Adding and subtracting negative numbers
Example: Adding negative numbers Adding Negative Numbers
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- Find the sum of -15 + (-46) + (-29)
- To do this, let's just first visualize
- what each of these numbers look like.
- So I'm going to draw a number line for each of them.
- -15 might look something like this.
- If this is 0, and let's say this is -15.
- I can represent -15 as an arrow that points from 0 to -15.
- The length of the arrow is the absolute value of this.
- It's the distance from 0.
- So the length here is 15
- and the negative says we are pointing to the left.
- So the absolute value is 15.
- That's the length of that arrow.
- Let's do the same thing for -46.
- Once again let me draw my number line.
- Put 0 is going to be right over there.
- and -46 is going to be someplace over here.
- Notice the same exact idea
- The distance between -46 and 0
- or another way to think that is the absolute value of -46.
- This distance right over here is going to be 46.
- And its direction is to the left.
- That's why we get to the number -46.
- The negative really tells you
- wether you're to the left or the right of 0.
- The absolute value says
- how far you are to the left or the right of 0.
- Finally let me do the same thing for -29.
- So once again, let me draw my number line.
- Use the yellow again.
- So got to draw my number line.
- Say this is 0, -29 is going to be roughly over here.
- Once again -29 is exactly 29 away from 0.
- So this length right over here is 29 and it is to the left
- that is why it is -29.
- If it was +29, it would be 29 to the right of 0.
- So we have represented all of these numbers
- and you can see what their absolute values are like.
- And now let's think what happens if you add them.
- One way to think about adding these numbers
- is as if you added these arrows.
- If you put this arrow on top of this arrow
- or to the left of this arrow
- If you started with this arrow leaves off,
- then you put this green arrow.
- And then you put the orange arrow.
- So let's do that. Let's draw that.
- It's going to be a longer arrow now.
- So we are going to start at 0.
- First we have -15.
- So we are going to move 15 spaces to the left,
- to get us to -15.
- Then we are going to go 46 spaces to the left,
- to get us to -15 + (-46)
- Let me draw that. Then we are going to go...
- We are going to figure out what that number is in a second.
- Then we're going to go 46 to the left. That's about that far.
- It's really just this arrow. I am now placing it.
- I am starting off where the -15 started
- or where the -15 left off.
- and then I am putting that arrow after that.
- And we don't know yet what number this gets us to.
- We are going to have to do a little math here.
- That's actually the point of this problem.
- But we know the length of this arrow is 46.
- It is 46 to the left.
- We know the length of this magenta arrow is 15.
- Then finally we have this orange arrow
- which we know has a length of 29,
- although it's 29 to the left.
- It looks like that.
- It has a length of 29.
- So when all is said and done, where are we?
- Where do we end up on our number line?
- Well, the total length that we are to the left,
- is going to be 15+46+29.
- But it's going to be to the left.
- So it's going to be negative.
- So we can really view this as
- Since all of these have the same sign
- Look this is the same thing
- as abs(-15)+abs(-46)+abs(-29)
- But then take the negative value of it.
- Let's just add that.
- 15+46+29 is equal to...let's see.
- 5+6 is 11. 11+9 is 20. Carry the 2.
- 2+1 is 3. 3+4 is 7. 7+2 is 9. So you get 90.
- So this entire length here is 90.
- If you add up the arrows, you get a 90.
- But it is not 90 to the right.
- If it was 90 to the right, this would all be positive.
- Then we'll just get positive 90.
- But this is 90 to the left.
- So when you add these guys up,
- you don't get to positive 90.
- You get to negative 90.
- So one way to think about it is, these are all the same sign.
- So this is going to be equal to
- the -( abs(-15)+abs(-46)+abs(-29))
- and the reason why we are writing this.
- this might look fancy
- but this is really just the length of this purple arrow.
- This absolute value of 46,
- that's really just the length of the green arrow, 46.
- This is just the length of the orange arrow.
- So this is 15+46+29, gives us 90.
- But it's to the left. So that is why it is -90.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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