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# Absolute value as distance between numbers

## Thuyết minh video

let's say that I have two numbers on a number line so let me draw a little quick number line right over here and the two numbers on my number line that I care about the number a and the number represented by B here the way I've drawn it B is to the right of a on our number line and by our convention B is going to be greater than a so if I wanted to figure out the distance between a and B what is this distance going from a and draw a straight line here this distance going from A to B so this distance right over there how would I figure it out well I could just take the larger of these two numbers which is going to be B and then subtract out the smaller so I subtract out a and I'll be left with this distance and this will give me a positive value when I'm going to dip when I want a distance I just think in terms of a positive value how far apart are these two things but I was only able to know to do B minus a because I knew that B was greater than a this is going to give me a positive value what if I knew that a was greater than B well then I would do the other way around so let me draw that again let me let me draw another number line right over here and in this world in this world I'm going to make a greater than B so this is B that is a and if I wanted to calculate the distance between B and a here well now I would now I would take the larger of the two a remember I want the positive distance here and then I would subtract out the smaller I would do a minus B well so here I did B minus a here I did a minus B but what if I didn't know which one was greater if I didn't know whether B or a was greater what could I do well what you could do is just take either a minus B or B minus a and take the absolute value and if you do that it doesn't matter if you take B minus a or a minus B it turns out that regardless of whether a is greater than B or B is greater than a or they're equivalent that the absolute value of a minus B is equivalent to the absolute value of B minus a and this is equivalent either of these expressions is the distance between these numbers and encourage you to play around with the negative see if you can factor out some negatives and think about the absolute value and it'll actually make a lot of sense why this is true in another video I might do a little bit more of a rigorous justification for it but I think the important thing for this video is to see that this is actually that this is actually true so let's say we're in a world let's let's get a number line out and let's look at some examples so let's say that we want to figure out the distance between between let's say negative 2 the distance between negative 2 and positive 3 so we can look at the number line and figure out what that distance is to go from negative 2 to positive 3 or the distance between them we see is 1 2 3 4 5 actually we draw a straighter line here this distance right over here this distance right over here is equal to is equal to 5 you see it right over here 1 2 3 4 5 or you can go you have to go 5 backwards to go from 3 to negative 2 but let's see that what I just wrote actually applies right over here so if we took negative 2 to be our a and 3 to be our B then we could write this as the absolute value of negative 2 minus minus 3 minus 3 what is this going to be equal to well this is going to be equal to negative 2 minus 3 is negative 5 so it's the absolute value of negative 5 and so this indeed equals 5 so notice I subtracted the larger number from the smaller number I got a negative value but that took the absolute value of it that gave me the actual distance between these two numbers now what if I did it the other way around what if I took what if I took 3 minus negative 2 so it's going to be 3 the absolute value of 3 let me do the blue color the absolute value of 3 minus 3 minus and in parentheses I'll write the negative 2 negative 2 now if you take if you subtract a smaller number from a larger number you should get a positive value so the absolute value sign here is just kind of extra you don't really need it and let's verify that that's true this is going to be three minus negative 2 that's the same thing as 3 plus positive 2 or 5 so there's just going to be the absolute value 5 which of course is equal to 5 so hopefully this makes you feel good that if you want the distance between two numbers you subtract one from the other and it doesn't matter which order you do it you could subtract 3 from negative 2 or negative 2 from 3 be careful with the negative symbols here and then take the absolute value and then that is going to give you the distance between these two numbers and this is super important because later in your mathematical careers you're going to see a math professor just say oh you know I care about the distance between two variables you know a and B so the distance is a minus B and then later they might write it like this and then to realize that these are actually the same thing and these are they're going to give the same expression the same value and they represent the distance between these numbers