Significant figures
What are significant figures?
Significant figures, also known as "sig figs", are used when working with measurements only. They help you properly round a number. There are rules to determine whether a figure is significant or non-significant.
Rules:
1. Figures from 1 to 9 are significant.
Example:
a) 123 has 3 sig figs.
b) 67 has 2 sig figs.
c) 4953 has 4 sig figs.
2. Zeros between other significant figures are significant.
Example:
a) 309 has 3 sig figs.
b) 7 902 has 4 sig figs.
c) 12 034 has 5 sig figs.
3. Zero(s) to the right of both a decimal place and a significant figure are significant.
Example:
a) 42.0 has 3 sig figs.
b) 370.0 has 4 sig figs.
c) 504.00 has 5 sig figs.
4. Zero(s) after a zero and a decimal place are called placeholders. Zero(s) after a significant number with nothing to the right are also called placeholders. These are non-significant figures. A basic rule of thumb is that if you can shorten the number to the format of a number multiplied by 10x and still have the same amount, only the shortened numbers are significant (i.e 12 x 107. 12 is the significant number.)
Example:
a) 1.23 x 104 has 3 sig figs.
b) 0.055 has 2 sig figs.
c) 230 has 2 sig figs
5. Numbers that aren’t measurements, such as constant values, have an infinite number of sig figs. We ignore the sig figs in a constant number.
Example:
a) 17 cats has an infinite number of sig figs.
b) Gravity has an infinite number of sig figs.
c) 9 cookies has an infinite number of sig figs.
Adding and subtracting with sig figs
Round the answer to the least number of decimal places. Be careful when using scientific notation.
Example:
a) 4.1 + 7.00092 = 11.1 because 4.1 has one decimal place.
b) 33.7 + 51 = 84 because 51 has no decimal places.
c) 1.93 - 10-1 = 1.93 + 0.1 = 1.8 because 0.1 has one decimal place.
When multiplying or dividing
Round your answer so that it has the same number of significant figures as the number that has the least significant figures.
Examples:
a) 101 x 6.0 = 610 because 6.0 has 2 sig figs.
b) 6.35 x 3098 x 25 = 490 000 because 25 has 2 sig figs.
c) 4302 ÷ 590 = 7.3 because 590 has 2 sig figs.
Here are links to some significant figures practice worksheets:
http://misterguch.brinkster.net/PRA006.pdf
http://www.wellesley.edu/Chemistry/Chem105manual/Appendices/SigFigPractice.pdf
http://www.saddleback.edu/faculty/jzoval/worksheets_tutorials/ch1worksheets/worksheet_Sig_Fig_9_11_08.pdf
http://www.everettcc.edu/uploadedFiles/Student_Resources_and_Services/TRIO/Significant_figures_wksht.pdf
http://misterguch.brinkster.net/september2002.pdf
http://nemsphysci.wikispaces.com/file/view/SigDig+Practice+Wkst.pdf
http://www.beachwoodschools.org/Downloads/ex%20pract%20mult%20divd%20with%20sig%20figs.pdf
Significant figures, also known as "sig figs", are used when working with measurements only. They help you properly round a number. There are rules to determine whether a figure is significant or non-significant.
Rules:
1. Figures from 1 to 9 are significant.
Example:
a) 123 has 3 sig figs.
b) 67 has 2 sig figs.
c) 4953 has 4 sig figs.
2. Zeros between other significant figures are significant.
Example:
a) 309 has 3 sig figs.
b) 7 902 has 4 sig figs.
c) 12 034 has 5 sig figs.
3. Zero(s) to the right of both a decimal place and a significant figure are significant.
Example:
a) 42.0 has 3 sig figs.
b) 370.0 has 4 sig figs.
c) 504.00 has 5 sig figs.
4. Zero(s) after a zero and a decimal place are called placeholders. Zero(s) after a significant number with nothing to the right are also called placeholders. These are non-significant figures. A basic rule of thumb is that if you can shorten the number to the format of a number multiplied by 10x and still have the same amount, only the shortened numbers are significant (i.e 12 x 107. 12 is the significant number.)
Example:
a) 1.23 x 104 has 3 sig figs.
b) 0.055 has 2 sig figs.
c) 230 has 2 sig figs
5. Numbers that aren’t measurements, such as constant values, have an infinite number of sig figs. We ignore the sig figs in a constant number.
Example:
a) 17 cats has an infinite number of sig figs.
b) Gravity has an infinite number of sig figs.
c) 9 cookies has an infinite number of sig figs.
Adding and subtracting with sig figs
Round the answer to the least number of decimal places. Be careful when using scientific notation.
