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Significant Figures

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Significant Figures

- All non-zero digits are significant.
- Zeros located between 2 non-zero digits are significant.
- Leading zeros (those at the start of a number) are never significant.
- Trailing zeros (those at the end of a number) are never significant unless they are preceded by a decimal point somewhere in the number.

Practice Problems

- How many significant figures are present in each of the following measurements?
- 5.13
- 100.01
- 0.0401
- 0.0050
- 220,000
- 1.90 x 103
- 153.000
- 1.0050

- 5.13 contains 3 significant figures. All non-zero digits are significant.
- 100.01 contains 5 significant figures. All non-zero digits are significant. All zeros located between two non-zero digits are significant.
- 0.0401 contains 3 significant figures. Only the '401' digits are significant. Leading zeros (those at the start of a number) are not significant.
- 0.0050 contains 2 significant figures. Only the '50' digits are significant. Leading zeros are never significant. Trailing zeros (those at the end of a number) are significant when they are found to the right of the decimal point.
- 220,000 contains 2 significant figures. Only the "22" digits are significant. Since there is no decimal point in the number, the trailing zeros are not significant.
- 1.90 x 103 contains 3 significant figures. When a number is written in scientific notation, ignore the "x 10power" and look only at the first number. In this case 1.90 contains 3 significant figures since trailing zeros to the right of the decimal point are significant.
- 153.000 contains 6 significant figures. Trailing zeros that are to the right of a decimal point are significant.
- 1.0050 contains 5 significant figures. Zeros found between non-zero digits are always significant. Trailing zeros that are to the right of a decimal point are significant, too.

- Rules for Multiplication and Division
- In calculations involving measurements, only the measurements are considered when determining the correct number of significant figures for the answer. Ignore exact values such as conversion factors.
- The answer must contain the same number of significant figures as the measurement with the fewest significant figures.

Example:

- 11.510 g/7.85 mL = 1.46624204 g/mL = 1.47 g/mL
- The answer is rounded to three significant figures because 7.85 contains 3 significant figures while 11.510 contains five significant figures.

- Rules for Addition and Subtraction
- In calculations involving measurements, only the measurements are considered when determining the correct number of significant figures for the answer.
- The answer must contain the same number of decimalplaces as there are in the measurement with the fewestdecimal places.

Example:

- 125.1 g + 1.300 g + 0.27 g = 126.670 g = 126.7 g
- The answer is rounded to 1 decimal place because the three masses that are being added have 1, 3, and 2 decimal places, respectively. Report the answer to the fewest number of decimal places.

Practice Problems

- Perform the following calculations using the rules that apply to calculations involving measurements (i.e. apply the rules for significant figures.
- 2.501 + 12.40 - 3.996 =
- 25.3 x 1.0 x 2.75 =
- (2.503 - 2.303)/2.303 =
- 15.00/1.50 =

- 2.501 + 12.40 - 3.996 = 10.905 = 10.91 (Report your answer to 2 decimal places because 12.40 has the fewest (2) decimal places)
- 25.3 x 1.0 x 2.75 = 69.575 = 7.0 x 101 (Report to 2 significant figures because 1.0 has the fewest (2) significant figures)
- (2.503 - 2.303)/2.303 = 0.200/2.303 = 0.086843 = 0.0868 (Determine the number of decimal places to use in the numerator using the rules for addition and subtraction. Then count the number of significant figures in the numerator and denominator and use the fewest to report your answer.)
- 15.00/1.50 = 10.0 (Report your answer to 3 significant figures because 1.50 has the fewest (3) significant figures.)

