Section 1.5—Significant Digits

Counting significant digits Section 1.5 A Counting significant digits

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 how can the answer be known to and infinite number of decimal places? If the actual measurements were only taken to 1 or 2 decimal places… It can’t!

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

Significant Digit Rules 1 All measured numbers are significant-if it is not measured and is instead counted or known then we say that they have infinite significant figures. 2 All non-zero numbers are significant 3 Middle zeros are always significant Trailing zeros are significant if there’s a decimal place 4 5 Leading zeros are never significant

All the fuss about zeros Middle zeros are important…we know that’s a zero (as opposed to being 112.5)…it was measured to be a zero 102.5 g 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 mL 125.0 is between 124.9 and 125.1 125 is between 124 and 126 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 0.0127 m is the same as 12.7 mm

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

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

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

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

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

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

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

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

Calculations with significant digits Section 1.5 B Calculations with significant digits

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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