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In this video, we will learn about significant figures.

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We will discuss how to identify significant figures in a number and look at how to round

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numbers to significant figures.

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Let's see five rules that will help us decide whether a digit in a number is significant

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or not.

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Here they are.

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All nonzero digits are significant, zeros sandwiched between nonzero numbers are significant,

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zeros at the end of a number are significant if they are after the decimal point, zeros

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in the beginning of a number are not significant, and if there is no decimal point, zeros at

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the end of a number are not significant.

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Let's apply these rules in a few examples.

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Since all nonzero digits are significant, the 2, the 5 and the 6 in 256 are all significant.

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Zeros sandwiched between nonzero numbers are significant, so all four numbers in 4007

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are significant.

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Zeros at the end of a number are significant if they are after the decimal point, so the

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zero at 23.810 is significant, along with all other numbers.

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However, zeros in the beginning of a number are not significant, so the three zeros in

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0.00248 are not significant, because of the first rule, 2, 4 and 8 are significant.

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If there is no decimal point, zeros at the end of a number, in other words, trailing

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zeros are not significant, so in 58,000, 5 and 8 are significant, however, the three

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zeros at the end are not.

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There is a way to show that these trailing zeros are significant, which you see in 3200,

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as it has a decimal point at the end, even though there are no numbers after the decimal

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point.

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So here, 3 and 2 are of course significant, and because of that decimal point, the two

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zeros are also significant.

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This notation is rarely used, so there is no need to worry about it too much.

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Let's see some examples about rounding numbers to significant figures.

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SF here stands for significant figures.

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In the first number, we have to round to two significant figures, so let's find the first

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two significant figures in this number.

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These are 9 and 8.

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So we have to round the 8 based on the next number, which is 6.

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This means that the 8 must be rounded up, so we get 9, 9, and the common mistake here

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is to leave this number as 99, assuming that the 6 and the 7 become nonsignificant numbers,

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so they can simply be omitted.

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However, it clearly doesn't make sense to round the number that is close to 10,000 down

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to 99.

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So instead of the 6 and the 7 disappearing, they simply become zeros at the end of the

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number, because these zeros are not significant.

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In the next number, the first three significant figures are 4, 7, and 0, so we will round

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based on the 8.

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We need to round the 0 between the 7 and the 8 up, so we get 4, 7, 1, and the last three

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digits become zeros, so we end up with three significant figures in this number.

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The first three significant figures in the next number are 5, 3, and 0.

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We have to round based on the other 0, so we will leave the 0 after the 3 as it is,

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and get 5.30.

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Remember that a 0 after the decimal point at the end of the number is a significant figure,

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so this number now has three significant figures.

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The first significant figure in the next number is 2, so we have to round based on the 8,

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so round up, which gives us 0.03.

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Since here adding more zeros at the end of the number would add more significant figures,

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in this case we just leave the zeros off.

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In the next number, the first four significant figures are 2, 6, 7, and 7, so we will round

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based on the 8, which means rounding up, and we get 2, 6, 7, 8, and the remaining five

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numbers are all zeros.

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For our last number, the first two significant figures are 4 and 7, so we have to round based

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on the 5, which gives us 4, 8, and for the remaining two numbers we will add two zeros.

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Note that originally this number had four significant figures because of the decimal

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point at the end of the number, but since night has to have two significant figures,

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so only the 4 and the 8 are significant, we will not put a decimal point at the end to

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show that the two zeros are not significant.

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Let's see what we have learned about identifying and rounding to significant figures.

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All non-zero digits are significant, zeros that are sandwiched between non-zero numbers

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are significant, if zeros are at the end of a number and after a decimal point, they

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are significant, zeros in the beginning of a number are not significant, and if there

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is no decimal point, zeros at the end of a number are not significant.

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This wraps up our discussion about significant figures.

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In the next video, we will learn about orders of magnitude.