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## Significant Figures

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**Significant Figures**What is so significant about numbers??**Just which numbers are significant?**• All nonzero numbers (all numbers 1-9) – • 289 • 573 • 119**Just which numbers are significant?**• All nonzero numbers (all numbers 1-9) - 289 • All zeros b/w numbers – 909 • 501 • 3002 • 4000001**Just which numbers are significant?**• All nonzero numbers (all numbers 1-9) - 289 • All zeros b/w numbers – 909 • Zeros to the right of a number AND to the left of a written decimal point – • 250. • 700. • 10.**Just which numbers are significant?**• All nonzero numbers (all numbers 1-9) - 289 • All zeros b/w numbers – 909 • Zeros to the right of a number AND to the left of a written decimal point – 250. 700. 10. • Zeros to the right of a number AND to the right or a written decimal point – • 5.0 • 236.70 • 3.0 x 108**These are NEVER significant**• Zeros to the left of the decimal in numbers less than one • 0.8 • 0.222 *** these are placeholders only**These are NEVER significant**• Zeros to the left of the decimal in numbers less than one • Zeros to the right of a decimal, but to the left of the first number • 0.008 • 0.02**Exceptions to the rule**• Conversion factors – unlimited sig. fig. • 1 km/1000 m • 60 min/1 hr • 100 cm/1 m**Exceptions to the rule**• Conversion factors – unlimited sig. fig. • Counting numbers • 30 days • 12 in one dozen**Calculations with sig. figs.**• The following rules must be followed so that results reflect sig. figs. from your measurements taken originally…**Addition & subtraction with sig. figs.**• Your answer can’t be MORE precise than your LEAST precise measurement (what in the world does that mean??)**Addition & subtraction with sig. figs.**• Your answer can’t be MORE precise than your LEAST precise measurement • Your answer must be rounded to the same number of decimal places from the original problem… • 34.2 + 2.002 = ?**Addition & subtraction**• 34.2 + 2.002 = • Actual answer would be 36.202**Addition & subtraction**• Actual answer would be 36.202 • The least precise number is 34.2, therefore your answer MUST be to one decimal place**Addition & subtraction**• Actual answer would be 36.202 • The least precise number is 34.2, therefore your answer MUST be to one decimal place • Correct answer = 36.2**Addition & subtraction**• Example #2 • 3.999 – 1.77 = ?**Addition & subtraction**• 3.999 – 1.77 = • Actual answer would be 2.229**Addition & subtraction**• Actual answer would be 2.229 • The least precise number is 1.77, therefore your answer MUST be to two decimal places**Addition & subtraction**• Actual answer would be 2.229 • The least precise number is 1.77, therefore your answer MUST be to two decimal places • The correct answer would be 2.23**Multiplication & division**• Answer must be in the fewest number of sig. figs. from the original problem. • Example: • 45.6 x 1.009 = ?**Multiplication & division**• Answer must be in the fewest number of sig. figs. from the original problem. • Example: • 45.6 x 1.009 = ? • Actual is 46.0104**Multiplication & division**• Answer must be in the fewest number of sig. figs. from the original problem. • Example: • 45.6 x 1.009 = ? • Actual is 46.0104 • Smallest # of sig figs is 3… so answer must be in 3 sig figs.**Multiplication & division**• Answer must be in the fewest number of sig. figs. from the original problem. • Example: • 45.6 x 1.009 = ? • Actual is 46.0104 • Smallest # of sig figs is 3… so answer must be in 3 sig figs. • Correct answer is 46.0 (not just 46)**Multiplication & division**• Example #2 • 505 / 7 = ?**Multiplication & division**• Example #2 • 505 / 7 = ? • Actual answer is 72.14285714**Multiplication & division**• Example #2 • 505 / 7 = ? • Actual answer is 72.14285714 • Least number of sig figs is 1, so answer can only have one sig fig.**Multiplication & division**• Example #2 • 505 / 7 = ? • Actual answer is 72.14285714 • Least number of sig figs is 1, so answer can only have on sig fig. • Correct answer is 70 or it could be 7 x 101**Rounding…**• If the number right past the one you want to keep is: • Greater than 5 go up by one • 295.46 to 4 sig figs would be 295.5 • 999.97 to four sig figs would be 1000.