What is the point of "x" in math?

The title here is a sort of double meaning. I know the point of x. But lets make x the variable in the question so various math questions for asking some of the experts here. Thus first question:

WHAT IS THE POINT OF ABSOLUTE VALUE IN ALGEBRA?

I know what it is, distance from zero or value without regards to +/-. Might be some other ways to define it that I never heard of. It all means the same thing. |-7|=|7|. I doubt the teacher spent 10 minutes explaining it because even those who struggled with algebra as we teach it could understand it.

But.....what on earth is the point of it?

I have used algebra in life. Despite not one algebra teacher ever even once explaining why or how I would. However, I have not once used absolute value or ever seen a practical reason for it.

Is there a use for absolute value outside of a textbook?

pshh! c'mon man everyone knows x marks the spot.

Good for testing "close enough." I used it in some statistical modeling where a calculated value was sometimes different than the theoretical value by a very small amount due the limitations of representing floating point numbers in a processor. I added code to the program to test whether the absolute value of the difference between the calculated and theoretical value was small enough to be ignored. In other words, "close enough."

I'll give it a shot: I think it's more of a space saver than anything else. Saying |X| < 1 as opposed to -1 < X < 1. For example, the portion of a normal distribution curve within 1 standard deviation.

There's also a use in complex mathematics; the absolute value of a complex number is the distance on the complex plane from the origin.
|3 + 4i| = 5.

Measures distance. It doesn't make sense to say you're negative 7 miles from something.

In doing a variance calculation you could technically use it, although the standard definition of variance uses a square of the distance to avoid the negative numbers.

Quote: AZDuffman

The title here is a sort of double meaning. I know the point of x. But lets make x the variable in the question so various math questions for asking some of the experts here. Thus first question:

WHAT IS THE POINT OF ABSOLUTE VALUE IN ALGEBRA?

I know what it is, distance from zero or value without regards to +/-. Might be some other ways to define it that I never heard of. It all means the same thing. |-7|=|7|. I doubt the teacher spent 10 minutes explaining it because even those who struggled with algebra as we teach it could understand it.

But.....what on earth is the point of it?

I have used algebra in life. Despite not one algebra teacher ever even once explaining why or how I would. However, I have not once used absolute value or ever seen a practical reason for it.

Is there a use for absolute value outside of a textbook?

I think so. All it means is the difference between two things, and it doesn't matter which is larger or smaller, i.e. which you measured first. Does this watermelon weigh half a pound more than that one, or does that one weigh half a pound less than this one? Doesn't matter. The difference is +8oz. The difference is -8oz. Same thing.

I use absolute values all the time (okay, not ALL the time).

Maybe you want to find the difference in points in a football game. Team_A_Score - Team_B_Score = difference?

Well if it was 40 to 25, then the difference is 15, which makes sense.

What if it's 25 to 40, difference is -15, which doesn't make sense. Especially if you want to extrapolate from that data.

If you're trying to find the average difference between scores in a game, using ABS you'll get a real answer. Without using ABS, your averages should tend toward zero.

I honestly don't know or forget. Algebra itself is mostly just to weed people out from higher learning. Then in college they use calculus.

Quote: RS

If you're trying to find the average difference between scores in a game, using ABS you'll get a real answer. Without using ABS, your averages should tend toward zero.

This is one of the most important points. Absolute value is about examining the magnitude of numbers, not the sign. Whether it's differences between scores in a game, distances between cities, or changes in anything else, the absolute value allows you to properly measure the magnitude of the change without needing to consider in which direction that change went. For example, if you had three changes:

a) 4 -> 10 (+6)
b) 5 -> 2 (-3)
c) 6 -> 9 (+3)

and you took the average, it's +2. But if you just care about how much the change was, regardless of direction, it's +4. The second value, +4, is a better representation of the actual average change to the three initial values.

It's also a very convenient shortcut to something you can do with far more effort. For the three values above, you can calculate the average by simply taking the absolute value of the difference of each pair and averaging them. Without absolute value, you first need to check which is the greater value, subtract from it the smaller, and then use that in the summation. Notationally it's far shorter to write
f(x,y) = |x-y|
than
if x>y, f(x,y) = x-y, else f(x,y) = y-x

Quote: onenickelmiracle

I honestly don't know or forget. Algebra itself is mostly just to weed people out from higher learning. Then in college they use calculus.

Not necessarily. A lot of learning (IMO) has more to do with the understanding and logic behind "why it is what it is". It also opens up your mind / critical thinking, so when you're in an unknown situation, you can figure out HOW to solve it. The "how" meaning what formula do you use and the process.

A great example (IMO) of this is ThatDonGuy. Lots of times there'll be a relatively tough question asked or a math riddle. The real smart math guys who know all the calculus and trig stuff answer one way. Then TDG responds with the simplest and most elegant answer....like the dude who shoved a stick in the ground then figured out how far the sun is from earth (or the size of earth or something).


I think people use a lot more math day to day than they realize. Of course, some will use more and others less, some more advanced math and others less advanced.