Significant Figures. Rules for Determining the Number of Significant Figures 1.All non-zero digits are significant. 2.Zeros located between 2 non-zero.

Significant FiguresSignificant Figures

Rules for Determining the Number of Rules for Determining the Number of Significant FiguresSignificant Figures

1.1. All non-zero digits are significant. All non-zero digits are significant.

2.2. Zeros located between 2 non-zero Zeros located between 2 non-zero digits are significant. digits are significant.

3.3. Leading zeros (those at the start of a Leading zeros (those at the start of a number) are never significant. number) are never significant.

4.4. Trailing zeros (those at the end of a Trailing zeros (those at the end of a number) are never significant unless number) are never significant unless they are preceded by a decimal point they are preceded by a decimal point somewhere in the number. somewhere in the number.

Practice ProblemsPractice Problems• How many significant figures are How many significant figures are

present in each of the following present in each of the following measurements?measurements?

1.1. 5.13 5.13 2.2. 100.01100.013.3. 0.04010.04014.4. 0.00500.00505.5. 220,000220,0006.6. 1.90 x 101.90 x 1033

7.7. 153.000153.0008.8. 1.00501.0050

1.1. 5.13 contains 5.13 contains 3 significant figures3 significant figures. All non-zero digits are . All non-zero digits are significant. significant.

2.2. 100.01 contains 100.01 contains 5 significant figures5 significant figures. All non-zero digits are . All non-zero digits are significant. All zeros located between two non-zero digits are significant. All zeros located between two non-zero digits are significant. significant.

3.3. 0.0401 contains 0.0401 contains 3 significant figures3 significant figures. Only the '401' digits are . Only the '401' digits are significant. Leading zeros (those at the start of a number) are not significant. Leading zeros (those at the start of a number) are not significant. significant.

4.4. 0.0050 contains 0.0050 contains 2 significant figures.2 significant figures. Only the '50' digits are Only the '50' digits are significant. Leading zeros are never significant. Trailing zeros significant. Leading zeros are never significant. Trailing zeros (those at the end of a number) are significant when they are (those at the end of a number) are significant when they are found to the right of the decimal point. found to the right of the decimal point.

5.5. 220,000 contains 220,000 contains 2 significant figures.2 significant figures. Only the "22" digits are Only the "22" digits are significant. Since there is no decimal point in the number, the significant. Since there is no decimal point in the number, the trailing zeros are not significant. trailing zeros are not significant.

6.6. 1.90 x 101.90 x 1033 contains contains 3 significant figures3 significant figures. When a number is . When a number is written in scientific notation, ignore the "x 10power" and look written in scientific notation, ignore the "x 10power" and look only at the first number. In this case 1.90 contains 3 significant 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 figures since trailing zeros to the right of the decimal point are significant. significant.

7.7. 153.000 contains 153.000 contains 6 significant figures.6 significant figures. Trailing zeros that are to Trailing zeros that are to the right of a decimal point are significant. the right of a decimal point are significant.

8.8. 1.0050 contains 1.0050 contains 5 significant figures.5 significant figures. Zeros found between non- Zeros found between non-zero digits are always significant. Trailing zeros that are to the zero digits are always significant. Trailing zeros that are to the right of a decimal point are significant, too. right of a decimal point are significant, too.

Arithmetic and Significant FiguresArithmetic and Significant Figures

• Rules for Multiplication and Rules for Multiplication and DivisionDivision

– In calculations involving measurements, In calculations involving measurements, only the measurements are considered only the measurements are considered when determining the correct number of when determining the correct number of significant figures for the answer. Ignore significant figures for the answer. Ignore exact values such as conversion factors. exact values such as conversion factors.

– The answer must contain the same number The answer must contain the same number of significant figures as the measurement of significant figures as the measurement with the with the fewest significant figures.fewest significant figures.

Example:Example:• 11.510 g/7.85 mL = 1.46624204 11.510 g/7.85 mL = 1.46624204

g/mL = 1.47 g/mLg/mL = 1.47 g/mL• The answer is rounded to three The answer is rounded to three

significant figures because 7.85 significant figures because 7.85 contains 3 significant figures contains 3 significant figures while 11.510 contains five while 11.510 contains five significant figures.significant figures.

