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INTRODUCTION TO SIGNIFICANT FIGURES SIGNIFICANT FIGURES Scientist use significant figures to determine how precise a measurement is Significant digits in a measurement include all of the known digits plus one estimated digit FOR EXAMPLE… Look at the ruler below Each line is 0.1cm You can read that the arrow is on 13.3 cm However, using significant figures, you must estimate the next digit That would give you 13.30 cm UNCERTAINTY IN MEASUREMENT Measurements are made one place beyond where the measuring device is marked. For example, a ruler marked to the nearest tenth of a centimeter can be read to the hundredths place by estimating how far between the lines the measurement falls. Device is marked to the ONES place, so read to the TENTHS place Measurement: 6.2 cm the 2 in the tenths place is ESTIMATED --------------------------- cm .1 .2 .3 .4 .5 .6 .7 .8 .9 Device is marked to the TENTHS place, so read to the HUNDREDTHS place Measurement: 0.48 cm the 8 in the hundredths place is ESTIMATED ------------------------------------- cm 10 20 30 40 50 60 70 80 90 Device is marked to the TENS place, so read to the ONES place Measurement: 67 cm the 7 in the ones place is ESTIMATED LET’S TRY THIS ONE Look at the ruler below What can you read before you estimate? 12.8 cm Now estimate the next digit… 12.85 cm THE SAME RULES APPLY WITH ALL INSTRUMENTS The same rules apply Read to the last digit that you know Estimate the final digit LET’S TRY GRADUATED CYLINDERS Look at the graduated cylinder below What can you read with confidence? 56 ml Now estimate the last digit 56.0 ml ONE MORE GRADUATED CYLINDER Look at the cylinder below… What is the measurement? 53.5 ml RULES FOR SIGNIFICANT FIGURES RULE #1 All non zero digits are ALWAYS significant How many significant digits are in the following numbers? •274 •3 Significant Figures •25.632 •5 Significant Digits •8.987 •4 Significant Figures RULE #2 All zeros between significant digits are ALWAYS significant How many significant digits are in the following numbers? 504 3 Significant Figures 60002 5 Significant Digits 9.077 4 Significant Figures RULE #3 All FINAL zeros to the right of the decimal ARE significant How many significant digits are in the following numbers? 32.0 3 Significant Figures 19.000 5 Significant Digits 105.0020 7 Significant Figures RULE #4 All zeros that act as place holders are NOT significant Another way to say this is: zeros are only significant if they are between significant digits OR are the very final thing at the end of a decimal FOR EXAMPLE How many significant digits are in the following numbers? 0.0002 6.02 x 1023 100.000 150000 800 1 Significant Digit 3 Significant Digits 6 Significant Digits 2 Significant Digits 1 Significant Digit RULE #5 All counting numbers and constants have an infinite number of significant digits For example: 1 hour = 60 minutes 12 inches = 1 foot 24 hours = 1 day HOW MANY SIGNIFICANT DIGITS ARE IN THE FOLLOWING NUMBERS? 0.0073 100.020 2500 7.90 x 10-3 670.0 0.00001 18.84 2 Significant Digits 6 Significant Digits 2 Significant Digits 3 Significant Digits 4 Significant Digits 1 Significant Digit 4 Significant Digits RULES ROUNDING SIGNIFICANT DIGITS RULE #1 If the digit to the immediate right of the last significant digit is less that 5, do not round up the last significant digit. For example, let’s say you have the number 43.82 and you want 3 significant digits The last number that you want is the 8 – 43.82 The number to the right of the 8 is a 2 Therefore, you would not round up & the number would be 43.8 ROUNDING RULE #2 If the digit to the immediate right of the last significant digit is greater that a 5, you round up the last significant figure Let’s say you have the number 234.87 and you want 4 significant digits 234.87 – The last number you want is the 8 and the number to the right is a 7 Therefore, you would round up & get 234.9 ROUNDING RULE #3 If the number to the immediate right of the last significant is a 5, and that 5 is followed by a non zero digit, round up 78.657 (you want 3 significant digits) The number you want is the 6 The 6 is followed by a 5 and the 5 is followed by a non zero number Therefore, you round up 78.7 ROUNDING RULE #4 If the number to the immediate right of the last significant is a 5, and that 5 is followed by a zero, you look at the last significant digit and make it even. 2.5350 (want 3 significant digits) The number to the right of the digit you want is a 5 followed by a 0 Therefore you want the final digit to be even 2.54 SAY YOU HAVE THIS NUMBER 2.5250 (want 3 significant digits) The number to the right of the digit you want is a 5 followed by a 0 Therefore you want the final digit to be even and it already is 2.52 LET’S TRY THESE EXAMPLES… 200.99 (want 3 SF) 201 18.22 (want 2 SF) 18 135.50 (want 3 SF) 136 0.00299 (want 1 SF) 0.003 98.59 (want 2 SF) 99 SCIENTIFIC NOTATION Scientific notation is used to express very large or very small numbers I consists of a number between 1 & 10 followed by x 10 to an exponent The exponent can be determined by the number of decimal places you have to move to get only 1 number in front of the decimal LARGE NUMBERS If the number you start with is greater than 1, the exponent will be positive Write the number 39923 in scientific notation First move the decimal until 1 number is in front – 3.9923 Now at x 10 – 3.9923 x 10 Now count the number of decimal places that you moved (4) Since the number you started with was greater than 1, the exponent will be positive 3.9923 x 10 4 SMALL NUMBERS If the number you start with is less than 1, the exponent will be negative Write the number 0.0052 in scientific notation First move the decimal until 1 number is in front – 5.2 Now at x 10 – 5.2 x 10 Now count the number of decimal places that you moved (3) Since the number you started with was less than 1, the exponent will be negative 5.2 x 10 -3 SCIENTIFIC NOTATION EXAMPLES Place the following numbers in scientific notation: 99.343 9.9343 x 101 4000.1 4.0001 x 103 0.000375 3.75 x 10-4 0.0234 2.34 x 10-2 94577.1 9.45771 x 104 GOING FROM SCIENTIFIC NOTATION TO ORDINARY NOTATION You start with the number and move the decimal the same number of spaces as the exponent. If the exponent is positive, the number will be greater than 1 If the exponent is negative, the number will be less than 1 GOING TO ORDINARY NOTATION EXAMPLES Place the following numbers in ordinary notation: 3 x 106 6.26x 109 5 x 10-4 8.45 x 10-7 2.25 x 103 3000000 6260000000 0.0005 0.000000845 2250 SIG FIGS: ADDITION AND SUBTRACTION The answer can only be as precise as the least precise of the measurements to be added or subtracted. The answer must be rounded to the place of precision of the least precise of the measurements. 33.5 cm + 7.88 cm + 0.977 cm = 42.357 cm ROUNDED = 42.4 cm tenths hundredths thousandths Since TENTHS goes out the least far, round the answer to the nearest TENTH SIG FIGS: MULTIPLICATION AND DIVISION The answer can contain only as many significant figures as the measurement with the least number of significant figures. 67.23 cm X 9.22 cm = 619.584 cm2 ROUNDED = 620. cm2 4 sig figs 3 sig figs Rounding the answer to 3 sig figs, you need to put the decimal point to show that the 0 is a sig fig. Accuracy and Precision in Measurements Accuracy: how close a measurement is to the accepted value. Precision: how close a series of measurements are to one another or how far out a measurement is taken. A measurement can have high precision, but not be as accurate as a less precise one. 30