What is the use of Absolute value in math? why does it exist?

Question

I don`t get it.

yeah, I know this:

| x - a | > d

means the distance from "x" to "a" is less than "d"

but whats the use of this?

we can get the distance without suggesting a new concept...

can you give a practical use of this???

Answer 1

To find the distance from a number to zero.

Answer 2

I this form it wouldn't be considered a new concept, just a way to write it down or express it concisely. We can of course get the distance with a function such as abs(x,a) but that is not in the algebraic form. If we write it as you have above, we can manipulate the expression using the rules of algebra.

The absolute value is not just used for distance, more generally it is used to find the magnitude of something, such as the volume of a shape whose coordinates don't exclusively lie in the positive quadrant (in 2D space of course), etc. It's really quite important.

Answer 3

When you simply want know the difference between two numbers without actually knowing which one is bigger, then you use absolute value

For example: lets say :

absolute value if "a" is denoted by abs(a)

so abs(13-20) = 7 = abs(20-13). It simply removes the -ve sign from the answer you get.

Ok so what's the big deal in that? It looks like an unnecessary fancy term right?

actually NO

you must know that square root of a negative number does not exist in the set of real numbers.

So in a particular situation, you want to compute square root of the distance between two points on the x axis. Lets say we know that the velocity of an particle moving on the x axis is proportional to the distance between a reference point and the point at which the particle is at that moment.Say, one point is (a,0) and the other is (b,0)

then that quantity is expressed as sqrt(abs(a-b)) or sqrt(abs(b-a))

sqrt(x) stands for square root of x.

abs() is just a compact way of representing a logical operations. That's what "functions" are. They are just a compact way of representing "well defined" operations.

And as for the example you posted "| x - a | > d", you can see that you are simply interested in the distance between x and a, not caring that x is greater or a is greater.

So after this "long" post, I would simply summarise.

The practical use is "compactness". The notation saves a lot of space on paper.

You will not appreciate its importance until you do some hardcore problems in math, no matter how much I explain. Right now just take my word (I have done those tough problems).

If you're good with that, its cool. If not, then WAIT. It will come to you with a lot of HIGH LEVEL problem solving

Answer 4

Absolute value is very important. You can't notice where it's being used.

For example:

Your driving on car forward 1 km, the distance you've travelled is 1 km. But if you drive your car in reverse, you haven't travelled   -1 km, but rather the absolute value of -1 km.

A lot of things that involve negative numbers in daily applications are just ASSUMED to be positive and THOSE are the practical uses of absolute value functions.

Note:

Negative numbers are strictly a philosophical construct. They don't "exist" in a realistic.Thus, we use absolute value type mentalities to keep it all positive and simple and logical.