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Why is the absolute value function or modulus function $|x|$ used ? What are its uses?

For example the square of a modulus number will always be positive, but why is it used when for example the square of any number whether positive or negative is always positive ? For example, $X^2$, will give a positive number whether negative or positive where $X$ is any number positive or negative.

In the context of real numbers the absolute value of a number is used in many ways but perhaps very elementarily it is used to write numbers in a canonical form. Every real number $ane 0$ is uniquely equal to $pm left |aright|$. So if we define the sign function $scolon mathbb Rsetminus{0}to {+,-}$ given by $s(a)=+$ if $a>0$ and $s(a)=-$ if $a<0$, then: for all $ane 0$ in $mathbb R$ we have $a=sign(a)cdot left | a right |$. In a sense this is a way to build all the reals from the positive ones. This is all just a special case of the polar representation of complex numbers, a representation of utmost importance.

One use of it is to define the distance between numbers. For example, in Calculus, you may want to say "the distance between $x$ and $y$ is less than $1$". The way to write that mathematically is $|x-y|<1$. And you want to write it mathematically so you can work with it mathematically.

The notation $vert xvert$ for absolute value of $x$ was introduced by Weierstrass in 1841:

K. Weierstrass, Mathematische Werke, Vol. I (Berlin, 1894), p. 67.

Quoted from [1]

...There has been a real need in analysis for a convenient symbolism for
"absolute value" of a given number, or "absolute number," and the two
vertical bars introduced in 1841 by Weierstrass, as in $vert zvert$, have met with wide adoption;...

Extra information: Absolute is from the Latin absoluere, "to free from"; hence suggesting, to free from its sign.

[1] Florian Cajori, A History of Mathematical Notations (Two volumes bound as one), Dover Publications, 1993.

My take on a usage example of absolute value:
$$
min(x,y)=frac{1}{2}(|x+y|-|x-y|)
$$
$$
max(x,y)=frac{1}{2}(|x+y|+|x-y|)
$$

Because both of them are useful.

You explicitly mentioned the square function. Therefore, I want to give some examples. The main idea is that the non-differentiability of $|cdot|$ is useful in minimization problem.

Estimators

We know that the arithmetic mean $hat{mu}=sum_{i=1}^n x_i$ gives

$$min_{mu} ,(x_i-mu)^2$$

but it is less well-known that the median gives

$$min_{mu} , |x_i-mu|.$$

Signal Processing

Let's use image processing as an example. Suppose $g$ is a given, noisy image. We want to find some smoother image $f$ which looks like $g$.

The Harmonic L$^2$ minimization model solves

$$-bigtriangleup f + f = g $$

and it turns out to be equivalent to solving a minimization problem:

$$min_{f} ,(int_{Omega} (f(x,y)-g(x,y))^2 dxdy + int_{Omega} |nabla{f(x,y)}|^2 dxdy).$$

An enhanced version is the ROF model. It solves

$$min_{f} ,(frac{1}{2} int_{Omega} (f(x,y)-g(x,y))^2 dxdy + lambda int_{Omega} |nabla{f(x,y)}| dxdy).$$

Notice that for appropriate $lambda$, these two models only differ by a square. Another remark is that $|cdot|$ gives the Euclidean norm when the argument is a vector. However, the idea still applies since the norm is non-zero

Model Selection

In classical model selection problem, we are given a set of predictors and a response (in vector form). We want to decide which predictors are useful. One way is to choose a "good" subset of predictors. Another way is to shrink the regression coefficients.

The classical regression model solves the following minimization problem:

$$min_{beta_0,...,beta_p} sum_{i=1}^n (y_i-beta_0-sum_{j=1}^p beta_j x_{ij})^2$$

The Ridge Regression solves the following:

$$min_{beta_0,...,beta_p} sum_{i=1}^n (y_i-beta_0-sum_{j=1}^p beta_j x_{ij})^2+lambda sum_{j=1}^p {beta_j}^2$$

, so that larger $beta_j$ gives penalty.

Another version is Lasso, which solves

$$min_{beta_0,...,beta_p} sum_{i=1}^n (y_i-beta_0-sum_{j=1}^p beta_j x_{ij})^2+lambda sum_{j=1}^p |beta_j|.$$