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Measurements & Calculations: Mastering Scientific Notation

Dive into the world of quantitative observations through scientific notation, units, measurements of length, volume, and mass, significant figures, and more with plenty of examples and rules to enhance your understanding!

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Measurements & Calculations: Mastering Scientific Notation

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  1. Measurements & Calculations Warning!Lots of math(not tough math, but lots of it)

  2. Remember there are qualitative and quantitative observations • This chapter deals with the quantitative! called measurements

  3. these measurements are not just numbers • they have units • as in5 millimeters,75 people,16 mph, etc. • but first…

  4. scientific notation • Some numbers are just too darn big or too small to deal with reasonably • Scientific Notationis a method for making very large or very small numbers more compact and easier to write. • as in: 64,400,000,000 can be written 6.44 x 1010 • it’s easy! :)

  5. Description:scientific notation must be written as the product of a number between 1 and 10 and the appropriate power of 10 • just count how many times you have to move the decimal point to get a number between 1 and 10 • if the number is gettingsmaller the exponent willcompensate by gettingbiggerand vice versa

  6. examples • 238,000 2.38 x 105 • 1,500,000 1.5 x 106 • 0.00043 4.3 x 10-4 • 0.135 1.35 x 10-1 • 357 3.57 x 102

  7. units • go into a restaurant, sit down, just tell the waitress “two,” and see what you get • Units are used everyday to give meaning to numbers • people have used them since, like, forever…

  8. the English system is used in the US; the metric system is used everywhere else • scientists everywhere use metric and standardized it into the International System (SI)

  9. these are the basic units of SI • know them,love them,marry them

  10. and these are the prefixes we use to make them even more convenient • “1 mm” is easier to use & write than “one thousandth of a meter” • know them, love them, marry them m

  11. measurements of length, volume, and mass • length is based on the meter

  12. volumeis how much 3D space something takes up • SI unit is the m3 • one thousandth of that is the dm3, aka the liter • 0.001 of that is cm3 or ml

  13. we mostly measure volume with a graduated cylinder but also these critters, all of which are marked on the side

  14. Remember that when you use the Graduated cylinder to read the bottom of the meniscus

  15. mass is measured in grams (even though SI unit is kg) • measured with a balance

  16. 5.4 uncertainty in measurement • many measurements are made of objects that make us estimate • so, we’ll always argue about the last number or two • the ones we agree on are called certain, the argued ones uncertain

  17. 31.7 31.8 31.8 31.6 31.7 31.7 31.8 31.6 • every measuring device has some degree of uncertainty • the certain numbers+the one uncertain one are called significant…

  18. which instrument gives us more sigfigs? • that’s the one you want to use (but it probably costs a lot more!)

  19. which instrument gives us more significant digits? • Consider the uncertainty in trying to measure the length of an object.

  20. 5.5 significant figures • the whole rest of your science/math life must reflect the measuring devices so you’se all gotta know dese tings called sigfigs • so what about zero’s and what not?

  21. Rules for Counting Significant Figures - Details Nonzero integersalways count as significant figures. 3456has 4sig figs.

  22. Rules for Counting Significant Figures - Details • Zeros • -Leading zerosdo not count as • significant figures. • 0.0486has • 3sig figs.

  23. Rules for Counting Significant Figures - Details • Zeros • -Captive zerosalways count as • significant figures. • 16.07has • 4sig figs.

  24. Rules for Counting Significant Figures - Details • Zeros • Trailing zerosare significant only if the number contains a decimal point. • 9.300has • 4sig figs.

  25. Rules for Counting Significant Figures - Details Exact numbershave an infinite number of significant figures. 1inch =2.54cm, exactly

  26. examples • the mass of an eyelashis 0.000304 g • 3 • the length of the skidmark was 1.270 x 102 m • 4 • A 125-g sample of chocolate chip cookie contains 10 g of chocolate • 3, 1 • the volume of soda remaining in a can after a spill is 0.09020 L • 4 • a dose of antibiotic is 4.0 x 10-1 cm3 • 2

  27. Sig Fig Practice #1 How many significant figures in each of the following? 1.0070 m  5 sig figs 17.10 kg  4 sig figs 100,890 L  5 sig figs 3.29 x 103 s  3 sig figs 0.0054 cm  2 sig figs 3,200,000  2 sig figs

  28. rounding off • yes, there are rules even for this • remember to use only the first digit to the right of the last sigfig to help you decide

  29. determining sigfigs in calculations • there are only two basic rules here, one to do with multiplication and division, the other addition and subtraction…

