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In mathematics, an invariant is an aspect of something that remains the same when other aspects of the thing change.

Mathematicians say that a quantity is invariant "under" a transformation; some economists say it is invariant "to" a transformation.

The most fundamental example of invariance is expressed in our ability to count. For a finite collection of objects of any kind, there appears to be a number to which we invariably arrive regardless of how we count the objects in the set. The quantity – a cardinal number – is associated with the set and is invariant under the process of counting.

An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the value of their variables change.

Another simple example of invariance is that the distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other hand multiplication does not have this property so distance is not invariant under multiplication.

Angles and ratios of distances are invariant under scalings, rotations, translations and reflections. These transformations produce similar shapes, which is the basis of trigonometry. All circles are similar therefore they can be transformed into each other and the ratio of the circumference to the diameter is invariant and equal to pi.

Some more complicated examples:

• The real part and the absolute value of a complex number are invariant under complex conjugation.
• The degree of a polynomial is invariant under linear change of variables.
• The dimension of a topological object is invariant under homeomorphism.
• The number of fixed points of a dynamical system is invariant under many mathematical operations.
• Euclidean distance is invariant under orthogonal transformations.
• Euclidean area is invariant under a linear map with determinant 1 (see Equi-areal maps).
• The cross-ratio is invariant under projective transformations.
• The determinant, trace, and eigenvectors and eigenvalues of a square matrix are invariant under changes of basis.
• Invariants of tensors.
• The singular values of a matrix are invariant under orthogonal transformations.
• The Midline Theorem is an invariant, where line xy is 1/2 of base bc.
• Lebesgue measure is invariant under translations.
• The variance of a probability distribution is invariant under translations of the real line; hence the variance of a random variable is unchanged by the addition of a constant to it.
• The fixed points of a transformation are the elements in the domain invariant under the transformation. They may, depending on the application, be called symmetric with respect to that transformation. For example, objects with translational symmetry are invariant under certain translations.
• The Fatou set and Julia set generated by a complex function f are completely invariant under f.
• The integral

of the Gaussian curvature K of a 2-dimensional Riemannian manifold (M,g) is invariant under changes of the Riemannian metric g. This is the Gauss-Bonnet Theorem.

See also

• Erlangen program
• Invariant in physics
• Invariant estimator in statistics
• Invariant theory
• Symmetry in mathematics
• Topological invariant
• Invariant differential operator
• Invariant measure
• Invariant subspace
• Mathematical constant

External links

• The Beginning of Mathematics from cut-the-knot