Editing Invariant (mathematics) (section)

== Examples ==
A simple example of invariance is expressed in our ability to [[counting|count]]. For a [[finite set]] of objects of any kind, there is a number to which we always arrive, regardless of the [[total order|order]] in which we count the objects in the [[set (mathematics)|set]]. The quantity—a [[cardinal number]]—is associated with the set, and is invariant under the process of counting.

An [[List of mathematical identities|identity]] is an equation that remains true for all values of its variables. There are also [[List of inequalities|inequalities]] that remain true when the values of their variables change.

The [[distance]] between two points on a [[number line]] is not changed by [[addition|adding]] the same quantity to both numbers. On the other hand, [[multiplication]] does not have this same property, as distance is not invariant under multiplication.

[[Angle]]s and [[ratio]]s of distances are invariant under [[Scaling (geometry)|scalings]], [[Rotation (mathematics)|rotation]]s, [[Translation (geometry)|translation]]s and [[Reflection (mathematics)|reflection]]s. These transformations produce [[Similarity (geometry)|similar]] shapes, which is the basis of [[trigonometry]]. In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching). The sum of a triangle's interior angles (180°) is invariant under all the above operations.  As another example, all [[circle]]s are similar: they can be transformed into each other and the ratio of the [[circumference]] to the [[diameter]] is invariant (denoted by the Greek letter π ([[pi]])).

Some more complicated examples:
* The [[real part]] and the [[absolute value]] of a [[complex number]] are invariant under [[complex conjugation]].
* The [[tricolorability]] of [[Knot (mathematics)|knots]].<ref>{{Cite web |last=Qiao |first=Xiaoyu |date=January 20, 2015 |title=Tricolorability.pdf |url=https://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_problem_solving_w2015/Tricolorability.pdf |url-status=dead |archive-url=https://web.archive.org/web/20240525145629/https://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_problem_solving_w2015/Tricolorability.pdf |archive-date=May 25, 2024 |access-date=May 25, 2024 |website=Knot Theory Week 2: Tricolorability }}</ref>
* The [[degree of a polynomial]] is invariant under a linear change of variables.
* The [[topological dimension|dimension]] and [[homology group]]s of a topological object are invariant under [[homeomorphism]].<ref>{{harvtxt|Fraleigh|1976|pp=166–167}}</ref>
* The number of [[fixed point (mathematics)|fixed points]] of a [[dynamical system]] is invariant under many mathematical operations.
* Euclidean distance is invariant under [[orthogonal transformation]]s.
* [[Area]] is invariant under [[linear map]]s which have [[determinant]] ±1 (see {{slink|Equiareal map|Linear transformations}}).
* Some invariants of [[projective transformation]]s include [[collinearity]] of three or more points, [[concurrent lines|concurrency]] of three or more lines, [[conic section]]s, and the [[cross-ratio]].<ref>{{harvtxt|Kay|1969|pp=219}}</ref>
* The [[determinant]], [[Trace (linear algebra)|trace]], [[eigenvectors]], and [[eigenvalues]] of a [[linear endomorphism]] are invariant under a [[change of basis]]. In other words, the [[spectrum of a matrix]] is invariant under a change of basis.
* The principal invariants of [[tensors]] do not change with rotation of the coordinate system (see [[Invariants of tensors]]).
* The [[singular-value decomposition|singular values]] of a [[matrix (mathematics)|matrix]] are invariant under orthogonal transformations.
* [[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 after the addition of a constant.
* The [[fixed point (mathematics)|fixed points]] of a transformation are the elements in the [[domain of a function|domain]] that are invariant under the transformation. They may, depending on the application, be called [[symmetry|symmetric]] with respect to that transformation. For example, objects with [[translational symmetry]] are invariant under certain translations.
*The integral <math display="inline">\int_M K\,d\mu</math> of the Gaussian curvature <math>K</math> of a two-dimensional [[Riemannian manifold]] <math>(M,g)</math> is invariant under changes of the [[Riemannian metric]] ''<math>g</math>''.  This is the [[Gauss–Bonnet theorem]].

===MU puzzle===
The [[MU puzzle]]<ref>{{Citation | last1 = Hofstadter | first1 = Douglas R. | title = Gödel, Escher, Bach: An Eternal Golden Braid | publisher = Basic Books | year = 1999 | orig-year = 1979 | isbn = 0-465-02656-7 | url-access = registration | url = https://archive.org/details/gdelescherbachet00hofs }}
Here: Chapter I.</ref> is a good example of a logical problem where determining an invariant is of use for an [[impossibility proof]]. The puzzle asks one to start with the word MI and transform it into the word MU, using in each step one of the following transformation rules:

# If a string ends with an I, a U may be appended (''x''I → ''x''IU)
# The string after the M may be completely duplicated (M''x'' → M''xx'')
# Any three consecutive I's (III) may be replaced with a single U (''x''III''y'' → ''x''U''y'')
# Any two consecutive U's may be removed (''x''UU''y'' → ''xy'')

An example derivation (with superscripts indicating the applied rules) is
:MI →<sup>2</sup> MII →<sup>2</sup> MIIII →<sup>3</sup> MUI →<sup>2</sup> MUIUI →<sup>1</sup> MUIUIU →<sup>2</sup> MUIUIUUIUIU →<sup>4</sup> MUIUIIUIU → ...

In light of this, one might wonder whether it is possible to convert MI into MU, using only these four transformation rules. One could spend many hours applying these transformation rules to strings. However, it might be quicker to find a [[Predicate (mathematical logic)|property]] that is invariant to all rules (that is, not changed by any of them), and that demonstrates that getting to MU is impossible. By looking at the puzzle from a logical standpoint, one might realize that the only way to get rid of any I's is to have three consecutive I's in the string. This makes the following invariant interesting to consider:

:''The number of I's in the string is not a multiple of 3''.

This is an invariant to the problem, if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. Looking at the net effect of applying the rules on the number of I's and U's, one can see this actually is the case for all rules:

:{| class=wikitable
|-
! Rule !! #I's !! #U's !! Effect on invariant
|-
| style="text-align: center;" | 1 || style="text-align: right;" | +0 || style="text-align: right;" | +1 || Number of I's is unchanged. If the invariant held, it still does.
|-
| style="text-align: center;" |  2 || style="text-align: right;" | ×2 || style="text-align: right;" | ×2 || If ''n'' is not a multiple of 3, then 2×''n'' is not either. The invariant still holds.
|-
| style="text-align: center;" |  3 || style="text-align: right;" | −3 || style="text-align: right;" | +1 || If ''n'' is not a multiple of 3, ''n''−3 is not either. The invariant still holds.
|-
| style="text-align: center;" |  4 || style="text-align: right;" | +0 || style="text-align: right;" | −2 || Number of I's is unchanged. If the invariant held, it still does.
|}

The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it  will not be afterwards either.

Given that there is a single I in the starting string MI, and one is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three).