# RealAbs RealAbs[x]

gives the absolute value of the real number x.

# Details • RealAbs is also known as modulus.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• RealAbs[x] is effectively equivalent to Piecewise[{{x,x>=0}},-x].
• RealAbs is continuous and differentiable everywhere except at the origin.
• • RealAbs[x] is left unevaluated if x is not a numeric quantity.
• RealAbs can be used with Interval and CenteredInterval objects. »
• RealAbs automatically threads over lists.

# Examples

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## Basic Examples(4)

Positive numbers:

Negative numbers:

Plot RealAbs over a subset of the reals:

Derivative of RealAbs:

Indefinite integral:

## Scope(29)

### Numerical Evaluation(6)

Evaluate numerically:

RealAbs remains unevaluated for imaginary numbers:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

RealAbs threads elementwise over lists and matrices:

RealAbs can be used with Interval and CenteredInterval objects:

### Specific Values(5)

Values of RealAbs at particular points:

Value at zero:

Values at infinity:

Evaluate symbolically:

Find values of for which :

Substitute in the value of to create pairs:

Visualize the result:

### Visualization(3)

Plot :

Plot RealAbs along with its first derivative:

Plot RealAbs in three dimensions:

### Function Properties(10)

RealAbs is defined only for real inputs:

The range of RealAbs is the non-negative reals:

RealAbs is an even function:

RealAbs is not an analytic function:

It has a singularity at the origin but no discontinuities:

RealAbs is neither nondecreasing nor nonincreasing:

RealAbs is not injective:

RealAbs is not surjective:

RealAbs is non-negative:

RealAbs is convex:

### Differentiation and Integration(5)

First derivative:

Obtain an equivalent expression using the definition of derivative:

The function Abs of complex variables is not differentiable:

Higher derivatives:

Compute the indefinite integral using Integrate:

Verify the antiderivative:

Definite integral:

More integrals:

## Applications(7)

Color plots according to RealAbs:

Solve a differential equation with RealAbs:

Compute the Fourier cosine series of RealAbs:

Compute a series involving RealAbs:

Solve an equation involving RealAbs:

Prove an inequality containing RealAbs:

Simplify expressions containing RealAbs:

## Properties & Relations(8)

RealAbs is defined only for real numbers:

Abs is defined for complex numbers:

RealAbs is a differentiable function:

Abs is not differentiable:

RealAbs is an integrable function:

Abs is integrable only for real arguments:

RealAbs is idempotent:

Definite integration:

Integral transforms:

Convert into Piecewise:

Denest:

## Neat Examples(1)

Form nested functions involving RealAbs: