Is Absolute Value a Useless Concept?

[bballwaterboy]

Is this mathematical concept EVER used in real life or "higher" levels of math?

I just find it to be a practically useless thing. It's like some guys sat around and invented this math concept just for the sake of it. Or, am I wrong?

 

[bballwaterboy]

Oh...and same for inequalities (e.g., <, >, etc.)...Are these useless too in math?

 

[Mentallic]

It's used all the time!

In real life, if you have a map of a city that has roads which go perfectly horizontal and vertical and want to figure out the distance traveled to get from point A to point B, when you travel horizontally from A = (a_x,a_y) to P = (p_x,p_y) then a_y = p_y because the points are horizontal to each other, and the distance between them is |a_x-p_x|. This means we don't have to worry about which point is on the left and which is on the right, the distance or magnitude will always turn out positive.

In higher maths, the absolute value is extended to deal with complex numbers as well. It is also needed a lot in calculus and various other disciplines.

And the same goes for inequalities. There are actually important formulas such as the triangle inequality that we can't do without. Inequalities are a powerful tool in helping mathematicians make conclusive statements about a problem.
You might end up learning about proof by mathematical induction. A lot of statements about inequalities will arise here.

 

[Mentallic]

There are infinite series that also converge to some well known irrational numbers. An example:

\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...

The infinite series jumps above and below \pi/4 forever, but gets closer and closer as you calculate more and more terms. A valid question would be, how many terms do we need to calculate in the series (give its value 'p') such that

\left|\frac{\pi}{4}-p\right|&lt;0.01

Which is essentially saying, when will it get to within 0.01 of the value of \pi/4. We also don't care if the value turns out to be above or below it, we just want it to be close enough (i.e. the distance between the values).

And there you have it, this problem alone has involved both absolute values and inequalities in a practical way.

 

[jedishrfu]

These concepts are used all the time in programming as an application of math to the real world.

As a curious aside, you can define absolute value via < operator

If x < 0 then let x = -x

Why you might use absolute value is when you need the magnitude of a value and its sign.

 

[pwsnafu]

Oh...and same for inequalities (e.g., <, >, etc.)...Are these useless too in math?

In higher mathematics we use Opial's inequality to establish the existence of solutions to differential equations. Inequalities are very much an active research area.
Further, in real life applications, I'd argue they are more important because we don't have infinite precision of measurement and we need to be bound our errors. Or we have a lot of data but can't do that many transactions (such as the market share portfolio), so we have to estimate.
Another example. You are managing a retail warehouse and need to optimize your inventory. Certain stock will sell at higher profit so you want those, but you don't want to completely fill up your warehouse because there may be new products around the corner. You use inequalities.

 

[Philip]

In geometry, absolute values are used extensively, in the implicit surface function, to define a shape with flat sides. The higher level application is higher dimensional shapes. Here are a few, in 2D and 3D, along with examples of curved only, having no absolute values.

For 2D

CIRCLE : √(x² + y²) - a²

TRIANGLE : ||x| - 2y| + |x| - a

SQUARE : |x - y| + |x + y| - aFor 3D

SPHERE : √(x² + y² + z²) - a²

TORUS : (√(x² + y²) - a)² + z² - b²

CONE : |√(x² + y²) - 2z| + √(x² + y²) - a²

CYLINDER : |√(x² + y²) - z| + |√(x² + y²) + z| - a

TETRAHEDRON : |||x| - 2y| + |x| - 2z| + ||x| - 2y| + |x| - a

TRIANGLE PRISM : |||x| - 2y| + |x| - z| + |||x| - 2y| + |x| + z| - a

SQUARE PYRAMID : ||x - y| + |x + y| - 3z| + |x - y| + |x + y| - a

UNIT CUBE : ||x - y| + |x + y| - 2z| +||x - y| + |x + y| + 2z| - aFurther on is an explosion of new possible shapes, in 4D, 5D, 6D, etc. I dare not post those, not quite yet.

 

[late347]

Absolute value makes sense for the function of let's say a triangle. But for variable of a triangle's physical quantity (etc length width...). Measurable quantity of object.

But negative magnitudes of length and width are impossible. It doesn't make sense in real life to have negative variable for length. It's just impossible. Length is zero or longer, period.

I guess the abovementioned concept reinforces the usefulness of absolute value idea...

Because length is impossible to be negative, we must use concept of absolute value? I am talking about actual geometric length euclidean.

Debt and money balance and these kind of quantities are possible to be negative. But physical length... lengths are always zero or positive.

Absolute value is like a sleight of hand (dirty trick) for manipulating functions like that... to prevent something from being negative. (Because length is always positive?)

Number line is length...

Length is always positive or zero...

Absolute value is always positive magnitude or zero magnitude (because absolute value is line between ttwo points). And this line exists in the realm of number line.

Line between two points sounds like length to me. But I could be wrong.

 

[Philip]

This is true, negative length does not apply. For all functions above, the value of 'a' and 'b' must be positive. Anything negative, and shape will not appear. Only the coordinate dimensions are using the absolute values, not the size parameter 'a' and/or 'b' . These functions are the long-hand way of using the Maximum function, by restricting two different implicit regions.

 

[Fredrik]

Is this mathematical concept EVER used in real life or "higher" levels of math?

I just find it to be a practically useless thing. It's like some guys sat around and invented this math concept just for the sake of it. Or, am I wrong?

It's used all the time. The absolute value can be used to define a function that gives you the distance between two arbitrary real numbers by d(x,y)=|x-y| for all x,y. Generalizations of this idea are used to define limits of sequences, among other things.

Oh...and same for inequalities (e.g., <, >, etc.)...Are these useless too in math?

They too are used all the time. The two most useful inequalities in math are the triangle inequality and the Cauchy-Schwartz inequality (which by the way involves an absolute value). The concept of "partially ordered set" (a set with a relation very much like the usual $\leq$) is essential in the study of the mathematical foundations of quantum mechanics.

 

[late347]

This is true, negative length does not apply. For all functions above, the value of 'a' and 'b' must be positive. Anything negative, and shape will not appear. Only the coordinate dimensions are using the absolute values, not the size parameter 'a' and/or 'b' . These functions are the long-hand way of using the Maximum function, by restricting two different implicit regions.

Dang those absolute value markings started giving me headache already :(

I think that I understand the basic concept of absolute value but I somehow was never good at complicated equations like that. (With many absolute value clauses)

It was probably lack of practice and confidence at manipulation with absolute value problems in school.

Your last sentence... I must confess that I didn't understand what you meant by that. I've never seen those kind of functions- not that I remember : (

multivariable calculus course was the last math course I took in high school. This kind of course would deal with such implicit functionspresumably?

I think that the course content was spread to two parts in my school. I only attended the first part of multivariablre calculus course. This could in part be a source of confusion for me.

 

[Philip]

Your last sentence... I must confess that I didn't understand what you meant by that. I've never seen those kind of functions- not that I remember : (

You'll find the max function in a program like mathematica. It's a different way to express a surface, as a different kind of equation.

This kind of course would deal with such implicit functionspresumably?

Probably, I'm not sure. I learned this from self study. I don't think I've seen those implicits anywhere else, other than the forum I helped develop them on.