Incongruence

Incongruence refers to a lack of alignment or compatibility between two or more elements. The term is used across fields to describe mismatches between beliefs and reality, expectations and outcomes, or structures that do not fit together.

In mathematics, incongruence is most often discussed in modular arithmetic. Two integers a and b are congruent modulo n if their difference a−b is divisible by n; if not, a and b are incongruent modulo n. For example, 7 and 3 are incongruent modulo 5 because 7−3 = 4 is not divisible by 5.

In geometry, figures are congruent when one can be transformed into the other by rigid motions such as rotation, reflection, or translation. Incongruence means the figures do not have the same size and shape. For instance, circles of radii 3 and 4 are incongruent, whereas two triangles with identical side lengths are congruent.

In psychology, incongruence is a key concept in person-centered theory, particularly in the work of Carl Rogers. It describes a mismatch between an individual’s self-concept and their actual experiences. Greater incongruence is associated with distress, while therapeutic approaches such as unconditional positive regard aim to reduce this gap by supporting alignment between experience and self-perception.

Incongruence can also describe broader misfits between expectations, communications, or system components. Across disciplines, the notion highlights the importance of coherence and alignment for functioning and well-being.