# Definition:Minor (Euclidean)

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## Definition

Let $a, b \in \R_{>0}$ in the forms:

- $a = \dfrac \rho {\sqrt 2} \sqrt {1 + \dfrac k {\sqrt {1 + k^2} } }$
- $b = \dfrac \rho {\sqrt 2} \sqrt {1 - \dfrac k {\sqrt {1 + k^2} } }$

where:

- $\rho$ is a rational number
- $k$ is a rational number whose square root is irrational.

Then $a - b$ is a **minor**.

In the words of Euclid:

*If from a straight line there be subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the squares on them added together rational, but the rectangle contained by them medial, the remainder is irrational; and let it be called a***minor***.*

(*The Elements*: Book $\text{X}$: Proposition $76$)

## Terms of Minor

The **terms** of $a - b$ are the elements $a$ and $b$.

### Whole

The real number $a$ is called the **whole** of the minor.

### Annex

The real number $b$ is called the **annex** of the minor.