CHEM 105 & 106 UNIT ONE, LECTURE THREE 1 YESTERDAY WHEN WE LEFT OFF WE WERE TALKING ABOUT CHANGE AND OF COURSE ONE OF THE

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1 CHEM 105 & 106 UNIT ONE, LECTURE THREE 1 CHM 105/106 Program 3: Unit 1 Lecture 3 YESTERDAY WHEN WE LEFT OFF WE WERE TALKING ABOUT CHANGE AND OF COURSE ONE OF THE WAYS THAT WE DETERMINE THAT CHANGE, WHETHER IT BE PHYSICAL OR CHEMICAL, CHANGE IS OCCURRING IS TO MAKE SOME OBSERVATIONS OR MEASUREMENTS AND SO WE HAVE TWO WAYS OF MEASURING CHANGE, ONE WHICH WE REFER TO AS QUALITATIVE AND OF COURSE, QUALITATIVE IS SOMETHING THAT WE TALKED ABOUT YESTERDAY AS WE LOOKED AT SOME OF THE CHEMICAL REACTIONS IN THE WAY THAT WE DETERMINED THE FACT THAT THEY OCCURRED WAS QUALITATIVE INDICATIONS, FOR INSTANCE THE FORMATION OF A COLORED GAS FROM TWO UNCOLORED GASSES, THE FORMATION OF A SOLID WHEN WE COMBINED TWO SOLUTIONS, THE FORMATION OF A GAS FROM REACTING A SOLUTION AND A SOLID. WE WOULD REFER TO THOSE MEASUREMENTS AS QUALITATIVE SO SUCH THINGS AS COLOR, CHANGE IN PHYSICAL STATE ETC..WOULD THEN BE QUALITATIVE MEASUREMENTS, BUT MORE CASES WE HAVE TO DO THE MEASUREMENTS QUANTITATIVELY TO TELL WHAT CHANGES ARE OCCURRING, SO THIS IS WHAT WE ARE GOING TO LOOK AT QUICKLY HERE. THE UNITS THAT WE USE IN SCIENCE TO MAKE QUANTITATIVE MEASUREMENTS, THE FIVE THINGS THAT WE TYPICALLY ARE INTERESTED IN MEASURING ARE THE MASS OF A CHUNK OF MATTER, ITS VOLUME, AND YESTERDAY WE TALKED ABOUT DENSITY AS BEING THEN MASS OF MATTER DIVIDED BY ITS VOLUME, SO THOSE ARE TWO PROPERTIES THAT WE SHOULD BE ABLE TO MEASURE, MASS AND VOLUME. LENGTH, WHEN I SAY LENGTH I AM TAKING ABOUT ALL DIMENSIONS SO WE MEAN, LENGTH AND WIDTH AND HEIGHT SO JUST WHATEVER MEASURED DIMENSIONS THAT WE COULD MAKE IN THAT TYPE OF MEASUREMENT. TEMPERATURE BECAUSE AS I INDICATED IN A PREVIOUS LECTURE, THAT THE CHANGE IN MATTER, WHETHER PHYSICAL OR CHEMICAL, ALWAYS INVOLVES ENERGY. OF COURSE, ONE OF THE WAYS WE MEASURE WHETHER THAT ENERGY IS FLOWING OUT OR ENERGY IS FLOWING IN IS TO MEASURE THE TEMPERATURE OF A MATERIAL SO MEASURING TEMPERATURE, SO THEN OF COURSE TIME, IF WE ARE TALKING ABOUT FOR INSTANCE LATER ON WE WILL TALK ON A CHAPTER OF WHAT WE CALL THE RATES OF CHEMICAL REACTIONS AND THAT IS GOING TO INVOLVE TIME, HOW FAST IS SOMETHING PRODUCED, MOLES PER SECOND, MOLES PER LITER PER MINUTE, SO TIME IS A MEASURED QUANTITY. NOW IN THE METRIC SYSTEM THE SCIENTIFIC SYSTEM OF COURSE,

2 CHEM 105 & 106 UNIT ONE, LECTURE THREE 2 THESE HAVE THEN CERTAIN UNITS THAT WE ASSIGN TO THEM. THE BASE UNIT FOR MASS IS THE GRAM AND WE USE A LOWER G FOR MASS, FOR VOLUME THE BASIC UNIT THAT WE USE IS THE LITER AND THE SYMBOL THAT WE USE IS THE UPPER CASE L, FOR LENGTH OR DIMENSIONS THE BASE UNIT IS THE METER AND WE USE A LOWER CASE M FOR THAT, FOR TEMPERATURE MEASUREMENT THE STANDARD TEMPERATURE SCALE THAT WE USE MOST OF THE TIME IN MEASURING TEMPERATURE IS THE CELSIUS TEMPERATURE AND WE USE A CAPITAL C WITH A LITTLE SUPERSCRIPT KNOT TO INDICATE DEGREES CELSIUS. TIME WE USE S LOWER CASE S WHICH STANDS FOR SECOND, O.K.? NOW IN SCIENCE WE OFTEN ARE WORKING WITH EITHER LARGE QUANTITIES OR VERY SMALL QUANTITIES OF MATERIAL AND SO FREQUENTLY WE THEN HAVE TO EXPRESS LARGE AMOUNTS OR SMALL AMOUNTS OF THESE MEASURED QUANTITIES AND THAT IS WHAT MAKES THE METRIC SYSTEM SO NICE. THOSE BASIC UNITS THERE ARE THE UNITS THAT COVER ANY TIME WE ARE TALKING ABOUT MASS LARGE OR SMALL GRAM IS THE BASIC UNIT WE CAN THEN USE SOME PREFIX TO TALK ABOUT BIG AND SMALL, WHICH WE USE POWERS OF TEN THEN AS MULTIPLES OF THESE BASIC UNITS, SO EVERYTHING IS THE BASIC UNIT AND THEN ANY OTHER MEASUREMENT LARGE OR SMALL IS MERELY MULTIPLIED BY TEN A CERTAIN NUMBER OF TIMES OR DIVIDED BY TEN A CERTAIN NUMBER OF TIMES THERE ARE SOME PREFIXES THAT WE SHOULD BE FAMILIAR WITH THAT WE WILL NEED TO USE THROUGHOUT THE COURSE AND THESE ARE