21
Measurement & Calculations
Measurement amp Calculations
Overview of the Scientific MethodOBSERVE
FORMULATEHYPOTHESIS
TEST
THEORIZE
PUBLISH RESULTS
CONVERSION FACTORS
Uncertainty in MeasurementbullTwo kinds of numbers in scientific work
exact numbers(those whose values are known exactly) and inexact numbers
(those whose values have some uncertainty)
bullExact numbers counts (eg students in this classroom)numbers by definition (eg 1 m = 100 cm 1 in = 254 cm)
bullNumbers obtained by measurement are always inexact
PRECISION OF CALCULATIONS
How close are your calculated answers to the known or accepted value
1 Non-zero digits are always significant
2 Any zeros between two significant digits are significant
3 A final zero or trailing zeros after the decimal are significant
4 Zeros that are holding decimal places are NOT significant
Rules for Significant Figures
Significant Figures in Calculations
The result of calculation using measured numbers shouldreflect the precision of the original measurements Thesignificant figures in calculation should follow thefollowing rules
Addition and SubtractionThe answer cannot have more decimal places than themeasurement with the fewest number of decimal placeseg 4 + 145 +124 = 1785 round off to 18
Multiplication and DivisionThe answer cannot have more significant figures thanthe measurement with the fewest numbers ofsignificant figureseg 6221 x 52 = 323492 1048774 round off to 32
Examples (continued)Problem Perform the calculation and express your answerto the correct number of sig fig
(a)(124 ndash 945) 02212= 295 02212 = 13336hellip round off to 13
In sequential operations the correct rule must be applied to eachoperation Carry all numbers through the calculation and round at the end
(b) (00045 x 200000) + (2813 x 12)= 90 + 33756 =00090 x 104 + 33756 x 104 = 33846 x 104 round off to 34 x 104
Scientific NotationScientific notation is used to eliminate the potentialambiguity of whether the zeros at the end of a number aresignificant The significant figure information is in thecoefficient and decimal point information is in the exponent
eg 10300 is written 103x104 (3 sig fig)10300 is written 1030x104 (4 sig fig)10300 is written 10300x104 (5 sig fig)Note that in scientific notation a correctly written numberhas a single nonzero digit to the left of the decimal point
ExerciseHow many significant figures are in each of the followingmeasured numbers (a) 4003 (b) 6023 x 1023 (c) 5000
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
Overview of the Scientific MethodOBSERVE
FORMULATEHYPOTHESIS
TEST
THEORIZE
PUBLISH RESULTS
CONVERSION FACTORS
Uncertainty in MeasurementbullTwo kinds of numbers in scientific work
exact numbers(those whose values are known exactly) and inexact numbers
(those whose values have some uncertainty)
bullExact numbers counts (eg students in this classroom)numbers by definition (eg 1 m = 100 cm 1 in = 254 cm)
bullNumbers obtained by measurement are always inexact
PRECISION OF CALCULATIONS
How close are your calculated answers to the known or accepted value
1 Non-zero digits are always significant
2 Any zeros between two significant digits are significant
3 A final zero or trailing zeros after the decimal are significant
4 Zeros that are holding decimal places are NOT significant
Rules for Significant Figures
Significant Figures in Calculations
The result of calculation using measured numbers shouldreflect the precision of the original measurements Thesignificant figures in calculation should follow thefollowing rules
Addition and SubtractionThe answer cannot have more decimal places than themeasurement with the fewest number of decimal placeseg 4 + 145 +124 = 1785 round off to 18
Multiplication and DivisionThe answer cannot have more significant figures thanthe measurement with the fewest numbers ofsignificant figureseg 6221 x 52 = 323492 1048774 round off to 32
Examples (continued)Problem Perform the calculation and express your answerto the correct number of sig fig
(a)(124 ndash 945) 02212= 295 02212 = 13336hellip round off to 13
In sequential operations the correct rule must be applied to eachoperation Carry all numbers through the calculation and round at the end
(b) (00045 x 200000) + (2813 x 12)= 90 + 33756 =00090 x 104 + 33756 x 104 = 33846 x 104 round off to 34 x 104
