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The topic of this video is estimation, we will discuss how to use estimation when measuring

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something and when working with graphs and we'll talk about how to round when carrying

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out calculations.

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When you measure something and read your measuring instrument, you never actually get an exact

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value.

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Let's say we are measuring the length of this line using a ruler.

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When writing down the measured value, we will need to use estimation.

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Looking at the ruler, I would say that the length of this line is approximately 5.3 cm,

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assuming that the ruler is in centimeters.

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Of course your estimation might be slightly different, which is what often happens when

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we estimate something.

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In some cases, you might have to make an estimation based on a graph.

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This graph shows how the speed of an object varies with time.

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The area under this graph is a distance traveled by this object.

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No worries if you haven't learned this yet.

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Here we are simply interested in estimating the area under the graph.

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We can do this by working with the small squares.

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We have approximately 42 full squares and about 9 squares that we will count as half squares.

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Of course, some of these half squares are more and some less than half of a square.

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But overall, these differences approximately balance each around and give us an area of 4.5

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squares, which adds up to a total of 46.5 squares.

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Since the horizontal side of each square represents 2 seconds and the vertical side represents

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1 meter per second.

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The area of 1 square is the product of these two, which gives us 2 meters.

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So since we have 46.5 squares and 1 square represents 2 meters, the distance traveled

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by this object during these 18 seconds is 2 times 46.5, which is 93 meters.

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Rounding is a type of estimation, so let's see how do we round after we carry out

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a calculation.

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Then we add or subtract.

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We need to round to the same decimal place as in the least accurate number.

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For multiplication and division, we round to the same number of significant figures as

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in the least accurate number.

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And when we raise a number to a power or take its root, we need to round to the same number

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of significant figures as in this original number.

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Let's try to understand these rules a bit better through some examples.

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This 1,700 plus 247 is equal to 2,947.

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The last significant figure in the 2,700 is here and in 247 it's here.

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This 7 is in the 100's place and this 7 is in the 1's place.

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Because of this, 2,700 is the least accurate number in the calculation.

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So we will round our result to the 100.

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This gives us 2,900.

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1.2278 minus 0.036 is 1.1918.

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In the first number, 8 is the last significant figure and in the second number, it is 6.

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8 is at the 10,000's place and 6 is at the 1000's place.

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So 0.036 is the least accurate.

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Therefore we will round to the 1000's place in our answer, which gives us 1.192.

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Let's see multiplication and division.

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21.0 times 0.05 gives us 1.05.

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21.0 has 3 significant figures since the 0 after the decimal place is a significant figure.

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And 0.05 has only 1 significant figure, which is the 5.

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The 0's before the 5 to not count as significant figures.

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Now we have to round to the least amount of significant figures that we see in our numbers.

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This is 1 significant figure in 0.05.

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So 1.05 rounded to 1 significant figure is simply 1.

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5,400 divided by 107 is equal to 50.4672 and the number continues on infinitely.

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5,400 has 2 significant figures, the 0's at the end do not count as significant figures.

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While 107 has 3 significant figures, remember that 0's sandwiched between non-zero numbers count as significant figures.

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This means that we have to round our answer to 2 significant figures, and this will give us simply 50.

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To show that the 0 in the 50 is significant, we will put a decimal point after the 0.

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Using the decimal point this way is somewhat unusual, and you will rarely come across this situation.

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Let's finish with powers and routes.

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15 to the power of 3 is 3,375 and 15 has 2 significant figures.

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This is what we will round to, so we get 3,481 under the 4th route is 3, and 81 has 2 significant figures.

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Not 3 is already to 1 significant figure, but we still have to try to give it to 2 significant figures,

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which will mean that we are going to write it as 3.0.

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Again, this is a relatively rare case that you will not see to often.

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Let's summarize what we have learned about estimation.

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When we measure something using measuring equipment, we never get an exactly accurate result,

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so we need to use estimation when reporting the measured value.

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In physics, we often encounter graphs.

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In our working with graphs, we often have to estimate a certain property or value on this graph.

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In our example, we estimated the area under a graph.

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Rounding is a type of estimation.

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We discussed how to round after carrying out a calculation.

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So when we round after an addition or subtraction, we have to round to the same decimal place

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as in the least accurate number.

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After a multiplication or division, we round to the same number of significant figures

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as the number of significant figures in the least accurate number.

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And finally, when we raise a number to a power or take its root,

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we round to the same number of significant figures as in this original number.

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This wraps up our discussion of estimation, which is the final video in sub-topic 1.1.

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In the first video of sub-topic 1.2, we will discuss random and systematic errors.