WEBVTT
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In this video, we will learn about the ways we express precision with numbers.
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We’ll look at the difference between significant figures, decimal places, and the situations where we might use one over the other.
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We use numbers to express the values we get from measurements.
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I’m one meter and 86 centimeters tall, which can be expressed as 1.86 meters.
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But am I really?
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That value comes from a measurement, one I took shortly before making this video, and I used a tape measure.
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This tape measure had markings for meters, centimeters — which are one hundredth of a meter — and millimeters — which are thousandths of meters.
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So, one meter is equivalent to 100 centimeters or 1000 millimeters, and one centimeter is equivalent to 10 millimeters.
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But what if the tape measure had only had the markings every meter?
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I’d know I was taller than one meter, but shorter than two meters.
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We can estimate that my height is closer to two meters, but beyond that it’s just a guess.
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So, for now, we’d round to two meters, if only that was all it took to make me taller.
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Instead, what if the tape measure had the meter markings and markings every 10 centimeters?
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With this tape measure, I could see my actual height was above 1.8 meters, below 1.9 meters, but actually a little bit closer to 1.9 than 1.8.
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Using a tape measure with finer increments means I can make a more precise measurement, and I get my height closer to the true value.
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Next, we include the one-centimeter markings.
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So, we can measure my height to be 186 centimeters or 1.86 meters.
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This is even more precise than before.
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You may have noticed that this is also more accurate because it’s closer to my actual height.
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But accuracy is not the focus of this video, but we could have even used the millimeters.
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If we go further, we should get more precise, right?
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For cases like this, it’s hard to measure someone’s height correctly to the nearest millimeter because we grow and shrink with our breathing, people also slouch, or people could try to cheat and stand on tiptoes.
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For this purpose, centimeters are precise enough.
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Let’s take a look back at the measurements we made, two meters, 1.9 meters, and 1.86 meters.
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It would be helpful if we had a clear way to describe the differences between these numbers.
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This way we can talk to other people about how precise we’re being.
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One of the most important terms we use when describing the precision of numbers is significant figure.
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Here is the long definition of a significant figure.
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A significant figure is a digit of a number that’s necessary to understand the value and precision of the number.
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However, that’s a little awkward to start with.
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So, let’s go in with a few examples.
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Here are the numbers from the previous example.
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The more significant digits there are in the number, the more precise it is.
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So, let’s imagine for a minute that I am exactly two meters tall.
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I could write it as two meters, 2.0 meters, or 2.00 meters.
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The fact we have zeros there is still important because it means we’ve measured that value to that precision.
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Let’s take a closer look at the example of 2.00.
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Two is in the units column, so it’s describing two lots of a single unit.
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Zero is in the tenths column, so it’s describing zero-tenths.
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And there’s a zero in the hundredths column, so it’s describing zero hundredths.
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If we were talking about meters, this would mean we had two meters, zero-tenths of centimeters, and zero centimeters.
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If we threw away these zeros, the number we’ll be left with would be less precise.
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The two in this example gives us both an indication of the value and the precision, and the zeros help us know we know that value to a great degree of precision.
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So, we know a zero is a significant digit If it was part of the measurement.
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Now, let’s go a little bit deeper.
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Here are the digits we established to be significant.
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But what about this zero here?
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We know this number has no lots of 10, so we know we could put zero here and that would still be correct.
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We call this type of zero a leading zero.
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It’s a zero to the left of the first nonzero digit.
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We could write an unending number of these and it wouldn’t change the value or precision of our number.
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The two is our first significant digit going left to right because it’s the first one that starts to define the value of the number.
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Leading zeros are not significant.
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There’s another type of zero you might come across.
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Once we finished our measurement to three significant figures, we didn’t actually measure any further.
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So we don’t know what those values are.
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Quite often, you might see these values written as zeros.
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We call these trailing zeroes.
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Just like leading zeros, trailing zeros can theoretically go on forever.
