It's quite easy to show that they are exactly: e^(2pik/n) with 1<=r<=n and such that r and n have no common factor.

Thus, there are phi(n) primitive roots of unity, where phi is the Euler Phi function.

KANJI: GEN hara (primitive, original, meadow, field, plain)

ASCII Art Representation:

 
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Character Etymology:

The radical that surrounds two sides of the character is the radical for cliff while the radical inside is a variant of spring (the watery kind). Thus, cliffside spring, and by extension source/origin and primary/primitive/natural/plain/moor.

A Listing of All On-Yomi and Kun-Yomi Readings:

on-yomi: GEN
kun-yomi: hara

Nanori Readings:

Nanori: ta baru ra wata wara

English Definitions:

1. GEN: original, primitive.
2. hara: field; plain, prairie, tundra, moor, wilderness.
3. hara(ppa): plain.
4. gen-: original, primitive, primary, fundamental; raw.

Character Index Numbers:

New Nelson: 652
Henshall: 107

Unicode Encoded Version:

原

Unicode Encoded Compound Examples:

原人 (genjin): primitive man.
原文 (genbun): original text.
草原 (kusahara): grassy plain.
原生動物 (gensei doubutsu): protozoa.
原子 (genshi): atom.
原水爆 (gensuibaku): atom and hydrogen bombs.
原子物理学 (genshi butsurigaku): atomic physics.

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Prim"i*tive (?), a. [L. primitivus, fr. primus the first: cf. F. primitif. See Prime, a.]

1.

Of or pertaining to the beginning or origin, or to early times; original; primordial; primeval; first; as, primitive innocence; the primitive church.

Milton.

2.

Of or pertaining to a former time; old-fashioned; characterized by simplicity; as, a primitive style of dress.

3.

Original; primary; radical; not derived; as, primitive verb in grammar.

Primitive axes of coordinate Geom., that system of axes to which the points of a magnitude are first referred, with reference to a second set or system, to which they are afterward referred. -- Primitive chord Mus., that chord, the lowest note of which is of the same literal denomination as the fundamental base of the harmony; -- opposed to derivative. Moore (Encyc. of Music). -- Primitive circle Spherical Projection, the circle cut from the sphere to be projected, by the primitive plane. -- Primitive colors Paint., primary colors. See under Color. -- Primitive Fathers Eccl., the acknowledged Christian writers who flourished before the Council of Nice, A. D. 325. Shipley. -- Primitive groove Anat., a depression or groove in the epiblast of the primitive streak. It is not connected with the medullary groove, which appears later and in front of it. -- Primitive plane Spherical Projection, the plane upon which the projections are made, generally coinciding with some principal circle of the sphere, as the equator or a meridian. -- Primitive rocks Geol., primary rocks. See under Primary. -- Primitive sheath. Anat. See Neurilemma. -- Primitive streak ∨ trace Anat., an opaque and thickened band where the mesoblast first appears in the vertebrate blastoderm.

Syn. -- First; original; radical; pristine; ancient; primeval; antiquated; old-fashioned.

© Webster 1913.

Prim"i*tive, n.

An original or primary word; a word not derived from another; -- opposed to derivative.

© Webster 1913.