# Definition:Conjugate

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## Disambiguation

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**Conjugate** may refer to:

- Geometry:
- The conjugate of an angle $\theta$ is the angle $2 \pi - \theta$, that is, the angle that makes a full angle with it.
- The conjugate axis of a hyperbola.
- Conjugate diameters in a central conic.

- Group theory:
- The conjugate of an element or subset $X$ of a group $G$ by a given element $a$ is defined as $a X a^{-1}$.
- Generated normal subgroup of a subset of a group, also known as the conjugate closure of such a subset.

- Number theory:
- The conjugate of the quadratic irrational $r + s \sqrt n$ is $r - s \sqrt n$.

- Complex analysis:
- The (complex) conjugate of a complex number $a + i b$ is $a - i b$.

- Quaternions:
- The conjugate quaternion of a quaternion $a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ is $a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k$.