octonionn.

Brit.

/ɒkˈtəʊnɪən/, U.S.

/ɑkˈtoʊniən/

Forms: 1800s– octonion, 1900s– octonian.

Origin: A borrowing from Latin, combined with an English element. Etymons: Latin octōnī , -ion suffix¹.

Etymology: < classical Latin octōnī eight each, eight at a time (see octonal adj.) + -ion suffix¹, after quaternion n. Compare earlier biquaternion n.In the form octonian after -ian suffix.

Mathematics.

1. A. McAulay's name for: a type of complex number in which the coefficients are quaternions, i.e. a number having the form x = p + ωq, where p and q are real quaternions, and ω is a scalar with ω² being equal to either 1 or 0; = biquaternion n. 2. Now rare.

ΚΠ

1895–6 A. McAulay in Proc. Royal Soc. 59 169 Octonions is a name adopted for various reasons in place of Clifford's Bi-quaternions.

1898 A. McAulay Octonions 19 An octonion is a quantity which requires for its specification and is completely specified by a motor and two scalars.

1910 Encycl. Brit. XXII. 722/2 Clifford makes use of a quasi-scalar ω, commutative with quaternions... He considers two cases, viz. ω² = 1 suitable for non-Euclidean space, and ω² = 0 suitable for Euclidean space; we confine ourself to the second, and will call the indicated bi-quaternion p + ωq an octonion... Clifford considers an octonion p + ωq as the quotient of two motors ρ + ωσ, ρ′ + ωσ′.

2. A type of hypercomplex number with eight coefficients, having the form x = x₀ + x₁e₁ + x₂e₂ + x₃e₃ + x₄e₄ + x₅e₅ + x₆e₆ + x₇e₇, where the x_n are real numbers and the e_n are imaginary units which, in conjunction with 1 and −1, form a closed, non-commutative, non-associative system under multiplication. Cf. octad n. 2a, octave n.² 2d.

ΚΠ

1908 Amer. Jrnl. Math. 30 60 The factoring of a complex system of eight units into a two-unit system and a four-unit system presents no new difficulties... The octonian system is easily seen to be the compound of the ordinary complex system and the quaternion system.

1969 Bull. Amer. Math. Soc. 75 980 (title) Octonion planes in characteristic two.

1991 Jrnl. Math. Physics 32 1383 (title) The Dirac equation in a non-Riemannian manifold... An analysis using the algebra of quaternions and octonions.

2002 New Scientist 9 Nov. 32/4 Octonions have revealed themselves as the most important system of all. That's because they are crucial to string theory.

This is a new entry (OED Third Edition, March 2004; most recently modified version published online March 2022).

单词	octonion
释义	octonionn. Brit. /ɒkˈtəʊnɪən/, U.S. /ɑkˈtoʊniən/ Forms: 1800s– octonion, 1900s– octonian. Origin: A borrowing from Latin, combined with an English element. Etymons: Latin octōnī , -ion suffix¹. Etymology: < classical Latin octōnī eight each, eight at a time (see octonal adj.) + -ion suffix¹, after quaternion n. Compare earlier biquaternion n.In the form octonian after -ian suffix. Mathematics. 1. A. McAulay's name for: a type of complex number in which the coefficients are quaternions, i.e. a number having the form x = p + ωq, where p and q are real quaternions, and ω is a scalar with ω² being equal to either 1 or 0; = biquaternion n. 2. Now rare. ΚΠ 1895–6 A. McAulay in Proc. Royal Soc. 59 169 Octonions is a name adopted for various reasons in place of Clifford's Bi-quaternions. 1898 A. McAulay Octonions 19 An octonion is a quantity which requires for its specification and is completely specified by a motor and two scalars. 1910 Encycl. Brit. XXII. 722/2 Clifford makes use of a quasi-scalar ω, commutative with quaternions... He considers two cases, viz. ω² = 1 suitable for non-Euclidean space, and ω² = 0 suitable for Euclidean space; we confine ourself to the second, and will call the indicated bi-quaternion p + ωq an octonion... Clifford considers an octonion p + ωq as the quotient of two motors ρ + ωσ, ρ′ + ωσ′. 2. A type of hypercomplex number with eight coefficients, having the form x = x₀ + x₁e₁ + x₂e₂ + x₃e₃ + x₄e₄ + x₅e₅ + x₆e₆ + x₇e₇, where the x_n are real numbers and the e_n are imaginary units which, in conjunction with 1 and −1, form a closed, non-commutative, non-associative system under multiplication. Cf. octad n. 2a, octave n.² 2d. ΚΠ 1908 Amer. Jrnl. Math. 30 60 The factoring of a complex system of eight units into a two-unit system and a four-unit system presents no new difficulties... The octonian system is easily seen to be the compound of the ordinary complex system and the quaternion system. 1969 Bull. Amer. Math. Soc. 75 980 (title) Octonion planes in characteristic two. 1991 Jrnl. Math. Physics 32 1383 (title) The Dirac equation in a non-Riemannian manifold... An analysis using the algebra of quaternions and octonions. 2002 New Scientist 9 Nov. 32/4 Octonions have revealed themselves as the most important system of all. That's because they are crucial to string theory. This is a new entry (OED Third Edition, March 2004; most recently modified version published online March 2022). < n.1895
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