.06409 kg 34,400 s 0.0641 kg

Numbers that are not measurements Some numbers, like the number of people counted in a room, or the value for π, are exact and therefore have infinite significant figures. These numbers DO NOT affect the number of significant figures in a calculation.

Making measurements with Instruments Recording significant figures assumes that the instrument is calibrated, it is as precise as advertised, and you are reading it correctly. 2 steps: Record the digits for which you are certain Estimate the last digit to the best of your ability (it can be 0-9) KEEP YOUR EYES LEVEL WITH THE MARK!

Reading liquids in graduated cylinders For reading volumes of water, read at the bottom of the meniscus (curve), at eye level.

Scientific notation background… Really big (and small) numbers are hard to write, and are hard to use in calculations! (example: world population estimated to be 7,525,986,931 people (on Aug 17, 2017 in the am) Most big or very small numbers are measurements with uncertainty (we are confident that the world has 7.53 billion people)

Scientific notation is a solution that will make your life as a scientist EASIER!!!

Scientific notation A way to write very small or very large numbers Expressed in the form m x 10n where n is any integer m is a number with digits only in the ones place value and lower (examples: 5, 3.2, 9.005, but not 50)

Base 10 exponents 10n= a 1 with n zeros behind it 103 = 1,000 (thousand) 106 = 1,000,000 (million) 109 = 1,000,000 (billion) 100 = 1

Negative exponents 10-n = 1 divided by a 1 with n zeros behind it 10 −1 = 1 10 1 = 1 10 =0.1 (one tenth) 10 −3 = 1 10 3 = 1 1000 =0.001 (one thousandth)

Multiplying base 10 exponents For multiplying, simply add the exponents to find the answer’s exponent For dividing, subtract the bottom (denominator) exponent from the top (numerator) exponent. 10 16 ∗ 10 −27 = 10 16+ −27 = 10 −11 10 −34 10 −12 = 10 −34− −12 = 10 −22

Expanding from scientific notation If the exponent is positive, move the decimal over to the right n times. 2.31 x 103 = If the exponent is negative, move the decimal over to the left n times 4.578 x 10-4=

Converting to scientific notation What if the second zero was actually a sig fig? Express in scientific notation! Go backwards. 2.0 x 102 s

Practice (answer in s.n. with sig figs) 6.23∗ 10 3 7.952∗ 10 5 =