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Units and Conversions American (English) system - yards, pounds, tons, seconds Metric system (SI) o mks system => meter (m), kilogram (kg), second (s) o Metric prefixes for each power of 10 Prefix deci centi milli micro nano pico femto atto Abbreviation d c m m n p f a Value 10-1 10-2 10-3 10-6 10-9 10-12 10-15 10-18 Prefix decka hecto kilo mega giga tera Abbreviation da h k M G T Value 101 102 103 106 109 1012 American to SI conversions: 1m = 39.37 in 1 km = 0.6214 mi 1 kg = 2.205 lb 1L = 0.2642 gal o How to convert We can multiply anything by 1 in an equation Conversion factors can be written as fractions that equal 1 For Example: 7 = 7 , so 7 / 7 = 1 1m = 39.37 in , …SO 1 m / 39.37 in = 1 OR… 39.37 in / 1 m = 1 Example: Express 170 pounds in units of kg. 170 lbs x 1 kg ----------2.205 lbs = 170 x 1 lbs x kg ----------- x ------------- = 2.205 lbs o Try: Express 170 m/s in units of miles/hour. 77.1 kg Scientific Notation Convenient way to write huge numbers that deal with the macrocosmic and the microcosmic in Astronomy o Write in powers of 10 => 101 = 10 & 10-2 = 1/100 o The exponent tells you how many times to multiply or divide by ten o Written as: a.00 x 10n , where 'a' and 'n' are whole numbers 100 = 1 100 is written as 102, or 1 x 102 Converting from normal to scientific notation o Decimal moves right => negative exponent => 0.0304 = 3.04 x 10-2 o Decimal moves left => positive exponent => 3040 = 3.04 x 103 Converting from scientific to normal notation o Positive exponent => decimal moves right => 3.04 x 103 = 3040 o Negative exponent => decimal moves left => 3.04 x 10-2 = 0.0304 o Try: Convert 2,398,000 to scientific notation. o Try: Convert 6.549 x 104 to normal notation. Adding and Subtracting with Scientific Notation o Convert the numbers so the exponents are equal o Then add or subtract the numbers in the front part of the notation (6.7 x 109) + (4.2 x 109) = (6.7 + 4.2) x 109 = 10.9 x 109 = 1.09 x 1010 (4 x 108) - (3 x 106) = (4 x 108) - (0.03 x 108) = (4 - 0.03) x 108 = 3.97 x 108 o Try: (5.5 x 109) + (1.55 x 1010) o Try: (6.2 x 108) - (2.0 x 107) Multiplication and Division with Scientific Notation o Rearrange so you can multiply or divide the numbers in front of the powers of ten, and add (for multiplication) or subtract (for division) the exponents (6 x 102) x (4 x 10-5) = (6 x 4) x (102 x 10-5) = 24 x 102-5 = 24 x 10-3 = 2.4 x 10-2 (9 x 108) (9 x 108) / (3 x 106) = ----------- = (9/3) x (108/106) = 3 x 108-6 = 3 x 102 (3 x 106) o Try: (8 x 107) x (5 x 10-5) o Try: (4 x 108) / (6 x 108) Significant Digits Numbers should be given to the greatest accuracy that they are known Write answers with as many decimal places as the number in the problem with the least amount of decimal places Rules for Significant Digits o All non-zero numbers are significant. 943 => 3 o Trailing zeroes after a decimal point are significant. 9430 => 3 943.0 => 4 943.10 => 5 o Captive zeroes are significant. 4903 => 4 o Leading zeroes are not significant. 0001 => 1 0.005 => 1 o All measurements or any number with units must abide by these rules. o Any counting number or number with no units is significant. 5 = 5.000… 5 apples = 5.000… o Examples: 2.3 + 4.71 = 7.0 (NOT 7.01) Number Sig. Digits Number 2998 4 .1 10 1 .0100 190 2 190. 0.001 1 190.0 0.001000 4 1459 0.010 2 459.000 Sig. Digits 1 3 3 4 4 6 Basic Geometry Dimensions of Circles and Spheres o The circumference of a circle with radius R is 2πR o The area of a circle of radius R equals πR2 o The surface area of a sphere of radius R is given by 4πR2 o The volume of a sphere of radius R is (4/3)πR3 Measuring Angles - Degrees o 3600 in a circle o 60' (minutes) in one degree o 60" (seconds) in one minute Basic Trigonometry In a right triangle: o The longest side, opposite the right angle is the hypotenuse (c) o The side adjacent to the angle labeled θ is side (b) o The side opposite to the angle labeled θ is side (a) o Pythagorean Theorem: a2 + b2 = c2 o Trig functions: sin θ = a / c cos θ = b / c tan θ = a / b c a θ 900 b Basic Algebra You can move variables (unknowns) across an equals sign in an equation just by moving them from the top on one side, to the bottom on the other o Example: d = distance; v = velocity; t = time; a = acceleration d = vt , can be changed to v = at , can be changed to => t = d / v => t = v / a OR v = d / t OR a = v / t