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Scientific Notation- Why? Also used to maintain the correct number of significant figures. An alternative way of writing numbers that are very large or very small. characteristic will be positive Ex: 6.022X1023 602200000000000000000000 Method to express really big or small numbers. Format is Mantissa Decimal part of original number 6.02 x Base Power x Decimal you moved 23 10 We just move the decimal point around. 602000000000000000000000 Using the Exponent Key on a Calculator EE EXP EE or EXP means “times 10 to the…” How How to to type type out out 6.02 6.02 xx10 102323:: 6 0 . 2 EE 2 3 Don’t do it like this… 6 WRONG! 0 . yx 2 2 3 …or like this… 6 . 0 WRONG! 2 x 1 …or like this: 6 . 0 0 EE 2 3 TOO MUCH WORK. 2 x 1 0 yx 2 3 1.2 x 105 Example: 2.8 x 1013 Type this calculation in like this: 1 . 2 EE 5 2 . 8 EE 1 3 = Calculator gives… 4.2857143 –09 or… 4.2857143 E–09 This is NOT written… 4.3–9 But instead is written… 4.3 x 10–9 or 4.3 E –9 Converting Numbers to Scientific Notation 0.00002205 1 2 3 4 2.205 x -5 10 5 In scientific notation, a number is separated into two parts. The first part is a number between 1 and 10. The second part is a power of ten. Scientific Notation- How To convert TO scientific notation, move decimal to left or right until you have a number between 1 & 10. Count # of decimal places moved If original is smaller than 1 than characteristic will be negative If original is larger than 1 the If original number is negative, don’t forget to put the – back on the front! Example: If you move the decimal the characteristic will be If you move the decimal the characteristic will be to the left positive to the right negative Convert 159.0 to scientific notation 1.59 x 102 Convert -0.00634 -6.34 x 10-3 Your Turn 1. 17600.0 2. 0.00135 1.76 x 104 1.35 x 10-3 1.02 x 101 3. 4. 5. 6. 7. 8. 10.2 -67.30 4.76 - 0.1544 301.0 -0.000130 -6.730 x 101 4.76 x 100 -1.544 x 10-1 3.010 x 102 -1.30 x 10-4 May drop leading zeros - keep trailing Expand Scientific Notation If characteristic is positive move decimal to the right If the characteristic is negative move the decimal to the left Ex: 8.02 x 10-4 0.000802 -9.77 x 105 -977,000 7.5 x 10-6 - 8.7 x 10-4 = -6.525 x 10-9 4.35 x 106 1.23 x 10-3 = 5.3505 x 103 or 5350.5 report -6.5 x 10-9 (2 sig. figs.) report 5.35 x 103 (3 sig. figs.) 5.76 x 10-16 9.86 x 10-4 = 5.84178499 x 10-13 report 5.84 x 10-13 (3 sig. figs.) 8.8 x 1011 3.3 x 1011 = 2.904 x 1023 report 2.9 x 1023 (2 sig. figs.) 6.022 x 1023 - 5.1 x 10-8 = -3.07122 x 1016 report -3.1 x 1016 (2 sig. figs.) Correcting Scientific Notation The mantissa needs to have one place holder to the left of the decimal (3.67 not 36.7), look at the absolute value Count how many decimals places you move and then you will increase or decrease the characteristic accordingly If you must INCREASE the mantissa, DECREASE the characteristic If you must DECREASE the mantissa, INCREASE the characteristic Be careful with negative characteristics! If you decrease 10-3 by two the new value is 10-5 Confused? Example To correct 955 x 108 Convert 955 to 9.55 – (move decimal left 2 times). Did we increase or decrease 955? 955 is larger than 9.55 so we decreased it -so we must increase 8 by 2. 955 x 108 becomes 9.55 x 1010 -9445.3 x 10-6 Convert -9445.3 to -9.445 (move decimal left 3 times). Did we increase or decrease -9445.3? (absolute value) We decreased the absolute value by 3 decimal places, so we must increase the characteristic 955 x 108 becomes 9.55 x 1010 Your Turn 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 36.7 x 101 -0.015 x 103 75.4 x 10-1 -14.5 x 102 0.123 x 104 97723 x 10-2 851.6 x 10-3 94.2 x 10-4 -0.012 x 103 966 x 10-1 3.76 x 102 -1.5 x 10-5 7.54 x 100 -1.45 x 103 1.23 x 103 9.7723 x 102 8.516 x 10-3 9.42 x 10-4 -1.2 x 101 9.66 x 10-1 May drop leading zeros - keep trailing Rule for Multiplication Calculating with Numbers Written in Scientific Notation 1. MULTIPLY the mantissas 2. Algebraically ADD the characteristics 3. Correct the result to proper scientific notation when needed Sample Problem: (4 x 10-3) (3 x 10-3) (4) x (3) = 12 (-3) + (4) = 1 or 101 1.2 x 102 Rule for Division Calculating with Numbers Written in Scientific Notation 1. DIVIDE the mantissas 2. SUBTRACT the characteristic of the denominator from the characteristic of the numerator 3. Correct the result to proper sci. notation if needed Sample Problem: Divide 7.2 x 10-4 by -8 x 105 . (7.2) . (-8) = -0.9 (-4) - 5) = -1 or 10-9 -0.9 x 10-9 Correct: -9 x 10-10 Your Turn 1. 2. 3. 4. (2 x 104)(3 x 10-3) (5 x 10-3)(4 x 10-4) (6 x104)(-7 x 10-5) (-4.5 x 10-2)(2 x 10-7) 1. 2. 3. 4. 5. (8 x 10-5) / (2 x 10-3) (4 x 103) / (8 x 10-3) (6 x 10-7)/(3 x 108) (4.5 x 104) / (9.0 X 10-12) ((2 X 103)(4X10-2)) / ((6 X 10-9)(4 X 105)) 6 x 101 20 X 10-7 2 x 10-6 -42 x 10-1 4.2 x 100 -0 x 10-9 4 x 10-2 5.0 x 105 2 x 101 5 x 1015 3.3 x 104 May drop leading zeros - keep trailing Rule for Addition and Subtraction Calculating with Numbers Written in Scientific Notation In order to add or subtract numbers written in scientific notation, you must express them with the same power of 10. (Same characteristic). Then correct to proper scientific notation. Sample Problem: Add 5.8 x 103 and 2.16 x 104 27.4 x 103 2.74 x 104 Exercise: Add 8.32 x 10-7 and 1.2 x 10-5 1.28 x 10-5 (5.8 x 103) + (21.6 x 103) =