  30. Rules for Significant Figures in Mathematical Operations Multiplication and Division:# sig figs in the result equals the number with the least sig figs used in the calculation. 6.38 x 2.0 = 12.76 13 (2 sig figs)

  31. Sig Fig Practice #2 Calculation Calculator says: Answer 22.68 m2 3.24 m x 7.0 m 23 m2 100.0 g ÷ 23.7 cm3 4.22 g/cm3 4.219409283 g/cm3 0.02 cm x 2.371 cm 0.05 cm2 0.04742 cm2 710 m ÷ 3.0 s 236.6666667 m/s 240 m/s 5870 lb·ft 1818.2 lb x 3.23 ft 5872.786 lb·ft 2.9561 g/mL 2.96 g/mL 1.030 g ÷ 2.87 mL

  32. Rules for Significant Figures in Mathematical Operations Addition and Subtraction: The number of decimal places in the result equals the number of decimal places in the least precise measurement. 6.8 + 11.934 = 18.734  18.7 (3 sig figs)

  33. Sig Fig Practice #3 Calculation Calculator says: Answer 10.24 m 3.24 m + 7.0 m 10.2 m 100.0 g - 23.73 g 76.3 g 76.27 g 0.02 cm + 2.371 cm 2.39 cm 2.391 cm 713.1 L - 3.872 L 709.228 L 709.2 L 1821.6 lb 1818.2 lb + 3.37 lb 1821.57 lb 0.160 mL 0.16 mL 2.030 mL - 1.870 mL

  34. What about calculations involving both multiplication/division and addition/subtraction? Remember order of operations in Math. Rule 1:First perform any calculations inside parentheses. Rule 2:Next perform all multiplications and divisions, working from left to right. Rule 3:Lastly, perform all additions and subtractions, working from left to right. (2.32 + 3.2) = 5.5 =1.1 (5.0) 5.0

  35. problem solving and dimensional analysis • I have to buy 72 CostCo muffins, but they only sell them by the dozen. Do I just give up? May it never be! • I convert into dozens! • but I have to know the relationship b/t individuals and dozens! • called a conversion factor! • here 1 dozen = 12

  36. unit1x conversion factor = unit2 • we’ll…1) make astarting point,2) determinewhere we’re going,then…3) build a bridgeto it with theconversion factor

  37. 1) write down what you know (given),2) where you’re going, then3) build a bridge (your book calls the bridge an equivalence statement) between them… • Change 100 mm into m. bridge where you’re going given 100mm x 1 m 0.1 = m  1000 mm

  38. Change 546 cm into mm. bridge where you’re going given 546cm x 10 mm 5460 = mm  1 cm

  39. Convert 7.75g to µg. bridge where you’re going given µg 7.75g x 106 7.75x106 or 7,750,000 = µg  g 1

  40. Change 45mm into km. • (Hint: you might make this a 2-stepper.) 1 km 45mm 1 m = 1000 m 1000 mm 4.5x10-5 or .000045 km

  41. temperature conversions:an approach to problem solving • here we learn both the different temp scales and how to convert between them

  42. the Big Three Temp Scales are Fahrenheit, Celsius, and Kelvin • in science we use almost exclusively C and K

  43. converting between K and C • a degree C and K are the same amount; they just differ by their starting points • they only differ by 273 • thus, and simply • TC + 273 = TK

  44. examples • What is 70˚C in kelvins? • TC + 273 = TK • 70 + 273 = 343 K • Nitrogen boils at 77 K. What is that in C? • TC + 273 = TK • TC = TK - 273 • TC = 77 - 273 • TC = -196 ˚C

  45. Not required for us. It’s 28˚C outside. What is that in F? TF = 1.8TC + 32 TF = 1.8(28) + 32 TF = 50. + 32 TF = 82˚F It’s -40˚C in that lab freezer. What’s that in F? TF = 1.8TC + 32 TF = 1.8(-40) + 32 TF = -72 + 32 TF = -40˚F

  46. Complete the Table -89 310 58

  47. density • density is just how much stuff is crammed into a certain space in science speak it’s mass/volume. This is a physical property. D = m/V • finding mass is no problem—use a balance; how do you find volume?

  48. one can either use dimensions (like lxwxh) or volume displacement for irregular objects • take volume before, volume after - tada!the difference is the volume of your object

  49. Problem Solving Steps • Record the information given in problem. • Write the equation. • Substitute the numbers into the equation • Solve for the answer. (Do the math)

  50. 4. D = 19.3 g/mL Practice • Calculate the density of a gold brick that has a mass of 9650 g and a volume of 500. mL. • D = m • V 1. D=? m= 9650 g v = 500. mL • D = 9650 g • 500. mL

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