(Significant Digits are in BOLD type and the non-significant digits are underlined)
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1 Name Per. Date Significant Digits Worksheet Significant digits (or significant figures) are used to represent the accuracy of a measurement. In a measurement the significant digits represent all the reliable digits and the first digit with some uncertainty. RULES FOR DETERMINING THE NUMBER OF SIGNIFICANT DIGITS When Reading a Measurement (Significant Digits are in BOLD type and the non-significant digits are underlined) Examples: 1) ALL non-zero digits are significant. Example: cm has 5 significant digits. 2) ALL zeros between two non-zero digits are significant. Example: kg has 8 significant digits. 3) ALL zeros to the left of an understood decimal point and to the right of a nonzero digit (place holding zeros) are not significant. Example: mm has 3 significant digits. 4) ALL zeros to the left of an expressed decimal point and to the right of a nonzero digit are significant. Example: m/s has 7 significant digits. 5) ALL zeros to the right of an expressed decimal point and to the right of a nonzero digit are significant. Example: cm has 5 significant digits. 6) ALL zeros to the right of an expressed decimal point and to the left of a nonzero digit are not significant. (place holding zeros) Example: cl has 3 significant digits. 1) s There are 4 significant digits. 5, 1, and 2 are significant by rule 1. The zero is significant by rule 2. 2) kg There are 4 significant digits. 4, 3, and 2 are significant by rule 1. The zero at the end is significant by rule 5. The two zeros after the decimal point an before the 4 are not significant by rule 6.
2 ml There are 3 significant digits. 1 and 2 are significant by rule 1. The zero between 1 and 2 is significant by rule 2. The three zeros after the 2 are not significant by rule cm There are 6 significant digits. 7, 8, and 9 are significant by rule 1. The three zeros at the end are significant by rule 4. RULES FOR DETERMINING THE NUMBER OF SIGNIFICANT DIGITS When Taking a Measurement 1) Determine the accuracy of the instrument. (The value of the smallest separation between lines on a measuring device) example: 2) Take your measurement so that ALL certain digits are kept, They ARE significant. Example: It is certain that the measurement is greater than 10 cm and less than 11 cm. Therefore, the measurement has 2 certain digits 1 in the tens place and 0 in the ones place [10. cm]. (note: the 0 is significant because it was read from a line on the measuring device) Counting the lines between 10 cm and 11 cm, it is certain that 5 lines are passed but not 6. Thus another significant digit appears in the tenths place. [10.5 cm]. Thus, the measurement has 3 certain digits which ARE significant. 3) Estimate the last digit of the measurement. Divide the space between the known lines that the measurement falls, into tenths. Estimate this value one place beyond the accuracy of the instrument.
3 Example: It looks as if the measurement is less than half way between the lines, so it is possible to estimate a digit in the hundredths place. This digit is significant because the estimate was based on known lines and was not a blind guess. Thus there are 4 significant digits in the measurement. [10.53 cm] (Note: no digits in a place value smaller than the hundredths place would be significant, because the estimate would be based on two lines that are not on the measuring device. Hence, they would be blind guesses, which are NOT significant) 4) Record the measurement with ALL significant digits. Example: Length = cm There are no significant digits in place values higher than the tens place because the meter stick has a maximum length of 100 cm and there is no uncertainty that the item being measured is smaller than the meter stick. The digit 1 in the tens place is significant because there was a line on the measuring device that showed 1 ten and not 2. Thus, there is no uncertainty in the digit. The digit 0 in the ones place is significant because there was a line on the measuring device that showed 0 ones and not 1. Thus, there is no uncertainty in the digit. The digit 5 in the tenths place is significant because there was a line on the measuring device that showed 5 tenths and not 6. Thus, there is no uncertainty in the digit. The digit 3 in the hundredths place is significant because there is a line at the 5 tenths place and the 6 tenths place, so an estimate can be made on how far between these two lines that the measurement falls. Thus, there is some uncertainty in the digit but it was based on two visible reference marks and is not a blind guess. Any digits in place values smaller than the hundredths place would not be significant, because there are no lines on the measuring device to use as a reference point to make an estimation of the value. Thus, it would be a blind guess with no certainty at all!
4 Multiplication and Division Rule The general rule of thumb is that an answer can have only as many significant digits as the number with the least amount of significant digits used to calculate the answer. The following is a list of steps which will aid in calculating the number of significant digits for a multiplication or division problem. 1) Find the number of significant digits for each factor in the equation. 2) Determine which factor has the least number of significant digits. 3) Solve the equation 4) Round your answer so that it has the same number of significant digits as the factor found in step #2. Example: Step 1: Finding the number of significant digits in each term sig. figs sig. figs sig. figs. Step 2: Find the term with the fewest significant digits. Step 3: Do the math sig. figs. Step 4: Round the answer to number of digits determined in step 2. (3 significant digits) Start at the highest place value and move right rounds to 62.4 (Note: When rounding to a given place value (tenths place) look at the digit in the place value adjacent and to the right of that place value only (hundredths place) and truncate the rest of the digits. The general rule is if this digit is 0 to 4 leave it alone and if it is 5 to 9 round up.)
5 Addition and Subtraction Rule The general rule of thumb for adding and subtracting is that it is a place value rule. The answer cannot have a significant digit in a place value smaller than the smallest place value occupied by a significant digit of the terms to be added or subtracted. The following is a list of steps which will aid in calculating the number of significant digits for a addition or subtraction problem. 1) Find the significant digit occupying the smallest place value in each term and underline it. 2) Determine which of the underlined digits is in the highest place value. 3) Round each term to the place value determined in step 2 using the general rounding rule. 4) Solve the equation. This answer will have the proper number of significant digits. Example: Step 1: Finding the smallest place values Step 2: Determine which underlined digit is in the highest place value. 0 is in the thousandths place. 9 is in the ten thousandths place. 1 is in the hundredths place. 4 is in the millionths place. The hundredths place is greater than all the other place values. So the 1 in the third term occupies the highest place value. Step 3: Round each term to the place value found in step rounds to
6 Step 4: Do the math Significant digits are a way of determining how accurate measurements are and how accurate the quantities that are derived using the measurements are. Digits that are significant have meaning and dimension in the real world. While digits that are not significant are both dimensionless and meaningless. It is possible to divide to numbers and get an infinite number of digits on a calculator (3.0 m / 9.00 s). However only some of these digits have a physical meaning, the others are worthless ( repeating to infinity, only 2 digits are significant.33 m/s). Since science deals with real quantities and real measurements the use of significant digits is mandated.
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