GIVEN TO YOU ON TABLE 1.6 IN THE TEXT AND LET'S JUST LOOK AT THOSE QUICKLY. THE BASIC UNIT OF COURSE IS ONE SO WE ARE NOT SHOWING A ONE ON HERE, THAT WOULD BE THE GRAMS, THAT WOULD BE THE LITERS, THAT WOULD BE THE METERS, IF WE WANTED TO TALK ABOUT TEN UNITS OF SOMETHING LIKE TEN GRAMS WE USE THE PREFIX DECA, SO DECAGRAM WOULD MEAN THEN TEN GRAMS, OR IF WE WANTED A THOUSAND GRAMS THE PREFIX FOR ONE THOUSAND IS KILO AND THE SYMBOL THAT WE USE THERE IS A LOWER CASE K AND FOR DECA (DA), OR IF WE NEED VERY LARGE AMOUNTS IF WE ARE TALKING ABOUT FOR INSTANCE THE ENERGY RELEASED IN AN ATOMIC WEAPON EXPLOSION IT IS USUALLY EXPRESSED IN TERM OF MEGATONS WHICH MEANS THEN MILLIONS OF TONS EQUIVALENT DYNAMITE SO IF WE HAD A FIFTEEN MEGATON BOMB IT WOULD MEAN THAT THE BOMB WOULD PRODUCE THE SAME AMOUNT OF EXPLOSIVE FORCE AS FIFTEEN TONS OF DYNAMITE, THAT IS WHAT WE ARE TAKING ABOUT SO MEGA OF COURSE OBVIOUSLY A LARGE VALUE. IF WE TALK ABOUT ELECTRIC PRODUCTION AT POWER PLANTS, WE TALK ABOUT POWER

3 CHEM 105 & 106 UNIT ONE, LECTURE THREE 3 PLANTS IN TERMS OF MEGAWATTS OF ENERGY THAT THEY CAN PRODUCE. SO MEGA IS A LARGE UNIT THERE ARE BIGGER ONES THEN THIS OBVIOUSLY, BUT THESE ARE THE ONES THAT I EXPECT THAT YOU SHOULD BECOME FAMILIAR WITH. BELOW THE UNIT ONE THEN WE HAVE SMALLER DIVISIONS AND THE FIRST IS DECI WHICH IS ONE TENTH OF SOMETHING OR PUTTING IT INTO A DECIMAL REPRESENTATION (.1) SO A DECIMETER WOULD BE ONE TENTH OF A METER, A DECIGRAM WOULD BE ONE TENTH OF A GRAM ETC...SO THESE PREFIXES CAN BE USED WITH ANY OF THOSE MEASURED QUANTITIES THAT WE JUST TALKED ABOUT GRAMS AND LITERS AND METERS ETC...AND PUTTING THESE PREFIXES IN FRONT THEN GIVES US A LARGE OR SMALL MEASURED QUANTITY. CENTI IS ONE HUNDREDTH OR POINT ZERO ONE (.01), MIILI A LITTLE M FOR ONE THOUSANDTH OF A UNIT, MICRO WHICH IS GETTING QUITE SMALL NOW WHEN WE TALK ABOUT MICRO WHICH IS ONE MILLIONTH SO WE ARE TALKING ABOUT POINT ZERO ZERO ZERO ONE UNIT (.0001), AND NANO WHICH IS GETTING VERY SMALL AS YOU CAN SEE THEN ONE BILLIONTH OR TEN TO THE MINUS NINE. NOW, NANOMETERS ARE UNITS THAT WE USE WHEN WE TALK ABOUT THE LENGTH OF ELECTROMAGNETIC RADIATION OF LIGHT, VISIBLE LIGHT HAS A WAVE LENGTH OF ABOUT BETWEEN FOUR HUNDRED AND SEVEN HUNDRED NANOMETERS THAT IS THE TERM THAT WE USE TO TALK ABOUT THAT. MILLI, AS A MATTER OF FACT THESE THREE UNITS RIGHT HERE ARE BASICALLY OUR MONEY SYSTEM, THE UNITED STATES IS SORT OF AN ODD COUNTRY FROM THE STANDPOINT THAT WE TEND TO REFUSE TO USE THE METRIC SYSTEM FOR MEASUREMENT BUT WE ARE ONE OF THE FEW COUNTRIES IN THE WORLD THAT HAD A MONETARY SYSTEM THAT HAS ALWAYS BEEN BASED ON THE METRIC NUMBERS. WE HAVE A TENTH OF A DOLLAR, THAT IS CALLED A DECIDOLLAR OR DIME, WE HAVE A CENTIDOLLAR THAT IS CALLED A CENT OR A PENNY AND THAT IS JUST A CARRY OVER FROM THE OLD ENGLISH OUR RELATIONSHIP TO ENGLAND WHICH HAD A PENCE, BUT THE PENCE IN ENGLAND HAD NOTHING TO DO WITH BEING ONE HUNDREDTH OF ANYTHING IT WAS JUST A COIN CALLED A PENCE, BUT IN AMERICA THEN, A CENT IS ONE/ONE HUNDREDTH OF A DOLLAR (1/100), AND A MILL IS ONE/ONE THOUSANDTH OF A DOLLAR (1/1000) AND THAT IS EXACTLY WHAT YOU ARE CHARGED IF YOU ARE GOING TO PAY TAXES, YOU PAY A MILL LEVY, A MILL LEVY HAS NOTHING TO DO WITH GRINDING CORN, IT IS NOT A MILLING PROCESS THAT WE ARE TALKING ABOUT WE ARE TALKING ABOUT YOUR CHARGE ONE THOUSANDTH OF A DOLLAR OR SOME NUMBER OF