Scientific NotationScientific notation is used to eliminate the potentialambiguity of whether the zeros at the end of a number aresignificant The significant figure information is in thecoefficient and decimal point information is in the exponent
eg 10300 is written 103x104 (3 sig fig)10300 is written 1030x104 (4 sig fig)10300 is written 10300x104 (5 sig fig)Note that in scientific notation a correctly written numberhas a single nonzero digit to the left of the decimal point
ExerciseHow many significant figures are in each of the followingmeasured numbers (a) 4003 (b) 6023 x 1023 (c) 5000
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
CONVERSION FACTORS
Uncertainty in MeasurementbullTwo kinds of numbers in scientific work
exact numbers(those whose values are known exactly) and inexact numbers
(those whose values have some uncertainty)
bullExact numbers counts (eg students in this classroom)numbers by definition (eg 1 m = 100 cm 1 in = 254 cm)
bullNumbers obtained by measurement are always inexact
PRECISION OF CALCULATIONS
How close are your calculated answers to the known or accepted value
1 Non-zero digits are always significant
2 Any zeros between two significant digits are significant
3 A final zero or trailing zeros after the decimal are significant
4 Zeros that are holding decimal places are NOT significant
Rules for Significant Figures
Significant Figures in Calculations
The result of calculation using measured numbers shouldreflect the precision of the original measurements Thesignificant figures in calculation should follow thefollowing rules
Addition and SubtractionThe answer cannot have more decimal places than themeasurement with the fewest number of decimal placeseg 4 + 145 +124 = 1785 round off to 18
Multiplication and DivisionThe answer cannot have more significant figures thanthe measurement with the fewest numbers ofsignificant figureseg 6221 x 52 = 323492 1048774 round off to 32
Examples (continued)Problem Perform the calculation and express your answerto the correct number of sig fig
(a)(124 ndash 945) 02212= 295 02212 = 13336hellip round off to 13
In sequential operations the correct rule must be applied to eachoperation Carry all numbers through the calculation and round at the end
(b) (00045 x 200000) + (2813 x 12)= 90 + 33756 =00090 x 104 + 33756 x 104 = 33846 x 104 round off to 34 x 104
Scientific NotationScientific notation is used to eliminate the potentialambiguity of whether the zeros at the end of a number aresignificant The significant figure information is in thecoefficient and decimal point information is in the exponent
eg 10300 is written 103x104 (3 sig fig)10300 is written 1030x104 (4 sig fig)10300 is written 10300x104 (5 sig fig)Note that in scientific notation a correctly written numberhas a single nonzero digit to the left of the decimal point
ExerciseHow many significant figures are in each of the followingmeasured numbers (a) 4003 (b) 6023 x 1023 (c) 5000
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
Uncertainty in MeasurementbullTwo kinds of numbers in scientific work
exact numbers(those whose values are known exactly) and inexact numbers
(those whose values have some uncertainty)
bullExact numbers counts (eg students in this classroom)numbers by definition (eg 1 m = 100 cm 1 in = 254 cm)
bullNumbers obtained by measurement are always inexact
PRECISION OF CALCULATIONS
How close are your calculated answers to the known or accepted value
1 Non-zero digits are always significant
2 Any zeros between two significant digits are significant
3 A final zero or trailing zeros after the decimal are significant
4 Zeros that are holding decimal places are NOT significant
Rules for Significant Figures
Significant Figures in Calculations
The result of calculation using measured numbers shouldreflect the precision of the original measurements Thesignificant figures in calculation should follow thefollowing rules
Addition and SubtractionThe answer cannot have more decimal places than themeasurement with the fewest number of decimal placeseg 4 + 145 +124 = 1785 round off to 18
Multiplication and DivisionThe answer cannot have more significant figures thanthe measurement with the fewest numbers ofsignificant figureseg 6221 x 52 = 323492 1048774 round off to 32
Examples (continued)Problem Perform the calculation and express your answerto the correct number of sig fig
