Significant figures
Multiplying and Dividing with Significant Figures Multiplying and dividing with significant figures
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- Now that we have a decent understanding of
- how to figure out many significant figures that we are dealing with
- lets take a situation where,
- significant figures will or might become relevant.
- So let's say that I have a carpet here
- and I'm using a may be a meter stick to measure the carpet to nearest centimeter.
- And so I get the carpet as on to the the nearest centimeter I get to be 1.69 meters.
- So this is 9, obiviously this is to the nearest centimeter.
- This 9 hundredths of a meter as the same thing as 9 centimeters.
- And let's say I'm able to measure the width here as 2.09 meters.
- I use the same meter stick and you were to ask me:
- "Sal, what is the area of your carpet?"
- So - you know - just to the straight up calculation:
- the area is going to be the length times the width,
- so it would be 1.69 meters times 2.09 meters.
- We could do this by hand, but let me just get the calculator out
- to make things move along a little bit faster.
- And so we have 1.69 times 2.09 and they gives us 3.5321.
- Let me write that down 3.5321.
- So let me write this in a new color. So this gives us 3.5321,
- and we have a meters times a meters which gives us meters squared or square meters.
- I might very proudly to tell you that
- "Hey the area is 3.5321 square meters.
- And the problem here is that it when I give you this thing
- that has all of these numbers behind the decimal point
- and all of these are significant figures
- that implies that I had a really precise way of measuring the area.
- But in reality I only was able to measure the area to the nearest centimeter.
- So the way we would do this we don't...
- so I don't make it look like my measurement is more precise than it really is.
- So this calculation is derived from my measurements.
- I make sure that it has no more significant figures then either of the numbers that I multiplied.
- So in this situation I have 3 significant figures here,
- and over here I have 3 significant figures.
- And so in general you multiply or devide the significant figures in your product
- or the divisor, divident, the quotient , the quotient!
- The numbers, the significant figures in your product or your quotient can not be any more than
- the least number of significant digits,
- or whatever you are using to come up with that product quotient.
- So over here both of these have 3 significant figures.
- So I can only have 3 significant figures in my product.
- If one of these had 3 significant figures and this have 2 significant figures,
- I can have only have 2 significant figures in my product.
- So in order to be kind of legit here,
- I have to round this to 3 significant figures.
- I have to round it to 3 significant figures,
- I need to round it to the nearest hundredth here
- and so this 2 will round down, so we got
- this gets us to 3.53 meters squared.
- Now we're cool with the significant figures.
- Let's do another situation with division!
- Let's say that I'm...
- Let's say that I'm laying tiles down in my bathroom,
- and so the diagram will look very similar,
- and I measure... I measure the...
- I measure the width of my bathroom to be,
- let's say it is 10.1 feet now. 10.1 feet.
- And this is the precision that I'm able to measure.
- So I'm able to measure the 10th of a foot.
- And let's say that the length of my floor is ...
- The length of my floor - I just make up a number - is 12 point...
- And for whatever reason I was able to measure this with slightly higher precision
- So 12.07 feet. And let's say...
- Let's say that I have tiles... I have tiles.
- And the tile has an area, so someone else measure for me, it has an area of...
- Let's say that the area of this tile is 1.07 feet squared.
- What I want to do is to figure out how many tiles can fit on this bathroom floor.
- So what I would do I would figure out the area of this bathroom floor,
- and then divide by the area of the tiles.
- So the area of the bathroom floor, so floor area...
- Floor area is going to be equal to 10.1 feet times 12.07 feet,
- so that will give us, let's calculate it.
- It is 10.1 times 12.07 feet, it gives a 121.907.
- So this equal to - let's scroll over a little bit to the right -
- this is equal to - a little bit more over the right -
- this is equal to 121.907 feet squared, or squared feet.
- Now, we are not done with our calculation,
- but there might be a temptation right here,
- say look, I have 4 significant figures here,
- I have 3 significant figures over here,
- there would be a temptation to say,
- "Look, my area should not have more then 3 significant figures."
- And that temptation would be OK, if this is all you are looking for.
- If the final answer you are looking for was the area of the floor.
- But we are not done with our calculation,
- we want to figure out how many of these tiles will fit into this area.
- So the general rule of thumb,
- because you don't want to loose information,
- the general thumb is:
- don't round the significant figures until you are done with the calculation.
- Especially if you are just doing a bunch of multiplying and dividing
- because otherwise if you round here,
- you actually will introduce more error into your calculation than you'd want to.
- So what you do is you keep it as kind of a full number.
- Now you do the division. So let's do the division.
- So the tiles per floor... tiles per floor.
- I guess we could say my bathroom or tiles in the bathroom,
- tiles fitting in bathroom, on the floor of this bathroom.
- It would be the area of the bathroom, so 121.907 feet squared,
- divided by the area of the tile,
- divided by 1.07 feet squared.
- And ones again, let me get the calculator out.
- So we have 121.907 divided by 1.07,
- and you get this crazy thing with all of these digits,
- but this is going to be our final answer,
- so here we do care about significant figures.
- So tiles fitting in the bathroom,
- we get something that is actually just keeps going,
- so it's... Let me write this in a new color.
- We got 113.931775701 - and actually it just keeps going - feet squared.
- And this is the final answer.
- We cared about how many tiles will fit onto this bathroom floor,
- now the significant figures come into play.
- And the way to think about this:
- I have 4 significant figures over here, I have 2 significant figures here,
- I have 3 significant figures over here,
- and since we did just a bunch of multiplying and dividing, in general...
- Since we did a bunch of multiplying and dividing we have to have the minimum,
- whatever is the minimum significant figures of the things we computed with,
- that's how many significant figures we can have in our final answer.
- We'll make this clear: this has 2 significant figures, but this is 3: the 1, the 0 and the 1.
- So our final answer can only the 3 significant figures.
- 3 significant figures.
- So we need to round to the nearest foot. The next digit over is a 9, so we need to round up.
- So we are going to round up.
- So this would get us to 114. Actually this units aren't squared feet, this is in tiles.
- This is feet divided by feet, and so this is going to be 114, 114 tiles.
- Obviously it is not going to be exactly 114 tiles,
- but based on the precision of the measurements we have done,
- we can say 114 tiles.
- Now what I have just showed you right here what we multiply and divide measurements
- that has a certain number of significant figures.
- The general rule of thumb is whatever the minimum number of significant figures
- in any of the numbers calculated, that is how many significant figures...
- that is the least number is the number of significant figure
- in your final quotient or product or answer.
- When you do addition and subtraction, it's a little bit different and we will cover that in the next video.
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