4 CHEM 105 & 106 UNIT ONE, LECTURE THREE 4 THOUSANDTHS IF YOU MILL LEVY WAS A HUNDRED IT WOULD BE A HUNDRED THOUSANDTH OF A DOLLAR FOR EVERY X DOLLARS THAT YOU VALUE HAS THAT IS A MILL LEVY,M THAT IS HOW WE PAY TAXES ON REAL ESTATE. SO WE HAVE A METRIC SYSTEM FOR OUR MONETARY AND YET WE TEND TO RESIST USING THE METRIC SYSTEM IN TERMS OF OUR MEASURED QUANTITIES, WE ARE ABOUT THE ONLY COUNTRY THAT DOESN'T FULLY USE THE METRIC SYSTEM FOR ROADSIDE AND SPEEDOMETERS IN THE AUTOMOBILE. WE ARE MOVING THAT WAY BUT WE STILL TEND TO RESIST. YOU MIGHT, THIS RIGHT HERE, A MILLIMETER, THAT IS KIND OF AN INTERESTING ONE BECAUSE A METER IS ABOUT A YARD OKAY? IT IS A LITTLE BIT LONGER THEN THE YARD AND WE DON'T WORRY ABOUT THE ENGLISH METRIC RELATIONSHIPS IF WE NEED IT WE HAVE A TABLE IN THE BACK OF THE BOOK, BUT THE METER IS ABOUT A YARD, NOW THERE WAS A COMMERCIAL I DON'T KNOW HOW MAY HAVE SEEN IT, IT ISN'T TOO OLD, BENSON AND HEDGES CIGARETTES ANYBODY EVER REMEMBER SEEING BENSON AND HEDGES CIGARETTE COMMERCIAL, AT LEAST ONE PERSON, THEY USED TO ADVERTISE THEIR CIGARETTE AND TALK ABOUT THE CIGARETTE WAS A SILLY MILLIMETER LONGER NOW A TYPICAL KING SIZE CIGARETTE IS ONE HUNDRED MILLIMETERS IN LENGTH, THAT IS THE COMMERCIAL WAY, BENSON AND HEDGES TO HAVE A LITTLE UPMANSHIP IN THEIR SALES MADE THEIRS ONE HUNDRED AND ONE MILLIMETERS IN LENGTH. NOW, THAT MILLIMETER YOU WOULDN'T BE ABLE TO TELL THAT IS ABOUT ( NOT AUDIBLE) THE ESSENCE OF THE AD WAS THE PERSON WOULD PUT A BENSON AND HEDGES CIGARETTE IN AND HAPPEN TO TURN TO CLOSE THE CAR WINDOW AND THE CAR WINDOW WOULD CATCH IT OR THEY WOULD STEP ON TO THE ELEVATOR AND THE ELEVATOR DOORS WOULD SHUT ON IT, I MEAN IMPLYING THAT YOU WERE GETTING A WHOLE LOT MORE CIGARETTE BECAUSE THEIRS WAS ONE HUNDRED AND ONE MILLIMETERS BUT THEY ALSO SAID SILLY MILLIMETER LONGER AND THAT WAS TRUE, THAT ONE MILLIMETER HAD NOTHING TO DO WITH ALL OF THESE THINGS THAT THEY WERE APPLYING IT TO. SO, WE DO SEE THE USE OF THESE IN DIFFERENT AREAS AND THESE ARE THE ONES THAT ESPECIALLY CENTI, MILLI, AND KILO THAT WE USE VERY FREQUENTLY THAT WE SHOULD CERTAINLY BE FAMILIAR WITH. AS A MATTER OF FACT, THIS ONE HERE KILO WITH GRAM A KILOGRAM IS USUALLY ABBREVIATED IN THE NEWS AS A KILO, SO WHEN WE HEAR THAT THE POLICE STOPPED A CAR ON INTERSTATE I44 AND THEY FOUND TEN KILO'S OF MARIJUANA WHAT DOES THAT MEAN? THEY FOUND TEN THOUSAND GRAMS OF MARIJUANA, THAT IS WHAT IT MEANS, A KILO IS ONE

5 CHEM 105 & 106 UNIT ONE, LECTURE THREE 5 THOUSAND GRAMS, OR TO PUT IT INTO PERSPECTIVE A KILOGRAM, ONE THOUSAND GRAMS IS ABOUT 2.2 POUNDS SO A TEN KILOGRAM BUST WOULD BE THEN ABOUT TWENTY-TWO POUNDS OF MARIJUANA. ALRIGHT, NOW THE OTHER THING THAT YOU WILL NOTE OVER HERE IS THAT THESE NUMBERS HAVE ALSO BEEN WRITTEN IN WHAT WE CALL SCIENTIFIC NOTATION, AND WE FREQUENTLY USE SCIENTIFIC NOTATION IN CHEMISTRY TO WRITE LARGE AND SMALL NUMBERS. TO WRITE A NUMBER IN SCIENTIFIC NOTATION FOR INSTANCE THIS NUMBER HERE WRITTEN AS A DECIMAL AND THEN AS SCIENTIFIC NOTATION, SCIENTIFIC NOTATION IS WRITTEN WITH ONE DIGIT FIRST THE DECIMAL POINT AND THEN WHATEVER OTHER NUMBERS WOULD FALL AFTER IN THIS CASE ZEROS, MULTIPLIED BY TEN AND THEN EITHER TO A NEGATIVE OR A POSITIVE, IF IT IS A NEGATIVE POWER IT MEANS THAT TO CONVERT THEN THIS NUMBER RIGHT HERE IN FRONT OF THE TIMES SIGN BACK TO THIS ONE WE WOULD HAVE TO DIVIDE THAT IS WHAT THE NEGATIVE SIGN IMPLIES WE WOULD HAVE TO DIVIDE THE NUMBER BY TEN NINE TIMES OR IN OTHER WORDS WE ARE GOING TO NEED TO MOVE THE DECIMAL POINT NINE PLACES TO THE LEFT SO THE NEGATIVE SIGN WOULD MEAN WE ARE MOVING THE DECIMAL BACK TO THE LEFT NINE PLACES IN ORDER TO GET BACK TO OUR ORIGINAL NUMBER IF WE GO RIGHT HERE AND PUT THE DECIMAL HERE WE ARE GOING ONE, TWO, THREE, FOUR, FIVE, SIX, SEVEN, WIGHT, NINE PLACES BACK TO THE LEFT THEN TO GIVE US BACK THE ORIGINAL NUMBER. AS I SAY WE DO USE SCIENTIFIC NOTATION QUITE EXTENSIVELY IN WRITING NUMBERS IN CHEMISTRY SO YOU DO NEED TO BECOME FAMILIAR WITH THOSE. I THINK I HAD A PROBLEM TO LOOK AT, AT THE END OF THE CHAPTER, YES I DO. PROBLEM NUMBER SEVEN AT THE END OF THE CHAPTER AS AN EXAMPLE OF WRITING SOME NUMBERS IN SCIENTIFIC NOTATION AND I WILL JUST GO AHEAD AND DO THAT RIGHT ON THE BOTTOM HERE, THE FIRST NUMBER IS NOW IF I WANTED TO WRITE THAT NOW IN SCIENTIFIC NOTATION, I AM GOING TO WRITE IT AS MULTIPLIED BY 10, NOW IN THIS PARTICULAR CASE BECAUSE THE NUMBER IS BIGGER THEN ONE AND WE ARE MOVING THE DECIMAL IN THE FIRST PLACE TO THE LEFT THE POWER OF TEN IS GOING TO BE POSITIVE, HOW MANY PLACES DID WE MOVE IT? THREE, SO THIS IS TEN TO THE THIRD. IN OTHER WORDS IF I TAKE THIS NUMBER HERE AND MULTIPLY IT BY TEN THREE TIMES THE FIRST TIME I MULTIPLIED IT BY TEN WOULD BE HERE, THE SECOND TIME HERE, AND THE THIRD TIME HERE AND WE ARE BACK TO OUR ORIGINAL NUMBER. SO, A POSITIVE POWER OF TEN IMPLIES THAT WE ARE GOING TO MULTIPLY BY TEN OR IN OTHER WORDS WE