Example:
a) 4.1 + 7.00092 = 11.1 because 4.1 has one decimal place.
b) 33.7 + 51 = 84 because 51 has no decimal places.
c) 1.93 - 10-1 = 1.93 + 0.1 = 1.8 because 0.1 has one decimal place.
When multiplying or dividing
Round your answer so that it has the same number of significant figures as the number that has the least significant figures.
Examples:
a) 101 x 6.0 = 610 because 6.0 has 2 sig figs.
b) 6.35 x 3098 x 25 = 490 000 because 25 has 2 sig figs.
c) 4302 ÷ 590 = 7.3 because 590 has 2 sig figs.
Here are links to some significant figures practice worksheets:
http://misterguch.brinkster.net/PRA006.pdf
http://www.wellesley.edu/Chemistry/Chem105manual/Appendices/SigFigPractice.pdf
http://www.saddleback.edu/faculty/jzoval/worksheets_tutorials/ch1worksheets/worksheet_Sig_Fig_9_11_08.pdf
http://www.everettcc.edu/uploadedFiles/Student_Resources_and_Services/TRIO/Significant_figures_wksht.pdf
http://misterguch.brinkster.net/september2002.pdf
http://nemsphysci.wikispaces.com/file/view/SigDig+Practice+Wkst.pdf
http://www.beachwoodschools.org/Downloads/ex%20pract%20mult%20divd%20with%20sig%20figs.pdf
Uncertainty
Uncertainty is used when measuring something with an instrument. Uncertainty is expressed using two ways:
1) Absolute uncertainty which is written in the same units as the instrument is using.
Example: 30 cm ± 0.5 cm
2) Relative uncertainty which is written as a percentage of the measurement. It is calculated using this formula: (absolute uncertainty ÷ Value of measurement) x 100.
Example: 30cm ± 1.67%
You can find the uncertainty of a measurement using two ways.
1) It is written on the instrument already.
Example: On a balance, it might indicate that there is an uncertainty of 0.02 g.
2) When it is not already indicated on the instrument, simply take half of the smallest possible measurement on the instrument and that is your uncertainty.
Example: Let’s say a thermometer’s smallest measurement is 1°C. Half of 1 is 0.5 so the uncertainty of this thermometer is 0.5°C.
Always remember that uncertainty is written out as the value, then this sign (±), then the degree of uncertainty.
Example:
Absolute: 35mL ± 0.5mL
Relative: 35mL ± 1.43%
Here are links to some uncertainty practice worksheets:
http://www.nvsd44.bc.ca/en/Staff/MN/MauriceL10214/PH-11/~/media/Staff/MN/MauriceL10214/PDF/PH%2011/Unit%201%20%20%20Kinematics/Uncertainty%20Worksheet.ashx
http://wrean.disted.camosun.bc.ca/ph191/exercises/ph191_uncert_work.pdf
http://semichem11ib.wikispaces.com/file/view/5+-+Uncertainty+Worksheet+B+Key.pdf
http://www.wellesley.edu/Chemistry/Chem105manual/Appendices/UncertaintyPractice.pdf
1) Absolute uncertainty which is written in the same units as the instrument is using.
Example: 30 cm ± 0.5 cm
2) Relative uncertainty which is written as a percentage of the measurement. It is calculated using this formula: (absolute uncertainty ÷ Value of measurement) x 100.
Example: 30cm ± 1.67%
You can find the uncertainty of a measurement using two ways.
1) It is written on the instrument already.
Example: On a balance, it might indicate that there is an uncertainty of 0.02 g.
2) When it is not already indicated on the instrument, simply take half of the smallest possible measurement on the instrument and that is your uncertainty.
Example: Let’s say a thermometer’s smallest measurement is 1°C. Half of 1 is 0.5 so the uncertainty of this thermometer is 0.5°C.
Always remember that uncertainty is written out as the value, then this sign (±), then the degree of uncertainty.
Example:
Absolute: 35mL ± 0.5mL
Relative: 35mL ± 1.43%
Here are links to some uncertainty practice worksheets:
http://www.nvsd44.bc.ca/en/Staff/MN/MauriceL10214/PH-11/~/media/Staff/MN/MauriceL10214/PDF/PH%2011/Unit%201%20%20%20Kinematics/Uncertainty%20Worksheet.ashx
http://wrean.disted.camosun.bc.ca/ph191/exercises/ph191_uncert_work.pdf
http://semichem11ib.wikispaces.com/file/view/5+-+Uncertainty+Worksheet+B+Key.pdf
http://www.wellesley.edu/Chemistry/Chem105manual/Appendices/UncertaintyPractice.pdf