Scientific Measurements Significant Figures & Scientific Notation

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Large and Small Numbers Scientists often deal with numbers that are either extremely large or extremely small Examples the speed of light = 299792458 meters per second hydrogen atoms has a mass of 0.000000000000000000000000166g These numbers are cumbersome to write and use!

Scientific Notation Scientific Notation: a system used to write large or small numbers more compactly and precisely

Scientific Notation There are 2 parts to a number written in scientific notation: a coefficient - a number equal to or greater than 1 and less than 10 an exponential part - a base which is always 10, raised to an exponent (n)

Is it written in scientific notation? 19.5625 x 103

Is it written in scientific notation? No 19.5625 x 103 The coefficient is not between 1-10

Is it written in scientific notation? 837 x 102

Is it written in scientific notation? No 837 x 102 The coefficient is not between 1-10

Is it written in scientific notation? 2.894 x 10-3

Is it written in scientific notation? Yes 2.894 x 10-3 The coefficient is 1-10 and it is multiplied by 10 to a power

Is it written in scientific notation? 5.8938 x 52

Is it written in scientific notation? 5.8938 x 52 Must be multiplied by the base of 10

Scientific Notation Steps Put the decimal after the first significant digit Indicate how many places the decimal moved by the power of 10 (the number of places the decimal moved becomes the exponent) A positive power of 10 indicates the decimal moved to the left A negative power of 10 indicates the decimal moved to the right

Scientific Notation Steps

Scientific Notation Steps

Convert to scientific notation 0.00137 15237 59000005 123.025 0.00005025

Convert to scientific notation 0.00137 1.37 x 10-3 15237 59000005 123.025 0.00005025

Convert to scientific notation 0.00137 1.37 x 10-3 15237 1.5237 x 104 59000005 123.025 0.00005025

Convert to scientific notation 0.00137 1.37 x 10-3 15237 1.5237 x 104 59000005 5.9000005 x 107 123.025 0.00005025

Convert to scientific notation 0.00137 1.37 x 10-3 15237 1.5237 x 104 59000005 5.9000005 x 107 123.025 1.23025 x 102 0.00005025

Convert to scientific notation 0.00137 1.37 x 10-3 15237 1.5237 x 104 59000005 5.9000005 x 107 123.025 1.23025 x 102 0.00005025 5.025 x 10-5

Addition and Subtraction To add or subtract using scientific notation first write each quantity using the same exponent Add or subtract the quantity and leave the exponent the same Example: (4.3 x 104) + (3.9 x 103) =

Addition and Subtraction To add or subtract using scientific notation first write each quantity using the same exponent Add or subtract the quantity and leave the exponent the same Example: (4.3 x 104) + (3.9 x 103) (4.3 x 104) + (0.39 x 104) = 4.69 x 104

Multiplication and Division To multiply numbers expressed in scientific notation, multiply the numbers in the usual way, then add the exponents together. Example: (8.0 x 104) x (5.0 x 102) =

Multiplication and Division To multiply numbers expressed in scientific notation, multiply the numbers in the usual way, then add the exponents together. Example: (8.0 x 104) x (5.0 x 102) = 8.0 x 5.0 and 104 + 2 =

Multiplication and Division To multiply numbers expressed in scientific notation, multiply the numbers in the usual way, then add the exponents together. Example: (8.0 x 104) x (5.0 x 102) = 8.0 x 5.0 and 104 + 2 = = 40 x 106

Multiplication and Division To divide using scientific notation, divide the numbers as usual and subtract the exponents. Example: (6.9 x 107) = (3.0 x 10-5)

Multiplication and Division To divide using scientific notation, divide the numbers as usual and subtract the exponents. Example: 6.9 and 107 – (-5) (6.9 x 107) = 3.0 (3.0 x 10-5)

Multiplication and Division To divide using scientific notation, divide the numbers as usual and subtract the exponents. Example: 6.9 and 107 – (-5) (6.9 x 107) = 3.0 (3.0 x 10-5) = 2.3 x 1012

Calculator Time Try plugging these into your scientific calculator. Put all answers in scientific notation. 37,000 x 7,000 0.0008 x 0.0009 (7x106) x (8x105)

Calculator Time Try plugging these into your scientific calculator. Put all answers in scientific notation. 37,000 x 7,000 2.59 x 108 0.0008 x 0.0009 7.2 x 10-7 (7x106) x (8x105) 5.6 x 1012

