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
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
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
- 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
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
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