Where does root collide?

In algebra a root is a value of the variable that makes an equation or polynomial equal zero, so that substituting it satisfies the equation. For the polynomial x squared minus 4 the roots are 2 and negative 2, the numbers where the graph crosses the axis. The same word names the inverse operation, as in the square root or nth root, the number that raised to a given power returns the original. Dummit and Foote treat roots as the solutions a field is built to contain. The referent here is a number and the object is a formal expression, not a living thing and not a sound. A root in this meaning is found by calculation, not grown or played, and it exists whether or not anyone plants or sounds anything.[1]

Unit

A number that makes an expression equal zero

Recovery means

Not applicable

Measured by

Determined by solving the equation or factoring the polynomial, and confirmed by substituting the value back in

In morphology a root is the core part of a word that carries its basic lexical meaning and cannot be broken into smaller meaningful parts. In unfriendliness the root is friend, and prefixes and suffixes attach around it. Aronoff, and Katamba after him, treat the root as the base to which affixes and inflection apply, kept distinct from the stem and from the finished word. The referent is a unit of language, a morpheme, not a number and not an organ. Two words share a root when they descend from the same core form, which is a claim about meaning and history, not about zeros that solve an equation or organs that anchor a plant.[2][3]

Unit

A morpheme carrying a word's core meaning

Recovery means

Not applicable

Measured by

Identified by segmenting a word and stripping affixes down to the smallest form that still carries the core meaning

In tonal harmony the root is the note on which a chord is built and after which the chord is named, the foundation of its stacked thirds. A C major chord has C as its root whether or not C is the lowest sounding note. Rameau's Traite de l'harmonie introduced the basse fondamentale, the idea that every chord has a generating root even when the chord is inverted, and Piston carries that principle into modern teaching. The referent is a pitch inside a harmonic structure, not a number, a morpheme, or an organ. The root organizes how the chord functions, and it can be present in theory while a different note sits in the bass.[4][5]

Unit

The generating note of a chord

Recovery means

Not applicable

Measured by

Read off the chord's stacked thirds, naming the note the chord is built from even when that note is not in the bass