Scientific Notation

- Science deals with both very large and very small numbers. For example:
- Earth’s diameter is about 13,000,000 meters.
- The radius of a hydrogen atom is 0.00000000012 meters.
- Consequently, scientists and engineers use the so called scientific notation or exponential notation ("shorthand" way) to write very large or very small numbers involving powers of ten. Thus
- 1 = 10010 = 101100 = 1021000 = 10310,000 = 104100,000 = 1051,000,000 = 106
- 0.1 = 1/10 = 10-10.01 = 1/100 = 10-20.001 = 1/1000 = 10-30.0001 = 1/10,000 = 10-4

- In general, any number X can be written as the product of another number N and a power of ten.
- It's important to remember that 1 < N <10. In other words, N MUST be at least 1 but less than 10.
- The general format for a number written in scientific notation will be:
N x 10power

- Examples:20 = 2 x 10 = 2 x 1013500 = 3.5 x 1000 = 3.5 x 1030.0055 = 5.5 x 0.001 = 5.5 x 10-3

- Re-write those digits as a number with 1 digit in front of the decimal point and the rest of the digits after the decimal point (i.e. as a number greater than or equal to 1 but less than 10)
- Look at the new number you have written. Count the number of places you must move the decimal point in order to get back to where the decimal point was originally located.
- If you have to move the decimal point to the right to get the original number, then write the exponent as a positive number.
- If you have to move the decimal point to the left to get the original number, then write the exponent as a negative number.

Examples:

- Write 22 650 000 in proper scientific notation:
- Write all significant figures as a number > 1 but <10:22,650,000 = 2.265 x 10?
- To get back to the original number the decimal place must be moved 7 places to the right so the exponent will be positive 7.22,650,000 = 2.265 x 107

- Write 0.0004050 in proper scientific notation:
- Write all significant figures as a number > 1 but < 10:0.0004050 = 4.050 x 10?
- To get back to the original number, the decimal place must be moved 4 places to the left so the exponent will be negative 4.0.0004050 = 4.050 x 10-4

Practice Problems

- Express the following numbers using proper scientific notation.
- 13,000,000
- 7500.3
- 209,000
- 0.00970
- 0.0000605
- 0.00300

- 1.3 x 107
- 7.5003 x 103
- 2.09 x 105
- 9.70 x 10-3 Notice that the trailing zero is kept because it is a significant figure.
- 6.05 x 10-5
- 3.00 x 10-3 Notice that both trailing zeros are kept because they are significant figures.

In order to convert a number written in scientific notation to one written in standard or decimal notation, follow these steps.

- Write the number down without the "x 10power" part.
- Use the sign and numerical value of the exponent (power) to determine the direction and number of places to move the decimal place.
- Move the decimal point to the right if the exponent is positive.
- Move the decimal point to the left if the exponent is negative.
- Remember, numbers with an exponent of 0 are between 1 and 10. Numbers with a positive exponent are greater than or equal to 10 while those with a negative exponent are between zero and 1.

Examples:

- Convert 6.53 x 104 into decimal (standard) notation.
- Write the number without the "x 104" and add some extra zeros after in order to move the decimal point.6.53 x 104 becomes 6.5300
- Since the exponent is positive, move the decimal 4 places to the right.6.53 x 104 becomes 65300Notice that the decimal place doesn't actually appear in this case; it is understood to be at the end of the number. In science, however, placing a decimal point after the last zero in a number greater than or equal to 10 indicates that the zeros are significant figures.

- Convert 2.50 x 10-3 into decimal (standard) notation.
- Write the number without the "x 10-3" part and put some extra zeros in front of the number.2.50 x 10-3 becomes 0002.50
- Since the exponent is negative, move the decimal 3 places to the left.2.50 x 10-3 becomes 0.00250Remember that the number should have the same number of significant figures as the original number.

Practice Problems

- Convert the following numbers from scientific notation to standard (decimal) notation.
- 3.0900 x 103
- 6.55 x 10-5
- 2.455 x 102
- 1.9 x 10-4
- 8.008 x 102
- 2.05 x 10-3

- 3090.0 Notice that the number written in scientific notation included a positive exponent. Therefore, the decimal was moved to the right and a number greater than 10 was obtained. Also notice that the original number had 5 significant figures so my answer must also have 5 significant figures.
- 0.0000655 Notice that the number written in scientific notation included a negative exponent. Therefore, the decimal was moved to the left and a number between 0 and 1 was obtained.
- 245.5
- 0.00019
- 800.8
- 0.00205

Use three significant figures

for your answer

Use two significant figures

for your answer