• Rules for Addition and Rules for Addition and Subtraction Subtraction – In calculations involving measurements, In calculations involving measurements,

only the measurements are considered only the measurements are considered when determining the correct number when determining the correct number of significant figures for the answer. of significant figures for the answer.

– The answer must contain the same The answer must contain the same number of number of decimaldecimal placesplaces as there are as there are in the measurement with the in the measurement with the fewestfewest decimal places.decimal places.

Example:Example:• 125.1 g + 1.300 g + 0.27 g = 125.1 g + 1.300 g + 0.27 g =

126.670 g = 126.7 g126.670 g = 126.7 g• The answer is rounded to 1 The answer is rounded to 1

decimal place because the three decimal place because the three masses that are being added masses that are being added have 1, 3, and 2 decimal places, have 1, 3, and 2 decimal places, respectively. Report the answer respectively. Report the answer to the fewest number of decimal to the fewest number of decimal places.places.

Practice ProblemsPractice Problems• Perform the following Perform the following

calculations using the rules that calculations using the rules that apply to calculations involving apply to calculations involving measurements (i.e. apply the measurements (i.e. apply the rules for significant figures.rules for significant figures.

1.1. 2.501 + 12.40 - 3.996 = 2.501 + 12.40 - 3.996 =

2.2. 25.3 x 1.0 x 2.75 = 25.3 x 1.0 x 2.75 =

3.3. (2.503 - 2.303)/2.303 = (2.503 - 2.303)/2.303 =

4.4. 15.00/1.50 = 15.00/1.50 =

1.1. 2.501 + 12.40 - 3.996 = 10.905 = 10.91 2.501 + 12.40 - 3.996 = 10.905 = 10.91 (Report your answer to 2 (Report your answer to 2 decimal placesdecimal places because 12.40 has the fewest (2) decimal because 12.40 has the fewest (2) decimal places) places)

2.2. 25.3 x 1.0 x 2.75 = 69.575 = 7.0 x 101 25.3 x 1.0 x 2.75 = 69.575 = 7.0 x 101 (Report to 2 significant figures because 1.0 (Report to 2 significant figures because 1.0 has the fewest (2) significant figures) has the fewest (2) significant figures)

3.3. (2.503 - 2.303)/2.303 = 0.200/2.303 = (2.503 - 2.303)/2.303 = 0.200/2.303 = 0.086843 = 0.0868 (Determine the number 0.086843 = 0.0868 (Determine the number of decimal places to use in the numerator of decimal places to use in the numerator using the rules for addition and subtraction. using the rules for addition and subtraction. Then count the number of significant Then count the number of significant figures in the numerator and denominator figures in the numerator and denominator and use the fewest to report your answer.) and use the fewest to report your answer.)

4.4. 15.00/1.50 = 10.0 (Report your answer to 3 15.00/1.50 = 10.0 (Report your answer to 3 significant figures because 1.50 has the significant figures because 1.50 has the fewest (3) significant figures.) fewest (3) significant figures.)

Scientific NotationScientific Notation

• Science deals with both very large and very small numbers. For Science deals with both very large and very small numbers. For example:example:

• Earth’s diameter is about 13,000,000 meters. Earth’s diameter is about 13,000,000 meters. • The radius of a hydrogen atom is 0.00000000012 meters. The radius of a hydrogen atom is 0.00000000012 meters. • Consequently, scientists and engineers use the so called Consequently, scientists and engineers use the so called scientific scientific

notation notation or or exponential notationexponential notation ("shorthand" way) to write very ("shorthand" way) to write very large or very small numbers involving powers of ten. Thuslarge or very small numbers involving powers of ten. Thus– 1 = 101 = 1000

10 = 1010 = 1011

100 = 10100 = 1022

1000 = 101000 = 1033

10,000 = 1010,000 = 1044

100,000 = 10100,000 = 1055

1,000,000 = 101,000,000 = 1066

– 0.1 = 1/10 = 100.1 = 1/10 = 10-1-1

0.01 = 1/100 = 100.01 = 1/100 = 10-2-2

0.001 = 1/1000 = 100.001 = 1/1000 = 10-3-3

0.0001 = 1/10,000 = 100.0001 = 1/10,000 = 10-4-4

• In general, any number X can be In general, any number X can be written as the product of another written as the product of another number N and a power of ten. number N and a power of ten.