6 CHEM 105 & 106 UNIT ONE, LECTURE THREE 6 ARE GOING TO MOVE THE DECIMAL POINT TO THE RIGHT, POSITIVE MEANS THAT WE ARE MOVING THE DECIMAL TO THE RIGHT, A NEGATIVE MEANS THAT WE ARE MOVING THE DECIMAL TO THE LEFT THAT NUMBER OF PLACES ALRIGHT? ANY QUESTION ON THAT ONE? LET'S LOOK AT THE NEXT ONE HERE AND WE WANT TO WRITE IT IN SCIENTIFIC NOTATION NOW I AM GOING TO NEED TO MOVE THEN DECIMAL TWO PLACES TO THERE SO I AM GOING TO WRITE IT TIME TEN, NOW, IN ORDER TO PUT THE DECIMAL BACK WHERE IT WAS WE WOULD HAVE TO MOVE IT BACK TO THE LEFT WOULDN'T WE? SO IT IS A NEGATIVE POWER AND HOW MANY PLACES DID WE MOVE IT? TWO, SO TIMES TEN TO THE MINUS TWO WOULD BE THE CORRECT NUMBER WRITTEN IN SCIENTIFIC NOTATION. WE CAN JUST LOOK AT ONE MORE HERE THIS IS PART D ON NUMBER SEVEN AT THE END OF THE CHAPTER, THE DECIMAL REALLY IS THERE EVEN THOUGH IT ISN'T SHOWING AND SO WE ARE MOVING IT ONE, TWO, THREE, FOUR, TIMES TEN AND WE MOVED IT FOUR PLACES SO TIMES TEN TO THE FOURTH, ALRIGHT? ANY QUESTION ON WRITING NUMBERS IN TERMS OF SCIENTIFIC NOTATION? THERE ARE SEVERAL EXAMPLE PROBLEMS IN THE BOOK, THERE ARE ALSO SOME PROBLEMS AT THE END OF THE TEXT THAT YOU CAN TRY TO SEE IF YOU DO UNDERSTAND IT OR HAVE ANY QUESTIONS. OKAY! NOW, WHENEVER WE MEASURE SOMETHING THERE IS ALWAYS AN UNCERTAINTY RELATIVE TO THE MEASURED VALUE, IN OTHER WORDS NO SCIENTIFIC IS TOTALLY EXACT, NO SCIENTIFIC MEASUREMENT IS TOTALLY EXACT THERE IS SOME DEGREE OF UNCERTAINTY IN THAT MEASUREMENT. NOW, WHEN WE WRITE DOWN THE NUMBER THAT WE HAVE MEASURED WE SHOULD WRITE DOWN ONLY THE NUMBERS THAT WERE JUSTIFIED IN WRITING DOWN BASED ON THE MEASURING DEVICE THAT WE ARE USING, AND THOSE NUMBERS THAT WE CAN JUSTIFIABLY RECORD ARE REFERRED TO AS SIGNIFICANT FIGURES, SIGNIFICANT FIGURES ARE THEN ALL OF THE NUMBERS THAT WE HAVE ACTUALLY MEASURED PLUS ONE UNCERTAIN NUMBER, NOT FIVE UNCERTAIN NUMBERS, ONE UNCERTAIN NUMBER THOSE ARE CALLED SIGNIFICANT FIGURES WHAT CAUSES THIS UNCERTAINTY IN A MEASURED QUANTITY, WELL THERE ARE SEVERAL THINGS: NUMBER ONE, HUMAN ERROR IF WE TYPICALLY READ LOOKING DOWN OR UP AT A MEASURING DEVICE WE MAY READ AN ERROR, SOME OF US MAY HAVE A TENDENCY TO ALWAYS READ ON THE HIGH SIDE OF A MEASURING DEVICE AND SOME WOULD TEND TO ALWAYS READ ON THE LOW SIDE SO WE BUILT IN AGAIN, HUMAN ERROR. IN ADDITION TO THE HUMAN ERROR IF WE DON'T PROPERLY CALIBRATE THE MEASURING DEVICE WE

7 CHEM 105 & 106 UNIT ONE, LECTURE THREE 7 ALREADY HAVE A BUILT IN ERROR AND MECHANICAL LIMITATIONS, I THINK PROBABLY A GOOD EXAMPLE IS A BATHROOM SCALE, IF I HAVE A DIGITAL BATHROOM SCALE AND I STEP ON THAT SCALE IT MAY SHOW A WEIGHT OF SAY 160 POUNDS NOW I CAN'T RECORD MY WEIGHT AS BEING I DIDN'T MEASURE THAT, I MEASURED ONE HUNDRED, I MEASURED THE SIXTY, THE ZERO THE LAST NUMBER I SEE DOWN THERE IS STILL UNCERTAIN IT IS UNCERTAIN BECAUSE OBVIOUSLY THAT BATHROOM SCALE HAS SOME DEGREE OF MECHANICAL LIMITATION AND SOME DEGREE OF UNCERTAINTY OF CALIBRATION. I AM PRETTY SURE I WEIGH AT LEAST 159 POUNDS AND I PROBABLY WEIGH OR NO LESS THAN 159 AND I PROBABLY DO WEIGH AT THE MAXIMUM 161 POUNDS, A BATHROOM SCALE GIVES A THEN AN UNCERTAINTY OF PLUS OR MINUS ONE POUND AS A MEASURING DEVICE. IF I USE THAT TO CARRY OUT SCIENTIFIC EXPERIMENTS I CAN'T RECORD HUNDREDTHS OF POUNDS OR THOUSANDTHS OF POUNDS OR WHATEVER THE CASE MIGHT BE BECAUSE MY