**Rounding…**• If the number right past the one you want to keep is: • Greater than 5 go up by one • Less than 5 no change • 999.94 to four sig figs would be 999.9 • 564.44 to three sig figs would be 564**Rounding…**• If the number right past the one you want to keep is: • Greater than 5 go up by one • Less than 5 no change • 5 followed by a number go up by one • 2.352 to two sig figs would be 2.4 • 4.156 to two sig figs would be 4.2**Rounding…**• If the number right past the one you want to keep is: • Greater than 5 go up by one • Less than 5 no change • 5 followed by a number go up by one • 5 followed by nothing… look at the number before it… if it is ODD go up by one • 3.375 to three sig figs would be 3.38 • 0.035 to one sig fig would be 0.04**Rounding…**• If the number right past the one you want to keep is: • Greater than 5 go up by one • Less than 5 no change • 5 followed by a number go up by one • 5 followed by nothing… look at the number before it… if it is ODD go up by one • 5 followed by nothing… look at the number before it… if it is EVEN no change • 4.8785 to four sig figs would be 4.878 • 399.345 to five sig figs would be 399.34**Scientific notation**• Very large or very small numbers are expressed in scientific notation**Scientific notation**• Very large or very small numbers are expressed in scientific notation • M x 10n**Scientific notation**• Very large or very small numbers are expressed in scientific notation • M x 10n • “M” must be 1 or greater, but less than 10 • 2.2 x 105**Scientific notation**• Very large or very small numbers are expressed in scientific notation • M x 10n • “M” must be 1 or greater, but less than 10 • All numbers that represent “M” are significant • 3.10 x 109 • 9.98 x 103**Scientific notation**• Very large or very small numbers are expressed in scientific notation • M x 10n • “M” must be 1 or greater, but less than 10 • All numbers that represent “M” are significant • Numbers that represent “n” are whole numbers, positive or negative • 6.022 x 1023 • 2.1 x 10-5**Scientific notation**• Very large or very small numbers are expressed in scientific notation • M x 10n • “M” must be 1 or greater, but less than 10 • All numbers that represent “M” are significant • Numbers that represent “n” are whole numbers, positive or negative • Examples: • 270,000 would be 2.7 x 105 • The understood decimal is behind the last zero…move the decimal until it makes the number represent “M” • 0.000000505 would be 5.05 x 10-7**Calculations w/ scientific notation**• Addition and subtraction: • All numbers must have the same exponent before “M” can be added or subtracted • Answer will have the same exponent value as originals • Don’t forget rules w/ sig figs… they still apply!! • 2.4 x 102 + 5.7 x 102 = 8.1 x 102 • 9.05 x 105 – 5.5 x 105 = actual 3.55 x 105 • Correct is 3.6 x 105**Calculations w/ scientific notation**• Addition and subtraction: • All numbers must have the same exponent before “M” can be added or subtracted • The problem comes when exponents are DIFFERENT… • You must MAKE the exponents the same… • LL / RR = left larger or right reduce when you move the decimal it changes the exponent -- ALWAYS**Calculations w/ scientific notation**• Addition and subtraction: • 2.5 x 104 + 5.2 x 103 = • Choose which number you want to deal with… either will work out**Calculations w/ scientific notation**• Addition and subtraction: • 2.5 x 104 + 5.2 x 103 = • Choose which number you want to deal with… either will work out • I like to change the smaller exponent to the larger 5.2 x 103**Calculations w/ scientific notation**• Addition and subtraction: • 2.5 x 104 + 5.2 x 103 = • Choose which number you want to deal with… either will work out • I like to change the smaller exponent to the larger 5.2 x 103 • I want to make the exp. larger, so I move the decimal one place to the left**Calculations w/ scientific notation**• Addition and subtraction: • 2.5 x 104 + 5.2 x 103 = • Choose which number you want to deal with… either will work out • I like to change the smaller exponent to the larger 5.2 x 103 • I want to make the exp. larger, so I move the decimal one place to the left • This makes it .52 x 104**Calculations w/ scientific notation**• Addition and subtraction: • 2.5 x 104 + 5.2 x 103 = • Choose which number you want to deal with… either will work out • I like to change the smaller exponent to the larger 5.2 x 103 • I want to make the exp. larger, so I move the decimal one place to the left • This makes it .52 x 104 • Now “M” can be added because exp are the same • Actual answer = 3.02 x 104 • Correct answer = 3.0 x 104**Calculations w/ scientific notation**• Multiplication – this one is much easier • 3.0 x 103 x 4.0 x 104**Calculations w/ scientific notation**• Multiplication – this one is much easier • 3.0 x 103 x 4.0 x 104 • You will multiply your “M”s together (12)**Calculations w/ scientific notation**• Multiplication – this one is much easier • 3.0 x 103 x 4.0 x 104 • You will multiply your “M”s together (12) • Then add your “n”s together (7)**Calculations w/ scientific notation**• Multiplication – this one is much easier • 3.0 x 103 x 4.0 x 104 • You will multiply your “M”s together (12) • Then add your “n”s together (7) • CAREFUL… you will often need to move the decimal to keep correct scientific notation**Calculations w/ scientific notation**• Multiplication – this one is much easier • 3.0 x 103 x 4.0 x 104 • You will multiply your “M”s together (12) • Then add your “n”s together (7) • CAREFUL… you will often need to move the decimal to keep correct scientific notation • Actual answer would be 12 x 107