Significant Figures
References
- Textbook: Chapter 2
- Sig Fig Note Sheet
- Sig Fig Note Sheet_ANS KEY
Zeros: When are they significant?
- Zeros sandwiched between non-zero digits are significant (ex. 708 = 3 sig figs 1008 = 4 sig figs)
- Zeros at the end of a number are significant only if there is a decimal point (ex. 100 = 1 sig fig whereas 100. = 3 sig figs and 100.0 = 4 sig figs)
- Zeros at the beginning of a number are NEVER significant (ex. 0.00051 = 2 sig figs)
Tutorial Videos on Zeros
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Multiplication and Division Rules
- Multiply or divide the numbers
- Count the TOTAL number of sig figs in each number used in the calculation
- Round answer to LEAST number of TOTAL SIG FIGS used in the calculation
- Example: (135) (3.1) = 418.5 = 420 (final answer rounded to two sig figs since 135 contains three sig figs, but 3.1 contains only two sig figs)
Tutorial Videos on Multiplication and Division
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Addition and Subtraction Rules
- Add or subtract the numbers
- Keep all digits to the left of the decimal point in the answer
- Count the number of decimal places in each number used in the calculation
- Round answer to the LEAST number of DECIMAL PLACES used in the calculation
- Example: 5.00 - 4.352 = 0.648 = 0.65 (final answer rounded to two decimal places since 5.00 contains two decimal places and 4.352 contains three decimal places)
Tutorial Videos on Addition and Subtraction
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Rounding Rules
- Determine the number of significant digits to keep
- Moving left to right, start counting significant digits with the first non-zero digit
- Stop once you reach the last significant digit to keep. Look at digit directly to the right of the last significant digit that you will keep.
- If the digit to the right is less than five, then the preceding digit remains the same (Example: 1.346 m = 1.3 m, rounded to two sig figs. Note: only the first number to the right of the decimal point is used to round, the "6" does not round the "4" to "5" and then the "3" to "4")
- If the digit to the right is equal to or greater than five, then the preceding digit is increased by one (Example: 1.37 m = 1.4 m, rounded to two sig figs)
- When the last significant digit is in the tens, hundreds, thousands place or any other multiple of ten, round according to Rules #1-3, but also write a zero, "0", for each eliminated digit between the last significant digit and the decimal point. DO NOT write a decimal point.
- Example: 34914.849 m = 35000 rounded to two sig figs
- In most cases, it will be easier to first convert to scientific notation and then round for sig figs.
- Example: 34914.849 m = 3.4914849 x 10^4 m = 3.5 x 10^4 m rounded to two sig figs (3.5 x 10^4 m = 35000 m)
Tutorial Videos on Rounding
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Exact (Counted) Values vs. Measured Values
- Numbers that are counted rather than measured are called exact numbers
- Exact numbers have zero uncertainty, thus have an infinite number of sig figs
- Do not use exact numbers when determining the number of sig figs in a calculation
- Example: you count three people or you perform five trials. There is no uncertainty in these numbers, thus they do not limit the number of sig figs to keep in a final answer of a calculation.
Tutorial Videos on Doing Calculations with Exact (counted) Values
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Calculating the Average of Measured Values
- The final answer will contain the same number of decimal places as the LEAST number of decimal places in the measured values
- Example: calculate the average of these measurements: 65.44 g, 62.1 g, 65.348 g
- Add the measurements together then divide by 3: (65.44 g + 62.1 g + 65.348 g) / 3 = 64.296 g
- Round to correct number of decimal places: 64.296 g = 64.3 g
- Since there were exactly three values, "3" is an exact number and is not used to determine the number of sig figs to keep in an answer