MEASURING DEVICE HAS THAT DEGREE OF LIMITATION. NOW, ONE OF THE MEASURING DEVICES THAT WE USE QUITE EXTENSIVELY AND YOU WILL BE USING IF YOU ARE IN 105, WHICH A GOOD SHARE OF YOU ARE, IS WHAT WE REFER TO AS A CENTIGRAM BALANCE, THIS BALANCE HAS A MECHANICAL LIMITATION OF WEIGHING ONLY TO THE NEAREST ONE HUNDREDTH OF A GRAM, SO IF I WERE TO WEIGH SOME CHEMICAL HERE, LETS SAY ON THIS CENTIGRAM BALANCE HERE UP HERE THE SCALE DOESN'T SHOW REAL WELL BUT I DO KNOW THAT, THAT ONE IS IN THE FIFTY POSITION AND THESE ARE UNITS OF ONE ACROSS HERE AND IT LOOKS SETTING ON THE NONE AND THESE ARE TENTHS AND IT IS A LITTLE HARD TO SEE IT AND IN BETWEEN THE TENTHS ARE ANOTHER DIVISION WHICH GIVES US THE HUNDREDTH SO WE COULD PROBABLY THEN SAY THAT THE MASS OVER HERE WAS FIFTY AND THE NINE IS CERTAIN, SO I AM CERTAIN OF THE FIFTY AND I AM CERTAIN OF THE NINE AND I AM CERTAIN THAT IT IS BETWEEN.2 AND.3 SO I PUT DOWN.2 AND THEN I AM GOING BY SOME LITTLE MARK IN BETWEEN AND I AM NOT QUITE SURE WHICH ONE IT IS CLOSEST TO WE WILL JUST SAY IN THIS CASE FOUR, AND SO I WOULD RECORD THEN THE MASS AS RAMS OR IN OTHER WORDS WE HAVE THREE NUMBERS HERE THAT WE ARE CERTAIN OF AND A FOURTH NUMBER WHICH IS MY FIRST UNCERTAIN AND ALL FOUR OF THOSE MEASURED NUMBERS ARE REFERRED TO AS SIGNIFICANT FIGURES OKAY? THE NUMBER AS SIGNIFICANT FIGURES THAT WE CAN PUT DOWN THEN IS LIMITED BY THE MEASURING DEVICE THAT WE ARE USING IF I USE A BALANCE CALLED A MILLIGRAM BALANCE THAT MEANS THAT IT CAN

8 CHEM 105 & 106 UNIT ONE, LECTURE THREE 8 WEIGH, MECHANICALLY IT CAN WEIGH TO THE NEAREST ONE THOUSANDTH OF A GRAM AND IF THAT WERE THE CASE I WOULD BE JUSTIFIED IN RECORDING THEN ANOTHER DIGIT THERE OKAY? NOW, OFTEN TIMES HOWEVER, IN PROBLEM SOLVING IN THE BOOK WE ARE BASING OUR NUMBER SIGNIFICANT FIGURES THEN OF THE MEASURED QUANTITIES, THE NUMBERS GIVEN TO US NOT ONES THAT WE HAVE GONE TO THE LAB AND MEASURED, AS A MATTER OF FACT, FOR THOSE IN THE 105 YOUR SECOND EXPERIMENT THAT YOU DO IS AN EXPERIMENT STRICTLY LOOKING AT LIMITATIONS AND USING SIGNIFICANT FIGURES IN SCIENTIFIC CALCULATIONS SO YOU WILL GET QUITE A BIT OF EXPERIENCE ON THIS IN EXPERIMENT NUMBER TWO. IF WE ARE DEPENDING UPON SOMEBODY ELSES MEASURED QUANTITIES WE DO HAVE SOME RULES THAT WE USE TO DETERMINE THE NUMBER OF SIGNIFICANT FIGURES IN A NUMBER. WHAT NUMBERS ARE SIGNIFICANT? FIRST OF ALL, RULE NUMBER ONE, ALL NON-ZERO DIGITS BETWEEN ONE AND NINE ARE SIGNIFICANT, AND HERE WE HAVE SOME NUMBERS GIVEN, THE NUMBER 123 THAT HAS THREE SIGNIFICANT FIGURES IN IT. THE NEXT NUMBER THAT WE SEE UP THERE, THAT HAS FIVE SIGNIFICANT FIGURES, ALL OF THOSE ARE ASSUMED TO BE MEASURED NUMBERS, WE DON;T KNOW WHAT THEY MEASURED IT ON SO IF IT WAS A RECORDED NUMBER WE ARE ASSUMING THAT THEY ARE APPLYING THE PROPER RULES AND SO WE WOULD THEN USE ALL DIGITS AS SIGNIFICANT. THE NUMBER NINE IF THAT IS ALL THAT WE RECORDED WE HAD NINE APPLES WE HAVE ONE SIGNIFICANT FIGURE O.K.? ALL ZEROS BETWEEN NON-ZERO DIGITS ARE ALSO SIGNIFICANT AND HERE ARE SOME EXAMPLES OF THAT, THE NUMBER NOTICE THAT THOSE TWO ZERO'S FALL BETWEEN DIGITS AND SO THEY ARE IN FACT MEASURED QUANTITIES, THEY ARE NOT JUST ACCIDENTAL, THEY ARE MEASURED QUANTITIES, WE MEASURED TEN THOUSAND TWO ZERO THREE UNITS OF WHATEVER IT WAS, GRAMS, METERS, OR APPLES, OR ORANGES, SO WE HAVE FIVE SIGNIFICANT FIGURES IN THAT NUMBER. THE THIRD RULE, ALL ZERO'S PRECEDING THE FIRST NON-ZERO DIGIT ARE NOT SIGNIFICANT. THIS OCCURS WHEN WE SEE A NUMBER LIKE THIS ALRIGHT? THOSE FIRST TWO ZERO'S DO NOT FALL BETWEEN DIGITS AND BY THE SAME ASPECT AS WE WERE APPLYING RULE TWO ZERO'S BETWEEN DIGITS ARE SIGNIFICANT ZERO'S NOT BETWEEN DIGITS AND NOT FOLLOWING A DIGIT ARE NOT SIGNIFICANT. NOW, THE WAY WE GET AROUND THIS AND THE BEST WAY TO LOOK AT IT IS TO ALWAYS WRITE OUR NUMBERS IN SCIENTIFIC NOTATION, NOTICE IF I REWROTE THIS NUMBER HERE IN SCIENTIFIC NOTATION WE ARE GOING TO MOVE THE DECIMAL TWO

