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Section 1.5—Significant Digits

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Section 1.5 A Counting significant digits

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Taking & Using Measurements You learned in Section 1.3 how to take careful measurements Most of the time, you will need to complete calculations with those measurements to understand your results 1.00 g 3.0 mL = 0.3333333333333333333 g/mL If the actual measurements were only taken to 1 or 2 decimal places… how can the answer be known to and infinite number of decimal places? It can’t!

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Significant Digits A significant digit is anything that you measured in the lab—it has physical meaning The real purpose of “significant digits” is to know how many places to record in an answer from a calculation But before we can do this, we need to learn how to count significant digits in a measurement

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Significant Digit Rules 1 All measured numbers are significant 2 All non-zero numbers are significant 3 Middle zeros are always significant 4 Trailing zeros are significant if there’s a decimal place 5 Leading zeros are never significant

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All the fuss about zeros 102.5 g Middle zeros are important…we know that’s a zero (as opposed to being 112.5)…it was measured to be a zero 125.0 mL The convention is that if there are ending zeros with a decimal place, the zeros were measured and it’s indicating how precise the measurement was. 125.0 is between 124.9 and 125.1 125 is between 124 and 126 0.0127 m The leading zeros will dissapear if the units are changed without affecting the physical meaning or precision…therefore they are not significant 0.0127 m is the same as 127 mm

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Sum it up into 2 Rules 1 If there is no decimal point in the number, count from the first non-zero number to the last non-zero number 2 If there is a decimal point (anywhere in the number), count from the first non-zero number to the very end The 4 earlier rules can be summed up into 2 general rules

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Examples of Summary Rule 1 Example: Count the number of significant figures in each number 124 20570 200 150 1 If there is no decimal point in the number, count from the first non-zero number to the last non-zero number

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Examples of Summary Rule 1 Example: Count the number of significant figures in each number 124 20570 200 150 1 If there is no decimal point in the number, count from the first non-zero number to the last non-zero number 3 significant digits 4 significant digits 1 significant digit 2 significant digits

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Examples of Summary Rule 2 Example: Count the number of significant figures in each number 0.00240 240. 370.0 0.02020 2 If there is a decimal point (anywhere in the number), count from the first non-zero number to the very end

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Examples of Summary Rule 2 Example: Count the number of significant figures in each number 0.00240 240. 370.0 0.02020 3 significant digits 4 significant digits 2 If there is a decimal point (anywhere in the number), count from the first non-zero number to the very end

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Importance of Trailing Zeros Just because the zero isn’t “significant” doesn’t mean it’s not important and you don’t have to write it! “250 m” is not the same thing as “25 m” just because the zero isn’t significant The zero not being significant just tells us that it’s a broader range…the real value of “250 m” is between 240 m & 260 m. “250. m” with the zero being significant tells us the range is from 249 m to 251 m

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Let’s Practice Example: Count the number of significant figures in each number 1020 m 0.00205 g 100.0 m 10240 mL 10.320 g

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Let’s Practice Example: Count the number of significant figures in each number 1020 m 0.00205 g 100.0 m 10240 mL 10.320 g 3 significant digits 4 significant digits 5 significant digits

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Section 1.5 B Calculations with significant digits

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Performing Calculations with Sig Digs 1 Addition & Subtraction: Answer has least number of decimal places as appears in the problem 2 Multiplication & Division: Answer has least number of significant figures as appears in the problem When recording a calculated answer, you can only be as precise as your least precise measurement Always complete the calculations first, and then round at the end!

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Addition & Subtraction Example #1 Example: Compute & write the answer with the correct number of sig digs 15.502 g + 1.25 g This answer assumes the missing digit in the problem is a zero…but we really don’t have any idea what it is 1 Addition & Subtraction: Answer has least number of decimal places as appears in the problem 16.752 g

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Addition & Subtraction Example #1 Example: Compute & write the answer with the correct number of sig digs 15.502 g + 1.25 g 1 Addition & Subtraction: Answer has least number of decimal places as appears in the problem 16.752 g 16.75 g 3 decimal places 2 decimal places Lowest is “2” Answer is rounded to 2 decimal places

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Addition & Subtraction Example #2 Example: Compute & write the answer with the correct number of sig digs 10.25 mL - 2.242 mL This answer assumes the missing digit in the problem is a zero…but we really don’t have any idea what it is 1 Addition & Subtraction: Answer has least number of decimal places as appears in the problem 8.008 mL

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Addition & Subtraction Example #2 Example: Compute & write the answer with the correct number of sig digs 10.25 mL - 2.242 mL 1 Addition & Subtraction: Answer has least number of decimal places as appears in the problem 8.01 mL 2 decimal places 3 decimal places Lowest is “2” Answer is rounded to 2 decimal places 8.008 mL

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Multiplication & Division Example #1 Example: Compute & write the answer with the correct number of sig digs 10.25 g 2.7 mL = 3.796296296 g/mL 2 Multiplication & Division: Answer has least number of significant figures as appears in the problem

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Multiplication & Division Example #1 Example: Compute & write the answer with the correct number of sig digs 3.8 g/mL 4 significant digits 2 significant digits Lowest is “2” Answer is rounded to 2 sig digs 10.25 g 2.7 mL = 3.796296296 g/mL 2 Multiplication & Division: Answer has least number of significant figures as appears in the problem

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Multiplication & Division Example #2 Example: Compute & write the answer with the correct number of sig digs 1.704 g/mL 2.75 mL 4.686 g 2 Multiplication & Division: Answer has least number of significant figures as appears in the problem

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Multiplication & Division Example #2 Example: Compute & write the answer with the correct number of sig digs 4.69 g 4 significant dig 3 significant dig Lowest is “3” Answer is rounded to 3 significant digits 2 Multiplication & Division: Answer has least number of significant figures as appears in the problem 1.704 g/mL 2.75 mL 4.686 g

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Let’s Practice #1 Example: Compute & write the answer with the correct number of sig digs 0.045 g + 1.2 g

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Let’s Practice #1 Example: Compute & write the answer with the correct number of sig digs 1.2 g 3 decimal places 1 decimal place Lowest is “1” Answer is rounded to 1 decimal place 1.245 g Addition & Subtraction use number of decimal places! 0.045 g + 1.2 g

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Let’s Practice #2 Example: Compute & write the answer with the correct number of sig digs 2.5 g/mL 23.5 mL

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Let’s Practice #2 Example: Compute & write the answer with the correct number of sig digs 59 g 2 significant dig 3 significant dig Lowest is “2” Answer is rounded to 2 significant digits 2.5 g/mL 23.5 mL 58.75 g Multiplication & Division use number of significant digits!

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Let’s Practice #3 Example: Compute & write the answer with the correct number of sig digs 1.000 g 2.34 mL

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Let’s Practice #3 Example: Compute & write the answer with the correct number of sig digs 0.427 g/mL 4 significant digits 3 significant digits Lowest is “3” Answer is rounded to 3 sig digs 1.000 g 2.34 mL = 0.42735 g/mL Multiplication & Division use number of significant digits!

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