(a)(124 ndash 945) 02212= 295 02212 = 13336hellip round off to 13
In sequential operations the correct rule must be applied to eachoperation Carry all numbers through the calculation and round at the end
(b) (00045 x 200000) + (2813 x 12)= 90 + 33756 =00090 x 104 + 33756 x 104 = 33846 x 104 round off to 34 x 104
Scientific NotationScientific notation is used to eliminate the potentialambiguity of whether the zeros at the end of a number aresignificant The significant figure information is in thecoefficient and decimal point information is in the exponent
eg 10300 is written 103x104 (3 sig fig)10300 is written 1030x104 (4 sig fig)10300 is written 10300x104 (5 sig fig)Note that in scientific notation a correctly written numberhas a single nonzero digit to the left of the decimal point
ExerciseHow many significant figures are in each of the followingmeasured numbers (a) 4003 (b) 6023 x 1023 (c) 5000
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
PRECISION OF CALCULATIONS
How close are your calculated answers to the known or accepted value
1 Non-zero digits are always significant
2 Any zeros between two significant digits are significant
3 A final zero or trailing zeros after the decimal are significant
4 Zeros that are holding decimal places are NOT significant
Rules for Significant Figures
Significant Figures in Calculations
The result of calculation using measured numbers shouldreflect the precision of the original measurements Thesignificant figures in calculation should follow thefollowing rules
Addition and SubtractionThe answer cannot have more decimal places than themeasurement with the fewest number of decimal placeseg 4 + 145 +124 = 1785 round off to 18
Multiplication and DivisionThe answer cannot have more significant figures thanthe measurement with the fewest numbers ofsignificant figureseg 6221 x 52 = 323492 1048774 round off to 32
Examples (continued)Problem Perform the calculation and express your answerto the correct number of sig fig
(a)(124 ndash 945) 02212= 295 02212 = 13336hellip round off to 13
In sequential operations the correct rule must be applied to eachoperation Carry all numbers through the calculation and round at the end
(b) (00045 x 200000) + (2813 x 12)= 90 + 33756 =00090 x 104 + 33756 x 104 = 33846 x 104 round off to 34 x 104
Scientific NotationScientific notation is used to eliminate the potentialambiguity of whether the zeros at the end of a number aresignificant The significant figure information is in thecoefficient and decimal point information is in the exponent
eg 10300 is written 103x104 (3 sig fig)10300 is written 1030x104 (4 sig fig)10300 is written 10300x104 (5 sig fig)Note that in scientific notation a correctly written numberhas a single nonzero digit to the left of the decimal point
ExerciseHow many significant figures are in each of the followingmeasured numbers (a) 4003 (b) 6023 x 1023 (c) 5000
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
1 Non-zero digits are always significant
2 Any zeros between two significant digits are significant
3 A final zero or trailing zeros after the decimal are significant
4 Zeros that are holding decimal places are NOT significant
Rules for Significant Figures
Significant Figures in Calculations
The result of calculation using measured numbers shouldreflect the precision of the original measurements Thesignificant figures in calculation should follow thefollowing rules
Addition and SubtractionThe answer cannot have more decimal places than themeasurement with the fewest number of decimal placeseg 4 + 145 +124 = 1785 round off to 18
Multiplication and DivisionThe answer cannot have more significant figures thanthe measurement with the fewest numbers ofsignificant figureseg 6221 x 52 = 323492 1048774 round off to 32
Examples (continued)Problem Perform the calculation and express your answerto the correct number of sig fig
(a)(124 ndash 945) 02212= 295 02212 = 13336hellip round off to 13
In sequential operations the correct rule must be applied to eachoperation Carry all numbers through the calculation and round at the end
(b) (00045 x 200000) + (2813 x 12)= 90 + 33756 =00090 x 104 + 33756 x 104 = 33846 x 104 round off to 34 x 104