9 CHEM 105 & 106 UNIT ONE, LECTURE THREE 9 PLACES TO THE RIGHT SO TEN TO THE NEGATIVE TWO, 4.9 TIMES 10 TO THE NEGATIVE 2, WE JUST LOOKED AT THE NUMBER 4.9 AND WE CAN SEE THAT WE HAVE TWO SIGNIFICANT FIGURES, SO IT IS USUALLY BEST TO WRITE NUMBERS IN SCIENTIFIC NOTATION WHEN WE ARE TRYING TO WORK WITH SIGNIFICANT FIGURES. HERE IS ANOTHER EXAMPLE NOW IF WE REWRITE IT YOU SEE, THESE THREE ZERO'S HERE ARE NOT SIGNIFICANT THEY ARE NOT BETWEEN DIGITS, THEY ARE NOT PROCEEDED BY A DIGIT, SO THEY ARE NOT SIGNIFICANT BUT THAT ZERO IS SIGNIFICANT IT IS BETWEEN TWO DIGITS, RULE NUMBER TWO, AND SO WE WOULD HAVE THEN IF WE WROTE IT IN SCIENTIFIC NOTATIONS 7.09 TIMES TEN TO THE MINUS THREE, WE CAN EASILY SEE THEN THREE SIGNIFICANT FIGURES. THE FOURTH RULE ON THIS AND IT MAY BE THE FIFTH IN YOUR TEXT, I CAN'T REMEMBER WHICH WAY IT WAS WRITTEN IN THERE. ZERO'S FOLLOWING A DECIMAL POINT ARE SIGNIFICANT IF PRECEDED SOMEWHERE IN THE NUMBER BY A NON-ZERO DIGIT. WHEN WE LOOK AT THIS NUMBER HERE THOSE TWO ZERO'S ARE SIGNIFICANT, WHATEVER THE MEASURING TOOL WAS ALLOWED US TO RECORD EXACT ZERO'S ON THE END, IN OTHER WORDS IF WE WERE USING A MILLIGRAM BALANCE, A BALANCE THAT ALLOWED US TO WEIGH TO THE NEAREST THOUSANDTH OF A GRAM, IF IT WEIGHED EXACTLY 3.4 GRAMS IT WOULD GIVE US NUMBERS OF IN OTHER WORDS WE ARE CERTAIN OF THE FIRST ZERO AND UNCERTAIN OF THE SECOND BUT THAT IS STILL A SIGNIFICANT FIGURE, ALL KNOWN NUMBERS PLUS ONE UNCERTAIN NUMBER OKAY? THIS ONE HAS A BUNCH OF ZERO'S AND THIS ZERO IS NOT SIGNIFICANT, RULE NUMBER THREE. THIS ZERO IS SIGNIFICANT, RULE NUMBER TWO BECAUSE IT FALLS BETWEEN DIGITS AND THIS ZERO IS SIGNIFICANT BECAUSE OF RULE NUMBER FOUR, IT IS PRECEDED SOMEPLACE IN THERE BY A DIGIT AND IT OCCURS AFTER A DECIMAL POINT, DO IT IS A SIGNIFICANT SO THEREFORE WE WOULD HAVE FIVE SIGNIFICANT FIGURES IN THIS NUMBER, OKAY? ALRIGHT, A COUPLE OF OTHER RULES HERE, I THINK IN THE BOOK RULE NUMBER FIVE IS RULE NUMBER SIX, AN EXACT NUMBER AS DEFINED IN CERTAIN CONVERSION FACTORS HAS UNLIMITED NUMBER OF SIGNIFICANT FIGURES. ALRIGHT IF WE HAVE DEFINED SOMETHING, IN OTHER WORDS WE HAVE DEFINED THAT THERE ARE THREE FEET IN A YARD, THAT IS BY DEFINITION, THREE FEET IN A YARD, THAT MEANS THAT ARE EXACTLY THREE FEET IN A YARD, THERE ARE ONE HUNDRED CENTS IN A DOLLAR, SO THERE IS NOT 100.? THERE IS WHATEVER NUMBER OF ZERO'S YOU WANT TO INFINITY CENTS IN A DOLLAR, IT IS NOT A MEASURED QUANTITY IT IS A DEFINED QUANTITY THERE

10 CHEM 105 & 106 UNIT ONE, LECTURE THREE 10 ARE TWELVE INCHES IN A FOOT, THERE ARE ONE THOUSAND METERS IN A KILOMETER, THOSE ARE BY DEFINITION THEY HAVE AN UNLIMITED NUMBER OF SIGNIFICANT FIGURES. WELL, AS WE GO ALONG AND WORK PROBLEMS WE ARE GOING TO EMPHASIZE VERY STRONGLY THE IMPORTANCE OF RECORDING ANSWERS AND NUMBERS TO THE CORRECT NUMBER OF SIGNIFICANT FIGURES, ESPECIALLY IN YOUR LABORATORY WORK BECAUSE YOU WILL BE MAKING THE MEASUREMENTS, YOU WILL KNOW THE LIMITATION OF THE TOOL THAT YOU ARE USING AND SO THOSE NUMBERS ARE IMPORTANT WHEN YOU RECORD THEM. WE HAVE A QUESTION IN THE BACK THERE, THE QUESTION WAS, COULD I RE- EMPHASIZE THIS LAST ONE PERTAINING TO A DEFINED NUMBER, AGAIN IF WE SAY THERE ARE SIXTEEN OUNCES IN A PINT THAT IS NOT A MEASURED THAT IS BY DEFINITION SOMEBODY SAID, WE WILL SAY ONE SIXTEENTH OF A PINT IS AN OUNCE, BY DEFINITION THERE ARE TWO PINTS IN A QUART, AGAIN THAT IS STRICTLY BY DEFINITION. ANY NUMBER THAT IS SOMETHING THAT IS MERELY BY DEFINITION NOT A MEASURED QUANTITY HAS UNLIMITED NUMBER OF SIGNIFICANT FIGURES. WHATEVER YOU WANT TO CALL IT TEN, ONE HUNDRED, ONE THOUSAND, IT IS WHATEVER IT IS OKAY? ALRIGHT, DOES THAT ANSWER? ALRIGHT, ANY OTHER QUESTIONS ON THAT? OKAY. NOW, WHEN WE CALCULATE AN ANSWER AND OFTEN THIS IS WHAT WE ARE GOING TO BE DOING, THE CALCULATED ANSWER HAS TO PAY ATTENTION TO THE SIGNIFICANT FIGURES THAT WE