Accuracy and Precision Accuracy: how close a measurement is to the true value Example: weighing a 50g mass 50.00 g = accurate 32.18g = not accurate 49.99g = accurate

Accuracy and Precision Precision: how close multiple measurements are to each other Example: weighing a 50g mass 50.00g, 49.99g, 50.00g = precise 32.18g, 51.02g, 63.44g = not precise

Accuracy and Precision An Easy way to remember ACcurate = Correct Precision = Reproducibility

Percent Error Percent Error: the difference between the measured value and the accepted value in a %. Example: You measured the temperature of boiling water in the lab at 99.1oC. The accepted value of boiling water is 100oC.

Percent Error Formula | 99.1 oC - 100oC | x 100% 100oC You measured the temperature of boiling water in the lab at 99.1oC. The accepted value of boiling water is 100oC. | 99.1 oC - 100oC | x 100% 100oC

Percent Error Formula | 99.1 oC - 100oC | x 100% 100oC = 0.9% You measured the temperature of boiling water in the lab at 99.1oC. The accepted value of boiling water is 100oC. | 99.1 oC - 100oC | x 100% 100oC = 0.9%

Significant Figures You can read the temperature to the nearest degree. You can also estimate the temperature to the nearest tenth of a degree by noting the closeness of the liquid inside the lines – but this would involves some amount of uncertainty.

Significant Figures Significant Figure: the number that is known precisely in a measurement, plus a last estimated digit. When using significant numbers, the last digit is understood to be uncertain

Significant Figures Example We might measure the volume to be 6mL. The actual volume is the range of 5mL to 7mL (6 + 1)

Significant Number Rules

Significant Figure Rules 1 All nonzero digits are significant. Examples: 1.05 0.00110 Significant Figure Rules 1

Significant Figure Rules 2 Sandwiched zeros (zeros between two numbers) are significant. Examples: 4.0208 50.1 Significant Figure Rules 2

Significant Figure Rules 3 Leading zeros (zeros to the left of the 1st nonzero number) are not significant. Examples: 0.0005 only has 1 significant digit, the 5 Significant Figure Rules 3

Significant Figure Rules 4 Trailing zeros (zeros to the right of a nonzero number) after the decimal are significant. Examples: 5.10 3.00 Significant Figure Rules 4

Significant Figure Rules 5 Trailing zeros (zeros to the right of a nonzero number) before the decimal are significant. If they are at the rightmost end of an understood decimal as a place holder, they are not significant. Examples: 50.00 significant 1700.24 significant 300 not significant Significant Figure Rules 5

Significant Figure Rules 6 A number that is counted is exact and can have unlimited significant figures. Significant Figure Rules 6

How many significant digits? 5.703 70 100. 395830 0.0101 21.0

How many significant digits? 5.703 4 70 100. 395830 0.0101 21.0

How many significant digits? 5.703 4 70 1 100. 395830 0.0101 21.0

How many significant digits? 5.703 4 70 1 100. 3 395830 0.0101 21.0

How many significant digits? 5.703 4 70 1 100. 3 395830 5 0.0101 21.0

How many significant digits? 5.703 4 70 1 100. 3 395830 5 0.0101 3 21.0

How many significant digits? 5.703 4 70 1 100. 3 395830 5 0.0101 3 21.0 3

Significant Figures in Calculations Rounding Round down if the last digit is 4 or less Round up if the last digit is 5 or more

Significant Figures in Calculations Addition and Subtraction The answer is written with the same number of decimal places as the measurement with the fewest decimal places 89.332 + 1.1 one digit after the decimal point 90.432 round answer off to 90.4

Significant Figures in Calculations Multiplication and Division The answer is expressed with the same number of significant figures as the number with the fewest significant figures 2.8 x 4.5039 = 12.61092 2 significant figures round off to 13 6.85 112.04 = 0.0611388789 3 significant figurers round off to 0.0611

Significant Figures in Calculations Example: 3.489 x (5.67 - 2.3) Complete the subtraction first 5.67 – 2.3 = 3.37 Use the subtraction rule to determine the significant figurers 3.489 x 3.4 = Complete the multiplication 3.489 x 3.4 = 11.8626 Use the multiplication rule to determine the significant figures 12 Multiplication/Division and Addition/Subtraction In calculations involving both multiplication/division and addition/subtraction, do the steps in the () first.