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Now, the definition of a trailing zero is simply any zero that’s to the right of the last nonzero digit.
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Since we only have one nonzero digit in our number, all the zeros to the right of our two are trailing zeroes.
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The two zeros that were part of our measurement are trailing zeroes.
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But since they define the precision of our measurement, they are significant, While the trailing zeros we added are not significant.
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So, trailing zeros can be significant if they’re part of the measurement; otherwise, they are not significant.
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But there’s one more possibility we need to discuss.
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In this example, the two in the ones position is both the first significant figure and the last nonzero digit.
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But what if, instead, we were dealing with a measurement of 2.01 of something?
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The zero in the middle isn’t leading because it’s not in front of the first nonzero digit, and it’s not trailing because it’s not to the right-hand side of the last nonzero digit.
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We call this a sandwiched zero.
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It’s a zero between two nonzero digits.
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And sandwiched digits are always significant.
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If we have a nonzero digit, that must mean it was part of our initial measurement.
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Therefore, even if it’s a zero, any numbers in between must have been part of the measurement as well.
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So, a zero between two nonzero digits, a sandwiched zero, or a middle zero is always significant.
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And it doesn’t matter how many zeros there are that are sandwiched or where they are in the sequence.
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Now, we get to the important part where we’re dealing with rounding according to a specific number of significant figures.
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When we do a calculation on our calculator, it doesn’t know how many significant figures we want, so it just gives them all.
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Nine divided by eight in perfect arithmetic is equal to 1.125.
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We can imagine the unimportant leading zeros stretching off to the left forever.
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And because this is an exact mathematical process, we get an exact answer going off into ∞.
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So, we could think of every single trailing zero as being significant.
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But instead we’re going to think about this as a measured number and assume that these are the significant digits.
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To work out how many significant figures we have, we first need to find the first nonzero digit.
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That’s the one here.
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All the digits to the left are zeros.
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The next thing we want to do is find the last significant digit.
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In this example, that’s this one here, the five to the far right.
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And then what we want to do is round that off, and we do that by looking at the next digit along.
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If that digit is a zero, one, two, three, or four, we round down.
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If it’s a five, six, seven, eight, or nine, we round up.
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Here, we can see an example where the digit following our last significant digit is either a one or a seven.
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With the one, we round down; with the seven, we round up.
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In this case, we have a zero.
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So, we round 1.25000 and so on to simply 1.125.
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But what if we wanted to round to three significant figures?
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Well, that would make our last significant digit the two because we’ve got one, one, and two as our three significant figures.
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But when it comes to rounding, we have this five.
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So, we round up to 1.13.
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If we only wanted two significant figures, it would be even easier.
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The digit after our last significant digit is this two.
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So, we round down to 1.1.
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And finally, if we only wanted one significant digit, we would round to one because the digit after the one is one.
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We round down and we’re left with one to one significant figure.
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Remember, none of the other digits count when you’re rounding, just the one immediately after your last significant digit.
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Of course, talking about significant figures, we’d probably need to know about decimal places as well.
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When we express a round number, it’s easy to do.
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We write the ones, the tens, the hundreds, and so on.
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When we write a number that’s not a whole number, we have to use the decimal point.
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After the decimal point, we get the tenths, the hundredths, the thousandths, and so on.
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But we can also have numbers that are smaller than one that may need leading zeros.
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These leading zeros are never significant, but we do need to write them to put the nonzero digit in the right place.
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If we remove them, we wouldn’t be expressing the right value.
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This number has three decimal places in total.
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We don’t see any digits after the two, so we stop counting.
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But this number is also to three decimal places.
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We’ve got a one in the units column, a three in the tenths column, and a four in the hundredths column.
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When we express something to a fixed number of decimal places, we just keep on writing digits after the decimal place until we’ve hit that many digits.
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If the number goes beyond that, like in this example, we just need to round in the same way we did with significant figures.
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We find the third decimal place because we’re rounding to three decimal places.
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And we round the number up if that digit is a five, six, seven, eight, or nine.