• It's important to remember that 1 It's important to remember that 1 << N N <10. In other words, N MUST be at <10. In other words, N MUST be at least 1 but less than 10.least 1 but less than 10.

• The general format for a number The general format for a number written in scientific notation will be:written in scientific notation will be:

N x 10N x 10powerpower

• Examples:Examples:20 = 2 x 10 = 2 x 1020 = 2 x 10 = 2 x 1011

3500 = 3.5 x 1000 = 3.5 x 103500 = 3.5 x 1000 = 3.5 x 1033

0.0055 = 5.5 x 0.001 = 5.5 x 100.0055 = 5.5 x 0.001 = 5.5 x 10-3-3

Converting a Number into Proper Converting a Number into Proper Scientific NotationScientific Notation

• Re-write those digits as a number with 1 Re-write those digits as a number with 1 digit in front of the decimal point and the digit in front of the decimal point and the rest of the digits after the decimal point rest of the digits after the decimal point (i.e. as a number greater than or equal to (i.e. as a number greater than or equal to 1 but less than 10)1 but less than 10)

• Look at the new number you have written. Look at the new number you have written. Count the number of places you must Count the number of places you must move the decimal point in order to get move the decimal point in order to get back to where the decimal point was back to where the decimal point was originally located.originally located.

• If you have to move the decimal point to If you have to move the decimal point to the right to get the original number, then the right to get the original number, then write the exponent as a positive number.write the exponent as a positive number.

• If you have to move the decimal point to If you have to move the decimal point to the left to get the original number, then the left to get the original number, then write the exponent as a negative number.write the exponent as a negative number.

Examples:Examples:• Write 22 650 000 in proper scientific Write 22 650 000 in proper scientific

notation:notation:– Write all significant figures as a number Write all significant figures as a number >> 1 but 1 but

<10:<10:22,650,000 = 2.265 x 1022,650,000 = 2.265 x 10??

– To get back to the original number the decimal To get back to the original number the decimal place must be moved 7 places to the right so the place must be moved 7 places to the right so the exponent will be positive 7.exponent will be positive 7.22,650,000 = 2.265 x 1022,650,000 = 2.265 x 1077

• Write 0.0004050 in proper scientific notation:Write 0.0004050 in proper scientific notation:– Write all significant figures as a number Write all significant figures as a number >> 1 but < 1 but <

10:10:0.0004050 = 4.050 x 100.0004050 = 4.050 x 10??

– To get back to the original number, the decimal To get back to the original number, the decimal place must be moved 4 places to the left so the place must be moved 4 places to the left so the exponent will be negative 4.exponent will be negative 4.0.0004050 = 4.050 x 100.0004050 = 4.050 x 10-4-4

Practice ProblemsPractice Problems• Express the following numbers Express the following numbers

using proper scientific notation.using proper scientific notation.

1.1. 13,000,00013,000,000

2.2. 7500.37500.3

3.3. 209,000209,000

4.4. 0.009700.00970

5.5. 0.00006050.0000605

6.6. 0.003000.00300

1.1. 1.3 x 101.3 x 1077

2.2. 7.5003 x 107.5003 x 1033

3.3. 2.09 x 102.09 x 1055

4.4. 9.70 x 109.70 x 10-3-3 Notice that the trailing Notice that the trailing zero is kept because it is a zero is kept because it is a significant figure. significant figure.

5.5. 6.05 x 106.05 x 10-5-5

6.6. 3.00 x 103.00 x 10-3-3 Notice that both trailing Notice that both trailing zeros are kept because they are zeros are kept because they are significant figures. significant figures.

