Significant Figures Name: Per: aka: Significant Digits Certain Digits: Numbers you are sure of. Uncertain Digits: Last number is an estimate. Certain Digits & Uncertain digits Measurement made with instruments have some known values, and some uncertain (guesses). In This Packet Rules for determining the number of significant digits in a measurement. Adding & Subtracting with significant digits. Multiplying & Dividing with significant digits. Rules for Determining Significant Digits 1) All non zero numbers are significant Eg. 982.15 kgrams contains 5 significant figures 2) Leading zeros are not significant Eg..00472 contains 3 significant digits. The zeros in the front are important but act as placeholders. 3) Captive zero s are significant regardless of how many captive zero s there are and where the decimal is located. Eg. 10.3 contains 3 significant digits. Eg..9008 contains 4 significant figures. 4) Trailing zeros are not significant unless a decimal is present. If no decimal is present the trailing zeros are placeholders. Eg. 100 contains 1 significant figure Eg. 100.0 contains 4 significant figures. placeholder) Eg. 1900 contains 2 significant figures Eg. 100. contains 3 significant figures. Eg..0800 contains 3 significant figures (the first zero is a Eg. 1900.0 contains 5 significant figures Oh but there is an easier way! Turn the page for the Oceans Technique 1
The Oceans Technique Determining Significant Figures (Digits) Draw a mental arrow over the number. The first non-zero number and all numbers after it are significant. Question: Do we draw the arrow from left to right, or right to left? Answer: It depends on whether a decimal point is present or absent in the number. If a decimal is resent, start at the acific ocean draw the arrow from left to right. Example How many sig figs are there in 32.500? Since the decimal is present, we start from the pacific and draw the line from left to right. 32.500 The first non-zero number is 3, so it is significant and all numbers after it are also. There are 5 sig figs! If a decimal is bsent, start at the tlantic ocean draw the arrow from right to left. Example How many sig figs are there in 32500? Since the decimal is absent, we start from the atlantic and draw the line from right to left. 32500 The first non-zero number is 5, so it is significant and all numbers after it are also. There are 3 sig figs! 2
Counting Sig Figs Determine the number of significant figures in each number below. Use the arrow Oceans Technique. Circle the decimal if it is present. Draw an arrow through the number from the appropriate ocean. Underline the first non-zero number and all after it (in the direction of the arrow). Write the number of underlined digits this is the number of sig figs. How many significant figures in the following numbers? Show your work! Number of Sig Figs? 18.00 22000 49,001 49,001. 2.00 600 3910.010 54500.0004 Calculate the number of sig figs in the following numbers, but DO NOT show your work. Mentally draw your arrows and count sig figs. Number How many Sig Figs? Number How many Sig Figs?.00001.1000 22,005 22,000 162.003 4,500. 89.0.002.1004 30 18000 22.00 3
Adding and Subtracting Measurements w/ Sig Figs Arrange the numbers in column form. Like this. Do the math like normal. The final answer can contain only as many decimal places as found in the measurement with the fewest number of decimal places. In this example, the first measurement (4.8) has only one decimal place, so the answer is rounded to only one decimal place. Answer = 0.8 Do the following calculations. Show your work. Circle your final answer. 1. 790 +.195 6. 0.003 + 0.100 + 0.002 11. 3.2 0.1-1.54 2. 5.1123 + 6.2 + 64.1 7. 1.1 + 1.2 +2 12. 5.2234 2.2-1.2 3. 4565.2331 + 1.111 + 42.902 8. 1.1 + 1.2 + 2.0 13. 0.002-0.002 4. 39 + 1.234 + 2.234 9. 49.00 + 1 + 1.00 14. 80 0.1 5. 0.003 + 0.1 + 0.002 10. 49.00 + 1.00 + 1.00 15. 80.0 0.1 4
Review Multiplying & Dividing w/ Sig Figs How many significant figures are there in each of the following measurements? Write your answer on the line. Measurement How many Measurement How many Measurement How many Sig Figs? Sig Figs? Sig Figs? 40.20 m 4020 m.0004 cm 0.001 g 32 L 6,000,000 Kg Multiplying & Dividing Measurements An answer cannot be more precise than any one of the measurements used in the problem. The Rule: ROUND OFF! Do the math, then round the answer so that the number of significant figures is equal to the least number of significant figures found in any one measurement in the problem. Example: We measure a room. I measure the length very carefully, estimating the last digit, and write Length = 8.542 m George measures the width very quickly and lazily and writes Width = 6 m The formula for the area of a rectangle is A = L x W so we use our calculator: A = 8.542m x 6 m The answer on the calculator is 51.252 So you would be tempted to write 51.252 m² YOUR ANSWER WOULD BE WRONG! Since George s measurement has only 1 sig fig, you must round the answer to only 1 sig fig. The correct answer is 50 m². Examples without units (we are assuming all numbers are measurements): 4.1 x 2 = 8.2 but we must round to 1 sig fig so the correct answer is 8 4.1 x 20 = 82 but we must round to 1 sig fig so the correct answer is 80 4.1 x 20. = 82 2 sig figs are correct (use decimal in answer) answer is 82. 4.1 x 200 = 820 but we must round to 1 sig fig so the correct answer is 800 4.100 x 200.0 = 820 but we must round to 4 sig figs so the correct answer is 820.0 5 Note that the last 2 examples have the same answer if you don t apply the sig. fig. rules. The reason we do this rounding is to indicate to our audience how accurate our measurements were.
Multiplying & Dividing w/ Sig Figs Practice The Rule: ROUND OFF! Do the math, then round the answer so that the number of significant figures is equal to the least number of significant figures found in any one measurement in the problem. Apply Sig Fig rules to solve the following problems. Although there are no units, assume all numbers are measurements. You may use a calculator. # Problem Final Answer # Problem Final Answer 1. 2.2 x 1 = 6..001 x.015 = 2. 34.5 x 20 = 7. 9.000 x 1.000 = 3. 12 / 2.2 = 8. 12.003 x 2.0006 = 4..02 / 3.5 = 9. 49.0 / 7.000 = 5. 100.0 x 2.000 = 10. 30 x 6 = 11. To find the area of a room, one student measures the width. They write W = 6.2 m. Another student measures length. They write L = 8.540 m. Calculate the area (A=LxW) and write your answer on the line Did you include units? Did you round your answer using sig fig rules? Note: When multiplying or dividing several measurements, you do not round until the final answer. Only round the final answer! # Problem Final Answer 12. 1.5 / 33.0 = 13. 1.5 / 33.0 x 2.0 = 14..001 x.050 x.333 = 15..001 x.050 x.333 x 90.5 = 16. Three students wish to calculate the total combined volume of 2 cardboard boxes. They know the formula is V=LxWxH. Each students measures a different dimension of Box A. Their measurements are as follows: L = 40.0 cm What is the volume of the box? W = 60.0 cm The volume of Box B is known to be 1,200 cm³, what is the combined H = 50.20 cm volume of the two boxes added together? 17. In the last problem, did you redo all the math rounding ONLY the final answer? 6