Scientific NotationScientific notation is used to eliminate the potentialambiguity of whether the zeros at the end of a number aresignificant The significant figure information is in thecoefficient and decimal point information is in the exponent
eg 10300 is written 103x104 (3 sig fig)10300 is written 1030x104 (4 sig fig)10300 is written 10300x104 (5 sig fig)Note that in scientific notation a correctly written numberhas a single nonzero digit to the left of the decimal point
ExerciseHow many significant figures are in each of the followingmeasured numbers (a) 4003 (b) 6023 x 1023 (c) 5000
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
Significant Figures in Calculations
The result of calculation using measured numbers shouldreflect the precision of the original measurements Thesignificant figures in calculation should follow thefollowing rules
Addition and SubtractionThe answer cannot have more decimal places than themeasurement with the fewest number of decimal placeseg 4 + 145 +124 = 1785 round off to 18
Multiplication and DivisionThe answer cannot have more significant figures thanthe measurement with the fewest numbers ofsignificant figureseg 6221 x 52 = 323492 1048774 round off to 32
Examples (continued)Problem Perform the calculation and express your answerto the correct number of sig fig
(a)(124 ndash 945) 02212= 295 02212 = 13336hellip round off to 13
In sequential operations the correct rule must be applied to eachoperation Carry all numbers through the calculation and round at the end
(b) (00045 x 200000) + (2813 x 12)= 90 + 33756 =00090 x 104 + 33756 x 104 = 33846 x 104 round off to 34 x 104
Scientific NotationScientific notation is used to eliminate the potentialambiguity of whether the zeros at the end of a number aresignificant The significant figure information is in thecoefficient and decimal point information is in the exponent
eg 10300 is written 103x104 (3 sig fig)10300 is written 1030x104 (4 sig fig)10300 is written 10300x104 (5 sig fig)Note that in scientific notation a correctly written numberhas a single nonzero digit to the left of the decimal point
ExerciseHow many significant figures are in each of the followingmeasured numbers (a) 4003 (b) 6023 x 1023 (c) 5000
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
Examples (continued)Problem Perform the calculation and express your answerto the correct number of sig fig
(a)(124 ndash 945) 02212= 295 02212 = 13336hellip round off to 13
In sequential operations the correct rule must be applied to eachoperation Carry all numbers through the calculation and round at the end
(b) (00045 x 200000) + (2813 x 12)= 90 + 33756 =00090 x 104 + 33756 x 104 = 33846 x 104 round off to 34 x 104
Scientific NotationScientific notation is used to eliminate the potentialambiguity of whether the zeros at the end of a number aresignificant The significant figure information is in thecoefficient and decimal point information is in the exponent
eg 10300 is written 103x104 (3 sig fig)10300 is written 1030x104 (4 sig fig)10300 is written 10300x104 (5 sig fig)Note that in scientific notation a correctly written numberhas a single nonzero digit to the left of the decimal point
ExerciseHow many significant figures are in each of the followingmeasured numbers (a) 4003 (b) 6023 x 1023 (c) 5000
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
Scientific NotationScientific notation is used to eliminate the potentialambiguity of whether the zeros at the end of a number aresignificant The significant figure information is in thecoefficient and decimal point information is in the exponent
eg 10300 is written 103x104 (3 sig fig)10300 is written 1030x104 (4 sig fig)10300 is written 10300x104 (5 sig fig)Note that in scientific notation a correctly written numberhas a single nonzero digit to the left of the decimal point
ExerciseHow many significant figures are in each of the followingmeasured numbers (a) 4003 (b) 6023 x 1023 (c) 5000
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
Accuracy amp Precision
bull Accuracy ndash How close you are to the
actual value ndash Depends on the person
measuring ndash Calculated by the formula
Error = (YV ndash AV) x 100 divide AVWhere YV is YOUR measured Value amp AV is
the Accepted Value
bull Precision ndash How close the
measurements are to each other
ndash Depends on the measuring tool
ndash Determined by the number of significant digits
