If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Identify and order absolute values

Use a number line to find absolute value and then order absolute values from least to greatest.   Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• An absolute value always makes a number positive. Is there a way to make a number always negative?
(159 votes)
• imagine a function like this:
abs(x)
it will always give the positive value of x,
if you put a minus in front of it, it will always be negative:
-abs(x)

If you plot the two functions you will see that they mirror each other through the x-axis. When two functions mirror each other like that, they are said to be conjugate.
For example, when x=2, abs(x) will give 2, so:
(x,y)=(2,2)
the conjugate of two is (the second term is flipped):
(x,y)=(2,-2)
and this is also what you would expect from:
-abs(2)= -(+2) = -2
so that:
(x,y)=(2,-2)
(87 votes)
• Why is the absolute value of a number always positive?
(36 votes)
• As meanderingly said, absolute value basically represents the distance a number is from zero on the number line. If you can imagine, nobody would say "My car went negative thirty miles today," so you wouldn't use negative "distances" on the number line either. That's why absolute values are always positive.

I hope this helps!
(38 votes)
• Does l-1l = 1?
(10 votes)
• Yes, the absolute value of any negative number is the positive of that number.
(8 votes)
• Smallest to largest order, is that ascending or descending order ?
(5 votes)
• Smallest to largest is ascending order, because the numbers are ascending, or rising.
(5 votes)
• What happens if the number is -|-3|?
(3 votes)
• Learn to "read" the expression given as looking for the OPPOSITE of the absolute value of -3.
therefore the answer is -3 since the absolute value of -3 is 3, thus the opposite of 3 is -3
(2 votes)
• When do you use the absolute value rule?
(6 votes)
• Is there a -0?
(4 votes)
• -0 is the same as just 0. It's the only number which doesn't change when you put a negative sign in front, and that's because it's the only number which is neither negative nor positive.
(2 votes)
• Why can't there be negative distances?
(2 votes)
• even if you walk backwards, you still went somewhere...a negative distance would imply that you are traveling into yourself or (others think) backwards in time...it doesn't make sense. so its always positive and left in terms of absolute value within equations. It might be taken for granted and your "just suppose to know" this, but it needs to be positive. If you go somewhere, you go somewhere and travel some positive distance.
(6 votes)
• Like on do you have to have lines?
(3 votes)
• |x| or |-x| is the correct way to show that you must find the absolute value of that number.
(3 votes)
• So, basically an absolute value is making a number positive?
(3 votes)
• That's true except for zero which isn't positive or negative. Absolute value is a tool used to prevent expressions from being negative - for instance in finding distances. Since we don't consider distances negative, we can surround the expression for finding distance with absolute value bars to ensure our distance is not negative. It can still be zero or positive.
(2 votes)

## Video transcript

We're told to list in order from smallest to largest. So each of these quantities, it looks like we have expressions inside of absolute value signs. And just as a bit of review, absolute value just means your distance from 0. Or another way to think about it is, if it's a negative number inside of the absolute value sign, it becomes positive. If it's already positive, it stays positive. So let's think about these numbers. So the first one is the absolute value of 5. How far is 5 away from 0? Well, it's 5 away from 0. So it's equal to 5. So I were to actually draw a number line, you would see that. 0 is here, 5 is over here, this distance right there is 5. So the absolute value of 5 is 5. Now, the next quantity they want us to figure out is the absolute value of 9 minus 7. Well, this is the same thing as the absolute value of 2. 9 minus 7 is 2. And once again, 2 is just 2 units away from 0, so it's just 2. If you have a positive value in the absolute value sign, it just is itself. The absolute value of 2 is 2. Then we have the absolute value of 5 minus 15. Well, that's going to be the same thing as the absolute value. 5 minus 15 is negative 10, so it's the same thing as the absolute value of negative 10. Now, there's two ways you can think about it. If it's a negative number inside of the absolute value sign, it just becomes the positive version of it, so it just becomes 10. Another way to think about it is, if you had negative 10, you would plot it-- it would be out here someplace, negative 10, we'd have to extend the number line-- it is 10 to the left of 0. That's what the absolute value is telling us. Then we have the absolute value of 0. Well, 0 is just 0 away from the number line. Absolute value of 0 is just 0. That is just 0. That's just right there. It has no distance from the origin. And then finally, we have the absolute value of negative 3. That's 3 to the left of 0, or you can just think of it as getting rid of this negative sign, so it is equal to 3. So, now that we've expressed them all as just simple integers, let's list them in order from smallest to largest. So of all of these values, which is the smallest? This one is the smallest, the absolute value of 0. So let me write that down, absolute value of 0. What's the next smallest? What's the next one? Since we have a 2 there, this is the next smallest, right there. And that original expression was the absolute value of 9 minus 7, so the absolute value of 9 minus 7. And then what's the next smallest? We have this 3, and then we have a 5 and a 10. So the next smallest is this 3, right here. That 3, and that original expression was the absolute value of negative 3. Then we have this 5 over here, which is just the absolute value of 5. And then finally, we have this 10 over here, which was the absolute value of 5 minus 15. And we are done.