HAVE USED IN THE CALCULATION, THE GENERAL RULE OF THUMB IS THAT IF WE ARE MULTIPLYING AND DIVIDING, THAT THE ANSWER FOR THE PROBLEM CANNOT HAVE MORE SIGNIFICANT FIGURES IN IT THEN THE NUMBER WITH THE LEAST SIGNIFICANT FIGURES IN IT USED IN THE PROBLEM. NOW ANOTHER WAY THAT WE CAN KIND OF THINK ABOUT THIS LIMITATION ON OUR CALCULATION IS WE COULD SAY THAT A CHAIN IS NO STRONGER THAN ITS WEAKEST LINK, I DON'T CARE HOW BIG THE LINKS OF A CHAIN ARE, ITS STRENGTH IS GOING TO DEPEND UPON IT WEAKEST LINK, IF I TAKE A LITTLE PIECE OF FLIMSY WIRE AND TIE A COUPLE OF THE LINKS TOGETHER THEN I HAVE GREAT BIG LINKS THE REST OF THE WAY, THE STRENGTH OF THE CHAIN IS GOING TO DEPEND UPON THAT WEAKEST LINK. THE ACCURACY OF OUT ANSWER DEPENDS UPON THE WEAKEST NUMBER THAT WE HAVE USED, WEAKEST IN TERMS OF NUMBER OF SIGNIFICANT FIGURES USED IN THE CALCULATION. OUR ANSWER CAN'T BE BETTER THEN WHAT WE HAVE USED TO MAKE THE CALCULATIONS, THAT IS VERY IMPORTANT TO REMEMBER AGAIN, ESPECIALLY IN YOUR LABORATORY WORK. NOW LET'S TAKE A LOOK AT WHAT WE ARE SAYING HERE AND HOW WE APPLY

11 CHEM 105 & 106 UNIT ONE, LECTURE THREE 11 THAT THEN, LETS SUPPOSE HERE IN THIS FIRST EXAMPLE NOW, THAT WE ARE GOING TO MULTIPLY 3.63 BY NOW IF I USE MY CALCULATOR, IF YOU PUNCH THIS IN ON YOUR CALCULATOR WHAT YOU ARE GOING TO SEE ON THE DISPLAY IS A NUMBER , AND YOU MIGHT HAVE A TENDENCY TO WANT TO WRITE THAT DOWN, BUT THAT IS NOT CORRECT, THAT IS NOT A CORRECT ANSWER, BECAUSE WHAT THAT IS IMPLYING IS THAT WE HAVE, IF WE LOOK AT THE NUMBERS OF SIGNIFICANT FIGURES IN OUT ANSWER, WE ARE SHOWING THAT WE HAVE AN ACCURACY OF 1,2,3,4,5,6,7,8 SIGNIFICANT FIGURES IN OUT ANSWER WHICH WOULD IMPLY THAT THE NUMBERS THAT WE USE TO GET THE ANSWER HAD AN ACCURACY OF EIGHT SIGNIFICANT FIGURES EACH, AND IT DOESN'T. YES, WE HAVE A QUESTION AGAIN, WHAT WERE THE NUMBERS THAT WERE MULTIPLIED? 3.63 MULTIPLIED BY OKAY. MAY BE THAT IS A LITTLE OUT OF FOCUS THERE, ALRIGHT, NOW, HOW ARE WE GOING TO ACTUALLY WRITE OUR ANSWER? WELL OUR ANSWER SHOULDN'T BE ANY BETTER THAN THE POOREST NUMBER THAT WE USEE. HOW MANY SIGNIFICANT FIGURES IN THE FIRST NUMBER THAT WE USED? THREE OKAY, THEY ARE ALL DIGITS, RULE NUMBER ONE SAYS THAT ALL DIGITS ARE SIGNIFICANT. HOW MANY SIGNIFICANT FIGURES IN THE SECOND NUMBER WE USED? FIVE SIGNIFICANT FIGURES IN THE SECOND NUMBER, OUR ANSWER SHOULD BE LIMITED TO HOW MANY SIGNIFICANT FIGURES? THREE, GOOD, THE POOREST NUMBER THAT WE USED. NOW IF WE DO THAT THEN WHAT WE ARE GOING TO HAVE TO DO IS FIRST OF ALL, REWRITE THIS NUMBER IN SCIENTIFIC NOTATION SO NOTICE THAT I HAVE MOVED THE DECIMAL NOW 1, 2, 3, 4, 5 PLACES OVER REWROTE THE NUMBER TIMES TEN TO THE FIFTH AND MAYBE I CAN GO AHEAD AND WRITE THAT AGAIN HERE IF IT ISN'T REAL CLEAR, LET ME REWRITE IT SO WE CAN SEE IT THEN TIMES TEN TO THE FIFTH. WE SAID THAT WE ARE GOING TO BE LIMITED TO THREE SIGNIFICANT FIGURES, SO I START FROM THIS SIDE AND COUNT OVER THREE SIGNIFICANT FIGURES THAT IS ONE, THAT IS TWO, AND THAT IS THREE. NOW I AM GOING TO DRAW A LINE RIGHT THERE, NOW WE APPLY THE RULES OF ROUNDING OFF, IF THE NUMBER AFTER THE LAST NUMBER WE ARE GOING TO KEEP, IN THIS CASE THE LAST NUMBER WE ARE GOING TO KEEP IS THE SIX, IF THE NUMBER AFTER THE LAST NUMBER THAT WE ARE GOING TO KEEP IS GREATER THEN FIVE WE ROUND UP, WE WOULD CALL IT 2.07, IF THE NUMBER AFTER THE LAST NUMBER THAT WE ARE GOING TO KEEP IS LESS THEN FIVE WE WILL DROP EVERYTHING AFTER THE LAST NUMBER, IN THIS CASE IT IS A ONE AND SO OBVIOUSLY WE ARE JUST GOING TO THROW AWAY EVERYTHING AFTER THE SIX. THERE ARE THOSE