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And we round it down if it’s a zero, one, two, three, or four.
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So, in this case, we round to 1.343 rather than 1.342 because the digit five came after the two and we were rounding to three decimal places.
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The last thing we need to deal with is how we know how to round if we’re doing a calculation.
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There’s a fairly simple golden rule for rounding.
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When you round your final answer, you do it to the same number of significant figures as the least precise value in the calculation.
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Well, let’s have a think about what this really means by trying to calculate the density of a penny.
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Maybe, we use some cheap kitchen scales accurate to the nearest gram.
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So, we got a value of three grams accurate to one significant figure.
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This isn’t particularly precise.
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But we’ve used a piece of equipment that tells us the volume of the penny very, very accurately.
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So, we have the value of 0.360 cubic centimeters to three significant figures.
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The trailing zero at the end is significant because it was part of our measurement.
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We calculate the density by dividing the mass by the volume.
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So, we divide three grams by 0.360 centimeters cubed, and our calculator tells us that the answer is 8.3 repeating.
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However, this is more precision than we can honestly say we have because the mass is accurate to only one significant figure.
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Therefore, we round our final answer to the same precision, one significant figure.
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If we use less precise numbers, we get a value we’re less confident in.
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We should never claim to know something to a greater precision than we actually do.
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That’s why we round so that other people know how precise our measurements were.
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There are occasional exceptions to the golden rule, but we won’t be dealing with those in this video.
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Instead, let’s have some practice.
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To how many significant figures is the value 0.0023 kilograms given?
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We can think of significant figures of a number as the digits that determine its value and precision.
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The first three digits in the number are zero, zero, and zero.
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None of these zeros contribute directly to the value or precision of the number to come.
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It’s the number two that’s the first nonzero digit.
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So, it’s the two that’s our first significant digit.
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The three has also been given, so that’s a significant digit as well.
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And we could write zeros after the three, but we haven’t actually been told what those digits are.
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The number could have been rounded, so these trailing zeroes are not significant.
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So, the value 0.0023 kilograms has been given to two significant figures.
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Next, let’s look at an example based in the lab.
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A student needs to use the mass balance apparatus shown below to measure out a certain number of moles of a substance.
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Using a calculator, the student determines that they need to weigh out 12.108225 grams of the substance.
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Why is this amount not a suitable value to measure using the mass balance apparatus?
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Well, the first thing we need to do is look at the mass balance apparatus.
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From the display, we can see it measures in grams.
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The student is looking out to weigh a mass in grams, which helps, but that’s not quite enough.
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The balance gives a value to a maximum of four decimal places.
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A gram mass balance accurate to four decimal places will be accurate to the nearest 0.0001 grams, one ten thousandths of a gram.
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But the mass the student calculated is to six decimal places.
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This is 100 times more precise than the balance is capable of.
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The balance would be able to read the mass of 12.1082 grams.
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And if the student added a little bit more, it would be able to read 12.1083 grams.
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But the number produced by the student’s calculator is much more precise than the mass measurements this mass balance is capable of.
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So, this amount is not a suitable value to measure using the mass balance apparatus because the mass balance apparatus only measures to four decimal places.
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What mass should the student try to weigh out instead?
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The number from the calculator was to six decimal places, but the balance only reads to four.
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So, to make sure we get as close to the calculated value as possible, we want to round that number to four decimal places.
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You may think you could just chop off the last two digits, but sometimes that won’t be as good as rounding.
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We know the first three decimal digits are one, zero, eight.
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And the digit in the fifth decimal place is a two.
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So, we round down.
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So, our final answer is 12.1082 grams.
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So, to finish up with the key points, significant figures or digits affect the value and precision of a number.
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Leading zeros are never significant.
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Trailing zeroes may be significant.
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And sandwiched zeros are always significant.
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Decimal places are the positions after the decimal point.
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And when you give your absolute final answer, you should round it to the same number of significant figures as the least precise value you used in the calculation.