What does invariant mean?
An invariant is something that remains unchanged despite changes in the conditions surrounding it. The word is especially important in mathematics, where an invariant is a quantity or property preserved under a specified transformation. This does not mean that the invariant never varies; it means that the relevant operation leaves it unaffected. For example, the sum of the parts of a number remains constant when the number is expressed in different ways. Outside mathematics, “invariant” is a formal word for a principle, condition, or attitude that persists over time or under changing circumstances. It can describe anything deliberately treated as fixed, though using it often implies that the stability is fundamental or difficult to alter. The related terms “invariance,” “invariable,” and “invariably” are common in scientific and mathematical writing. Together, they describe the preservation of a value, relationship, or principle despite transformations that might otherwise change it.
Unchanging or constant, especially under a specified condition, operation, or mathematical transformation.
"The total remains invariant when the sample is divided into smaller groups."
In mathematics, the conditions under which something remains invariant should be understood; the term does not imply that the quantity is independent of every possible change.
A quantity, property, principle, or other element that remains unchanged under a specified transformation or set of conditions.
"The researchers identified a useful invariant that characterized every stage of the process."
The noun is most common in mathematics and the physical sciences, but it can also refer more generally to an enduring principle.
The plural is used when referring to several preserved quantities, properties, or principles.
"The model has two useful invariants: total mass and total energy."
In mathematics, you can change the shape of the problem while its invariant stays the same.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of invariant
The word comes from Latin invariabilis, meaning “unchangeable,” formed from the prefix in-, meaning “not,” and variabilis, “liable to change.” English adopted it in the language of mathematics, where it came to denote quantities or properties preserved under specified transformations. Related English words such as variable, variance, and variation descend from the same Latin root associated with variation.
Related word forms
How invariant is actually used
In general use, the term is formal and often suggests something enduring and unalterable. Its mathematical sense is more technical: an invariant is invariant relative to a specified transformation, not necessarily absolutely unchanging in every context.
Easily confused with invariant
An invariant remains unchanged under a specified operation or transformation, whereas an invariable more generally means constant or unchanging.