Converting from Scientific Notation to Converting from Scientific Notation to

Decimal NotationDecimal Notation In order to convert a number written in scientific In order to convert a number written in scientific

notation to one written in standard or decimal notation to one written in standard or decimal notation, follow these steps.notation, follow these steps.

• Write the number down without the "x 10power" Write the number down without the "x 10power" part.part.

• Use the sign and numerical value of the exponent Use the sign and numerical value of the exponent (power) to determine the direction and number of (power) to determine the direction and number of places to move the decimal place.places to move the decimal place.

• Move the decimal point to the Move the decimal point to the rightright if the exponent if the exponent is is positivepositive..

• Move the decimal point to the Move the decimal point to the leftleft if the exponent is if the exponent is negativenegative..

• Remember, numbers with an exponent of 0 are Remember, numbers with an exponent of 0 are between 1 and 10. Numbers with a positive exponent between 1 and 10. Numbers with a positive exponent are greater than or equal to 10 while those with a are greater than or equal to 10 while those with a negative exponent are between zero and 1.negative exponent are between zero and 1.

Examples:Examples:• Convert 6.53 x 10Convert 6.53 x 1044 into decimal (standard) into decimal (standard)

notation.notation.• Write the number without the "x 10Write the number without the "x 1044" and " and

add some extra zeros after in order to add some extra zeros after in order to move the decimal point.move the decimal point.6.53 x 106.53 x 1044 becomes 6.5300 becomes 6.5300

• Since the exponent is positive, move the Since the exponent is positive, move the decimal 4 places to the right.decimal 4 places to the right.6.53 x 106.53 x 1044 becomes 65300 becomes 65300Notice that the decimal place doesn't Notice that the decimal place doesn't actually appear in this case; it is actually appear in this case; it is understood to be at the end of the understood to be at the end of the number. In science, however, placing a number. In science, however, placing a decimal point after the last zero in a decimal point after the last zero in a number greater than or equal to 10 number greater than or equal to 10 indicates that the zeros are significant indicates that the zeros are significant figures. figures.

• Convert 2.50 x 10Convert 2.50 x 10-3-3 into decimal into decimal (standard) notation.(standard) notation.

• Write the number without the "x 10Write the number without the "x 10-3-3" " part and put some extra zeros in front part and put some extra zeros in front of the number.of the number.2.50 x 102.50 x 10-3-3 becomes 0002.50 becomes 0002.50

• Since the exponent is negative, move Since the exponent is negative, move the decimal 3 places to the left.the decimal 3 places to the left.2.50 x 102.50 x 10-3-3 becomes 0.00250 becomes 0.00250Remember that the number should Remember that the number should have the same number of significant have the same number of significant figures as the original number.figures as the original number.

Practice ProblemsPractice Problems• Convert the following numbers Convert the following numbers

from scientific notation to from scientific notation to standard (decimal) notation.standard (decimal) notation.

1.1. 3.0900 x 103.0900 x 1033

2.2. 6.55 x 106.55 x 10-5-5

3.3. 2.455 x 102.455 x 1022

4.4. 1.9 x 101.9 x 10-4-4

5.5. 8.008 x 108.008 x 1022

6.6. 2.05 x 102.05 x 10-3-3

1.1. 3090.0 Notice that the number written in 3090.0 Notice that the number written in scientific notation included a positive scientific notation included a positive exponent. Therefore, the decimal was exponent. Therefore, the decimal was moved to the right and a number greater moved to the right and a number greater than 10 was obtained. Also notice that the than 10 was obtained. Also notice that the original number had 5 significant figures so original number had 5 significant figures so my answer must also have 5 significant my answer must also have 5 significant figures. figures.

2.2. 0.0000655 Notice that the number written 0.0000655 Notice that the number written in scientific notation included a negative in scientific notation included a negative exponent. Therefore, the decimal was exponent. Therefore, the decimal was moved to the left and a number between 0 moved to the left and a number between 0 and 1 was obtained. and 1 was obtained.

3.3. 245.5 245.5 4.4. 0.00019 0.00019 5.5. 800.8 800.8 6.6. 0.00205 0.00205

Use three significant figuresfor your answer

Use two significant figuresfor your answer