Significant Figures (How Exact is Exact ?)

Part 1: significant figures and Certainty

The type of ruler used DOES make a difference ! As you saw from our short activity, the graduations on the measuring instrument used affects the certainty of our measurement When you used the ruler with a scale of 0.1, your measurement was much more accurate than when you were using the ruler that was scaled by 10

The system of significant figures was devised so we can determine how reliable a measurement was. Definition Significant Figures refers to those digits in a number that are directly read in a measurement plus one uncertain digit

REMEMBER If the reading is 23.57 cm, this means Whenever you make any measurement, always include all the digits that you can directly read PLUS one uncertain digit. REMEMBER If the reading is 23.57 cm, this means that 2,3 and 5 are read directly on the ruler, while 7 is estimated. The number 23.57 has four significant figures.

Part 2 : Counting Significant Figures One easy method of finding the number of significant figures in a number is through the Pacific-Atlantic Rule

Pacific Rule P stands for the PRESENCE of a decimal point P also stands for PACIFIC Use this rule when you see that a number has a decimal point (decimal pt. is PRESENT) RULE: If a number has a decimal point, count all digits (including zero) from the first nonzero digit at the LEFT

Think OF How THE Pacific Ocean HITS THE US COAST… From LEFT

Examples Answer: 5 sig. figures Answer: 4 sig. figures Count from first nonzero digit at left (4) to the last digit at right (6) Answer: 5 sig. figures 1) 435.06 1 2 3 4 5 2) 0.005 320 Count from first nonzero digit at left (5) to the last digit at right (0) Answer: 4 sig. figures 1 2 34

Your Turn !! Find the no. of significant figures in: (1) 57,376. 205 (1) 57,376. 205 1 2 3 4 5 6 7 8 Answer: 8 sig. figs. (2) 4,005. 900 1 2 3 4 5 6 7 Answer: 7 sig. figs.

More Questions… (3) 0. 000 240 1 2 3 Answer: 3 sig. figs. (4) 0.088 000 1 2 3 4 5 Answer: 5 sig. figs

Atlantic Rule A stands for the ABSENCE of a decimal point A also stands for ATLANTIC Use this rule when the number has no decimal point ( decimal point is ABSENT) RULE : If a number has no decimal point, count all digits starting from the first nonzero digit at the RIGHT

THINK OF HOW THE ATLANTIC OCEAN HITS THE US COAST… From RIGHT

Examples Count from first nonzero digit at right (9) to the leftmost digit (8) Answer: 3 significant. figures 1) 879,000 3 2 1 Count from first nonzero digit starting at right (3) to the leftmost digit (7) Answer: 4 significant figures 762,300 4 3 2 1

NUMBERS IN SCIENTIFIC NOTATION How many significant figures are there in 6.02 x 1023 RULE: For numbers in scientific notation, all the numbers in the decimal part are significant… ANSWER: 3 significant figures

How many significant figures are there in each of these numbers? 1) 3300 2) 0.000 017 6 3) 110.50 4) 210.0 5) 0.90 6) 8.70 X 106 7) 45000. 8) 45000

Check your answers: Now, wasn’t that easy!!! 1) 2 2) 3 3) 5 4) 4 5) 2 6) 3 7) 5 8) 2 Now, wasn’t that easy!!!

PART 3: ROUNDING OFF Let’s Review !

Don’t lose it… The number of significant figures in a measurement indicates the reliability of that measure When you do a mathematical operation (add, subtract, multiply….) on that number be sure to ROUND OFF properly so the answer reflects the reliability of the original measurement

Say what? ROUND OFF !!! One skill you learned in grade school is how to round off a number Can you still remember how you solved such problems…

PROBLEM: Round off the number 64.0384 to the underlined place RULE: If the number following the underlined decimal place is 5 or greater than 5 , add 1 to the underlined digit. Drop all the numbers after it. SOLUTION: You must round up to 3 The digit following 3 is 8 8 is greater than 5, so add 1 to 3 and the underlined digit becomes 4 Drop all numbers after 4 ANSWER: 64.04

PROBLEM: Round off the number 8.3722 to the underlined place RULE : If the number following the underlined decimal place is less than 5, keep the underlined digit. Drop all the numbers after it. SOLUTION: You must round up to 7 The digit following 7 is 2 2 is less than 5 so keep the underlined digit (7) Drop all the numbers after ANSWER: 8.37

Now it’s YOUR turn! Round each number to the underlined digit: (1) 18.421 Think …must round up to 4 … 4 is followed by 2 Since 2 is less than 5, …must keep 4 Drop numbers after 4 Answer: 18.4

Next number (2) 7.2619 Think …must round up to 2 … 2 is followed by 6 (2) 7.2619 Think …must round up to 2 … 2 is followed by 6 Since 6 is greater than 5, …must add 1 to 2 So 2 becomes 3 Drop all numbers after 3 Answer: 7.3

And one more… (3) 8234.67 Think …must round up to 2 (3) 8234.67 Think …must round up to 2 … 2 is followed by 3 Since 3 is less than 5, …must keep 2 Drop all numbers after 2, but to indicate place value keep two zeros Answer: 8200

Don’t forget to connect ROUNDING OFF with SIGNIFICANT FIGURES Problem 1: Round off 56.18754 to three significant figures A decimal point is Present, so follow Pacific rule (count from the first nonzero digit at the left) 56.18754 1 2 3 4 56 7 You must round up to 1 ( 56.18754) ANSWER: 56.2

Problem 2: Round off 0.006392 to two significant figures A decimal point is Present, so follow Pacific rule (count from the first nonzero digit at the left) 0.06392 1 2 3 4 You must round up to 3 ( 0.06392) ANSWER: 0.064

Practice Problems: Round these measurements to three significant figures: 85.779 kg. 0.007 231 56 m 457.4932 cm 256,769 mg 5.9256 m 405,670,000 mL

Did you get these answers? 85.8 kg. 0.007 23 m 457 cm 257,000 mg 5.93 m 406,000,000 mL

CALCULATIONS INVOLVING SIGNIFICANT FIGURES PART 4 CALCULATIONS INVOLVING SIGNIFICANT FIGURES

MULTIPLICATION and DIVISION RULE: Round off the answer so that it has the same number of significant figures as the measurement with the fewest significant figures 5.3 x 1.54 x 3.21 = 26.20002 (2 sf) (3 sf) (3 sf) You must round off 26.20002 so that it has 2 sig. figures . ANSWER: 26

Now give these questions a try: 8.2 cm x 6.35 cm = 4,170 m2 ÷ 5.0 m = 9.3 mm x 0.025 mm x 200.3 mm =

Did you get the right answers? (1) 52.07 becomes 52 Answer: 52 cm2 (2) 834 becomes 830 Answer: 830 m (3) 46.56975 becomes 47 Answer: 47 mm3

ADDITION AND SUBTRACTION RULE: Round off the answer so that it has the same number of decimal places as the measurement with the least number of decimal places. 23.5 (one decimal place) + 2. 1 (one decimal place) 7.26 (two decimal places) 32 .86 Round off the answer to one decimal place ANSWER: 32.9

How about these problems? Find the sum of 6.730 cm 1.1 cm 3.45 cm Problem 2: Get the difference of 68.731 m - 14.22 m

So how well did you do? Problem 1: 6.730 cm (3 decimal places) Answer must be rounded off to 1 decimal place Answer: 11.3 cm

Problem 2: 68.731 m (3 decimal places) - 14.22 m (2 decimal place) Answer must be rounded off to 2 decimal places Answer: 54.51 m

So good luck on sig figs and let’s keep trying !!!