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# Absolute value as distance between numbers

In this video, we think about what |a-b| really means, and we verify that |a-b| = |b-a| by looking at an example.

## Want to join the conversation?

• why 3-(-2) = 5? Could you please give me a example in real world to explain this equation?
• i don't get it how does -5 equal 5
• Absolute value basically measures how far the number is from zero. If you think about a number line, -5 is the same distance away from 0 as 5 is.
Putting an absolute value on something isn't really saying that they are the same number, but it's saying that they are the same distance away from 0.
• The practice & exam problems bringing in fractions and decimals is really throwing me off. I looked through the comments, but I don't see any examples that actually address this nor does Sal give any examples.
• With fractions and decimals solving for absolute value is the same process. If I have l 3/4 l and
l -7/6 l being compared then the absolute value of 7/6 is greater.
• Can you help me further understand this concept? I don't really understand how to work it out.
• Every number that are inside the I I becomes positive.
ex)1. I -6 I = I 6 I
2. I -5-2 I = I -7 I = I 7 I
• Is there a special name for this phenomenon of a formula |a-b| = |b-a|?
• This phenomenon is called commutativity.
So the property |a-b| = |b-a| means that the distance between two numbers is a commutative operation.
• this is not really a question but how does he draw straight lines perfectly??
• i still don't get how the absolute value of a-b is the same as the absolute value of b-a. Please help.
• Consider the following:
You have one value, a, that is 3; so a = 3
You also have another, b, that is 7; so b = 7
So a - b = 3 - 7= -4, while b - a = 7 - 3 = 4

When you take the absolute value of either equation (|a - b| or |b - a|), you can see that both result in an answer of 4, as the negative result of a - b (-4) still has a positive absolute value (4).

Let me know if this clarifies, or if you have any further questions!
• What's the distance between 15,17,and 12 and what does a-b mean?
• I literally did the practice questions of "Absolute value as distance between numbers" and they make no sense at all. It literally showed : "Which of the following expressions represent the distance between -4/3 and 1/3?" It then showed me a number line and placed dots of where the fractions value was supposed to be. It then showed me 3 answers but I can only pick one:

A. |-1/3 -(-4/3)|
B. |1/3-4/3|
C. None of the above

I picked B and got it wrong. Turns out, it was C. Can someone please explain? Does absolute value affect negative fractions??
• The distance between points x and y on the number line is always the absolute value of their difference, that is, either of the two equivalent expressions |x-y| or |y-x|. This is true regardless of whether x and y are negatives, fractions, etc.

Substituting -4/3 for x and 1/3 for y, we find that the distance between -4/3 and 1/3 can be written as either of the two equivalent expressions |-4/3-1/3| or |1/3-(-4/3)|. The value of either expression is 5/3, but the values of options A and B are each 1, not 5/3. So the answer is indeed C.

Alternatively, if you visualize -4/3 and 1/3 on the number line, you can see that they are 4/3 + 1/3 = 5/3 apart (because -4/3 and 1/3 are on opposite sides of 0). Doing the calculations for options A and B would give 1 in either case, eliminating options A and B.

Have a blessed, wonderful day!