12 CHEM 105 & 106 UNIT ONE, LECTURE THREE 12 VERY FEW CASES WHERE YOU MIGHT PLUG IN AND IT COMES OUT TO BE AN EXACT FIVE AND THAT IS WHAT IT SHOWS ON THE CALCULATOR. MEANING THAT EVERYTHING ELSE WAS ZERO'S ALRIGHT. WHAT DO WE DO THEN? WELL THE GENERAL RULE OF THUMB FOR ROUNDING OFF NUMBERS IS THAT WE ALWAYS ROUND SO THAT IT IS AN EVEN NUMBER, SEEING THAT SIX IS ALREADY AN EVEN NUMBER, WE WOULD DROP THE FIVE. LET ME JUST WRITE UP TWO NUMBERS HERE TO SHOW WHAT I MEAN, IF WE HAD AN EXACT NUMBER HERE OF BECAUSE THE NUMBER PRECEDING HERE WAS ALREADY AN EVEN NUMBER WE WOULD DROP IT. IF OUR NUMBER WAS BECAUSE IT IS AN ODD NUMBER WE WOULD REWRITE IT AS 2.08 IN OTHER WORDS WE ARE ALWAYS GOING TO ROUND AN EXACT FIVE SO THAT WE HAVE AN EVEN NUMBER, YOU SEE WHAT WE HAVE DONE IS WE HAVE THROWN IT AWAY AND KEPT IT AN EQUAL NUMBER OF TIMES, SO STATISTICALLY WE ARE NOT IMPACTING OUR ANSWER. BACK TO THIS ONE HOWEVER THEN, BECAUSE IT WAS A ONE LESS THEN A FIVE THEN WE ARE GOING TO MERELY THROW THAT PART OF OUR ANSWER AWAY AND THE FINAL ANSWER PROPERLY WRITTEN, WOULD BE 2.06 TIMES TEN TO THE FIFTH, NOW IT SHOWS THREE SIGNIFICANT FIGURES THAT IS WHAT WE MUST HAVE IN THE ANSWER, OKAY. THE QUESTION WAS, WHAT DO WE DO IF IT IS NOT AN EXACT FIVE? IS THAT WHAT YOU WERE SAYING? THEN IT JUST GOES BACK TO THE BASIC RULE THAT IS, IF IT IS OVER FIVE ANYTHING OVER FIVE, OKAY, THEN WE ARE GOING TO ROUND UP, SO IN OTHER WORDS IF WE HAD AN NUMBER THAT LOOKED LIKE THIS YOU BETTER NOT HAVE A CALCULATOR THAT DOES THAT MUCH, BUT IF WE DID HAVE AND WE ARE GOING WITH THREE SIGNIFICANT FIGURES THAT IS GREATER THEN A FIVE, RIGHT? FIVE PLUS THE NUMBER OF ZERO'S AND A ONE IS STILL BIGGER THEN AN EXACT FIVE, WE WOULD ROUND UP OKAY? THAT IS NOT GOING TO HAPPEN VERY OFTEN BECAUSE YOUR CALCULATOR IS GOING TO HAVE A FIXED NUMBER OF SPOTS THAT IT CAN DISPLAY, TYPICALLY, USUALLY SEVEN TO NINE AND MOST CALCULATORS DO NOT DISPLAY ZERO'S, IN OTHER WORDS IN THE CALCULATOR IF I HAD A FOR A NUMBER IT WILL NOT SHOW ON THE OTHER PLACES SO IF NOTHING SHOWS UP ON THAT CALCULATOR SCREEN THAT MEANS THAT IS EXACT ZERO'S FROM THERE ON OUT OKAY? OTHERWISE IT WOULD DISPLAY SOMETHING. ALRIGHT, ANY QUESTION ON THAT IDEA OF SIGNIFICANT FIGURES INVOLVED IN MULTIPLYING AND DIVIDING. MULTIPLYING AND DIVIDING OUR ANSWER IS LIMITED TO THE POOREST NUMBER WE HAVE USED IN THE CALCULATION, MERELY MEANING THE ONE WITH THE

13 CHEM 105 & 106 UNIT ONE, LECTURE THREE 13 LEAST NUMBER OF SIGNIFICANT FIGURES. NOW, IN ADDITION AND SUBTRACTION WE HAVE A SLIGHTLY DIFFERENT RULE THAT IS APPLIED, WE NEED TO KEEP THAT IN MIND. IN ADDING OR SUBTRACTING IT REALLY DEPENDS UPON THE NUMBERS AFTER THE DECIMAL POINT THAT WE ARE CONCERNED WITH. WE MUST MAINTAIN THE SAME NUMBER OF DIGITS AFTER THE DECIMAL IN THE ADDED OR SUBTRACTED ANSWER, THE SAME NUMBER OF DIGITS AFTER THE DECIMAL POINT AS THE POOREST NUMBER THAT WE HAVE USED, NOT THE NUMBER OF SIGNIFICANT FIGURES, THE NUMBER OF DIGITS AFTER THE DECIMAL POINT OF THE POOREST NUMBER THAT WE HAVE USED. IN THIS PARTICULAR CASE THE FIRST EXAMPLE, EVERY NUMBER HAS ONE PLACE AFTER THE DECIMAL POINT, OUR ANSWER SHOULD HAVE ONE PLACE AFTER THE DECIMAL POINT ALL NUMBERS PRECEDING THE DECIMAL POINT ARE SIGNIFICANT AND ARE KEPT SO NOTE WHAT HAPPENED HERE IS THAT IN THIS PARTICULAR CASE WHEN WE ADDED THESE THREE NUMBERS TOGETHER THE FIRST NUMBERS HAD TWO SIGNIFICANT FIGURES THE SECOND NUMBER HAD ONE SIGNIFICANT FIGURE THE THIRD NUMBER HAD TWO SIGNIFICANT FIGURES WHEN WE ADDED THEM TOGETHER WE GOT AN ANSWER WHICH HAS HOW MANY? SIGNIFICANT FIGURES, THREE, 18.7 ALL OF THOSE NUMBERS WOULD BE KEPT. ONLY AFTER THE DECIMAL POINT THAT WE ARE LOOKING WHEN WE ADD AND SUBTRACT, AND VERY QUICKLY THEN BECAUSE WE NEED TO WRAP UP LETS JUST LOOK AT THIS ONE HERE FOR A SUBTRACTION, WE HAVE A.8 AND WE ARE SUBTRACTING A AND THIS IS THE NUMBER THAT WE WOULD GET, BUT HOW MANY PLACES ARE WE JUSTIFIED AFTER THE DECIMAL POINT IN OUR ANSWER, JUST ONE BECAUSE THAT IS ALL THAT IS IN THE FIRST NUMBER THE 10.8, AND SO MY ANSWER, MY CORRECTLY WRITTEN ANSWER WOULD BE 6.1 SO WE HAD THREE SIGNIFICANT FIGURES AND FOUR SIGNIFICANT FIGURES AND WE ENDED WITH AN ANSWER WITH TWO SIGNIFICANT FIGURES.