1. Using Numbers in Science Significant Digits You start by counting the first non-zero digit, and then all those that follow. Significant digits are all those digits that occupy places for which an actual measurement was taken. These include the last digit which should always be an estimated value Exceptions 1. If there is no decimal in the number, any zero(s) that the number ends with are not significant. 2. Numbers that are arrived at when counting and exactly defined quantities have an unlimited number of significant digits Determining the Number of Significant Digits

2. Examples 1. 5.3451 5 sig. dig. 2. 0.00845 3 sig. Dig. 3. 0.0431000 6 sig. dig. 4. 0.003200 4 sig. dig. 5. 70 000 1 sig. dig. 6. 4.821 x 10 5 4 sig. dig.

3. Try These 1. 1.98 2. 0.02850 3. 6.509 4. 900.00 5. 8.02 x 10 -9 6. 0.00016 7. 505 3 SD 4 SD 4 SD 5 SD 3 SD 2 SD 3 SD

4. Scientific Notation When writing a number in scientific notation, it is composed of two parts. The mantissa, and the exponent. 2.45 x 10 6 mantissa exponent The number is always written so that there is one non-zero digit in front of the decimal point. Converting from Scientific to Standard Form If the exponent becomes larger the mantissa must become smaller and if the exponent become smaller the mantissa must become larger to compensate for the change to the exponent. The overall value of the number cannot change.

5. ie. 3.45 x 10 5 345 000 4.285 x 10 -3.004285 6.9 x 10 7 69 000 000 * Keep the same number of SD Converting Standard Form to Scientific Form If the mantissa becomes larger the exponent must become smaller and if the mantissa become smaller the exponent must become larger to compensate for the change to the mantissa. The overall value of the number cannot change. ie. 0.000 663 6.63 x 10 -4 36 297.63.62976 x 10 4 0.000 000 0004394.39 x 10 -10

6. Try These 1.25 387 2.6.92 x 10 7 3.0.000 03659 4.1.00 x 10 -5 5.88 543 756 6.5.21 x 10 -7 7.3 974 000 8.0.000 000 241 2.538 7 x 10 4 69 200 000 3.659 x 10 -5 0.0000100 8.854 375 6 x 10 7 0.000 000 521 3.974 x 10 6 2.41 x 10 -7

7. Rounding Off Rules 1. If the number is larger than 5, round up. ie. 52548 = 52550 2.If the number is smaller than 5, round down. ie. 0.047325 =.047 3. If the number is a 5 followed by all zeroes, or no digits, round to the nearest even number. ie. 8 775 = 8 780 4.If the number is a 5 followed by any non zero digits, round up. ie. 0.00445002 = 0.0045

8. Try These 1)34 783 2).0021754 3)750 043 4)68 500 5)29 384 6)5 550.1 7)0.9735 34 800.002175 800 000 68 000 29 000 5 600 0.974

9. Conversions terra T 10 12 gigaG 10 9 megaM 10 6 kilo k 10 3 hectah 10 2 decada 10 1 -10 0 decid 10 -1 centic 10 -2 millim 10 -3 microµ 10 -6 nanon 10 -9 pico p 10 -12 femto f 10 -15 Multiply Divide

10. When doing conversions, if the unit is getting larger, the number must get smaller and if the unit is getting smaller, the number must get larger. However, you must maintain the same number of significant digits. ie. 1000 mm = 1 m The meter is 1 000 times larger than the millimeter, therefore the number must become 1 000 times smaller. therefore the number must become 1 000 times smaller..00462 km = 46.2 dm The dm is 10 000 times smaller than the km, therefore the number is 10 000 times bigger. therefore the number is 10 000 times bigger. ie. ie..193 g = 19.3 cg.193 g = 19.3 cg 923 ps = 9.23 x 10 -10 s 923 ps = 9.23 x 10 -10 s 0.0446 Mm = 4.46 x 10 5 dm 0.0446 Mm = 4.46 x 10 5 dm

11. Try These 1).905 hg = mg 2)822 cm = dm 3)81.4 pm = m 4).00775 g = µ g 5)3.76 x 10 6 kg = dg 90500 82.2 8.14 x 10 -11 7.75 x 10 3 3.76 x 10 10

12. Multiplying and Dividing When multiplying or dividing, your answer must be rounded off so that it contains the same number of SD as the value with the least number of SD. ie. 1. 17 / 42 =.404761904 (must contain 2 SD) =.40 2. 3.125 x.11 =.34375 (must contain 2 SD) =.34 3. (7.58 x 10 4 ) (8.32 x 10 -8 ) / (4.18 x 10 -5 ) = 150.8746411 (3 SD) = 151

13. Try These 1. (35.72) (0.00590) 9. (7.79 x 10 4 ) (6.45 x 10 4 ) (5.44 x 10 6 ) 2. (707000) (3.1) 10. (4.87 x 10 6 ) (9.69 x 10 1 ) 3. (0.05432) (62000) ( 9.765 x 10 -3 ) 4. (6.090 x 10 -1 ) (9.08 x 10 5 ) 11. (2.1 x 10 3 ) (2.593 x 10 -2 ) (5.23 x 10 -3 ) (6 x 10 -5 ) 5. (1.101 x 10 9 ) (4.75 x 10 9 ) 6. (6810.12) / (2.4) 7. (.4832) / (5.12) 8. (1.18 x 10 -2 ) / (2.2 x 10 3 )

14. Answers 1).211 2) 2.2 x 10 6 3)3400 4)5.53 x 10 5 5)5.23 x 10 18 6)2800 7).0944 8) 5.4 x 10 -6 9) 924 10) 4.83 x 10 10 11) 2 x 10 8 Adding and Subtracting When you add or subtract, the answer must be rounded off to the place value of the least accurate value. ie. 677 39.2 39.2 6.23 6.23 722.43 Must be rounded off to the ones place 722.43 Must be rounded off to the ones place 722 Correct rounded off answer 722 Correct rounded off answer

15. 2) 201 3) 5.32 - 0.75938 = 4.56062 3.57 = 4.56 (hundredths) 3.57 = 4.56 (hundredths) 98.493 98.493 303.063 = 303 (ones) 303.063 = 303 (ones) Try These 1) 35.6 + 4.987 + 0.09135 2) 2798 + 33.00 + 45.991 3).98 - 629.9 + 5300 Answers40.728774700

16. Rearranging Equations To rearrange an equation for a new variable it must be isolated in the numerator To rearrange an equation for a new variable it must be isolated in the numerator. Rules To move a number or variable to the opposite side of an equation, you must perform the opposite operation to that number or variable, and then do the same thing to both sides of the equation. In order to invert both sides of an equation, you must have a single denominator on both sides of the equation.

17. Rearrange for r 2. Take the square root of both sides.

18. Using and Expressing Measurements Measurement – is a quantity that has both a number and a unit. - it is fundamental to experimental science. - it is important to be able to make measurements and decide if they are correct Accuracy, Precision and Error Accuracy – a measure of how close a measurement is to the actual. Precision – is a measure of how close a series of measurements are to one another. Determining Error Error = experimental value – accepted value Percent Error = |error|_____ x 100% accepted value