The topic of this video is estimation, we will discuss how to use estimation when measuring
something and when working with graphs and we'll talk about how to round when carrying
out calculations.
When you measure something and read your measuring instrument, you never actually get an exact
value.
Let's say we are measuring the length of this line using a ruler.
When writing down the measured value, we will need to use estimation.
Looking at the ruler, I would say that the length of this line is approximately 5.3 cm,
assuming that the ruler is in centimeters.
Of course your estimation might be slightly different, which is what often happens when
we estimate something.
In some cases, you might have to make an estimation based on a graph.
This graph shows how the speed of an object varies with time.
The area under this graph is a distance traveled by this object.
No worries if you haven't learned this yet.
Here we are simply interested in estimating the area under the graph.
We can do this by working with the small squares.
We have approximately 42 full squares and about 9 squares that we will count as half squares.
Of course, some of these half squares are more and some less than half of a square.
But overall, these differences approximately balance each around and give us an area of 4.5
squares, which adds up to a total of 46.5 squares.
Since the horizontal side of each square represents 2 seconds and the vertical side represents
1 meter per second.
The area of 1 square is the product of these two, which gives us 2 meters.
So since we have 46.5 squares and 1 square represents 2 meters, the distance traveled
by this object during these 18 seconds is 2 times 46.5, which is 93 meters.
Rounding is a type of estimation, so let's see how do we round after we carry out
a calculation.
Then we add or subtract.
We need to round to the same decimal place as in the least accurate number.
For multiplication and division, we round to the same number of significant figures as
in the least accurate number.
And when we raise a number to a power or take its root, we need to round to the same number
of significant figures as in this original number.
Let's try to understand these rules a bit better through some examples.
This 1,700 plus 247 is equal to 2,947.
The last significant figure in the 2,700 is here and in 247 it's here.
This 7 is in the 100's place and this 7 is in the 1's place.
Because of this, 2,700 is the least accurate number in the calculation.
So we will round our result to the 100.
This gives us 2,900.
1.2278 minus 0.036 is 1.1918.
In the first number, 8 is the last significant figure and in the second number, it is 6.
8 is at the 10,000's place and 6 is at the 1000's place.
So 0.036 is the least accurate.
Therefore we will round to the 1000's place in our answer, which gives us 1.192.
Let's see multiplication and division.
21.0 times 0.05 gives us 1.05.
21.0 has 3 significant figures since the 0 after the decimal place is a significant figure.
And 0.05 has only 1 significant figure, which is the 5.
The 0's before the 5 to not count as significant figures.
Now we have to round to the least amount of significant figures that we see in our numbers.
This is 1 significant figure in 0.05.
So 1.05 rounded to 1 significant figure is simply 1.
5,400 divided by 107 is equal to 50.4672 and the number continues on infinitely.
5,400 has 2 significant figures, the 0's at the end do not count as significant figures.
While 107 has 3 significant figures, remember that 0's sandwiched between non-zero numbers count as significant figures.
This means that we have to round our answer to 2 significant figures, and this will give us simply 50.
To show that the 0 in the 50 is significant, we will put a decimal point after the 0.
Using the decimal point this way is somewhat unusual, and you will rarely come across this situation.
Let's finish with powers and routes.
15 to the power of 3 is 3,375 and 15 has 2 significant figures.
This is what we will round to, so we get 3,481 under the 4th route is 3, and 81 has 2 significant figures.
Not 3 is already to 1 significant figure, but we still have to try to give it to 2 significant figures,
which will mean that we are going to write it as 3.0.
Again, this is a relatively rare case that you will not see to often.
Let's summarize what we have learned about estimation.
When we measure something using measuring equipment, we never get an exactly accurate result,
so we need to use estimation when reporting the measured value.
In physics, we often encounter graphs.
In our working with graphs, we often have to estimate a certain property or value on this graph.
In our example, we estimated the area under a graph.
Rounding is a type of estimation.
We discussed how to round after carrying out a calculation.
So when we round after an addition or subtraction, we have to round to the same decimal place
as in the least accurate number.
After a multiplication or division, we round to the same number of significant figures
as the number of significant figures in the least accurate number.
And finally, when we raise a number to a power or take its root,
we round to the same number of significant figures as in this original number.
This wraps up our discussion of estimation, which is the final video in sub-topic 1.1.
In the first video of sub-topic 1.2, we will discuss random and systematic errors.