Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Journal of Theoretics Vol.3-6 Comparative Study of Quaternions and Mixed Numbers Author: Md. Shah Alam Department of Physics Shahjalal University of Sciences and Technology Sylhet, Bangladesh <[email protected]> Abstract: The general quaternion is X = a + bi + cj + dk where a, b, c, and d are real and the mixed number which is the sum of a scalar and a vector (i.e. α = x + A where x is a scalar quantity and A is a vector quantity). In this paper we compare the mixed number with quaternion. Keywords: quaternions, mixed numbers. Introduction The algebra of quaternions discovered [1] by the Irish mathematician Sir William Rowan Hamilton. Quaternions are “hypercomplex” [2] numbers with “imaginary” units i, j, k which satisfy the relations i2 = j2 = k2 = -1, ij = k, jk = i, ki = j, ij + ji = 0, ik + ki = 0, jk + kj = 0 . The general quaternion is X = a + bi + cj + dk where a,b,c,d are real. Mixed number [3] α as a sum of a scalar x and a vector A: α = x + A Quaternion Algebra Definition: For X = α0 + α1i + α2j + α3k , Y = β0 + β1i + β2j + β3k the addition is defined [4] as X+Y = (α0 + β0) + (α1 + β1)i + (α2 + β2)j + (α3 + β3)k . Definition: For X = α0 + α1i + α2j + α3k , Y = β0 + β1i + β2j + β3k the multiplication is defined [4] as X.Y = (α0 + α1i + α2j + α3k).(β0 + β1i + β2j + β3k) = (α0β0 - α1β1 - α2β2 - α3β3) + (α0β1 + α1β0 + α2β3 - α3β2 )i + (α0β2 + α2β0 + α3β1 - α1β3)j + (α0β3 + α3β0 + α1β2 - α2β1)k . [Using the relations: i2 = j2 = k2 = -1, ij = k, jk = i, ki = j, ij + ji = 0, ik + ki = 0, jk + kj = 0 .] Definition: For X = α0 + α1i + α2j + α3k in Q the adjoint of X, denoted by X *, is defined [4] by X * = α0 - α1i - α2j - α3k . Lemma 1. The adjoint in Q satisfies [4] (1) X * * = X 1 (2) ( δX + γY ) * = δX * + γY * (3) (XY) * = Y * X * for all X, Y in Q and all real δ and γ . Definition: If X ∈ Q then the norm of X, denoted by N(X), is defined [3] by N(X) = XX* . Lemma 2. For all X, Y ∈ Q, N(XY) = N(X)N(Y) . Definition: Real Quaternions consist of all matrices of the form [5] 1 0 0 1 a1 + bi + cj + dk = a i 0 +b 0 1 -1 0 +c 0 a + bi c + di -c + di a - bi -i 0 i i 0 + d = . Definition: The first known noncommutative division algebra was Hamilton’s quaternions.[6] Definition: The product of real quaternoins are associative, i.e. (xy)z = x(yz) . Mixed Number Algebra Definition: For α = x + A and β = y + B the addition is defined as α + β = (x + y) + (A + B) . Definition: For α = x + A and β = y + B the product is defined as αβ = (x + A)(y + B) = xy + A.B + xB + yA + iAxB . Definition: The adjiont of α, denoted by α*, is defined by α* = x – A . Lemma 1. The adjoint in α satisfies (i) α* * = α (ii) (α + β )* = α* + β* (iii) (αβ) * = β* α* . 2 Proof. (i) For α = x + A, α* = (x + A)* = x – A ∴α* * = (x – A)* = x + A = α Proof. (ii) For α = x + A and β = y + B, α + β = (x + y) + (A + B) ∴ (α + β )* = [(x + y) + (A + B)]* = (x + y) - (A + B) = (x – A) + (y – B) = α* + β* Proof. (iii) For α = x + A and β = y + B αβ = (x + A)(y + B) = xy + A.B + xB + yA + iAxB (αβ)* = [(x + A)(y + B)]* = [(xy + A.B) + (xB + yA + iAxB)]* = (xy + A.B) – (xB + yA + iAxB) ................................... (1) Now β* α* = (y - B)( x - A) = (yx + B.A – xB – yA + iB×A) = [(xy + A.B) – ( xB + yA + iA×B)] ................................. (2) From equation (1) and (2) we can write (αβ) * = β* α* . Definition: The norm of a Mixed number α, denoted by N(α), is defined by N(α) = αα* = (x + A) (x - A) = x2 – A2 – xA + xA - iA×A = x2 – A2 . Lemma 1. For all α and β N(αβ) = N(α)N(β) . Proof. For α = x + A and β = y + B αβ = (x + A)(y + B) = xy + A.B + xB + yA + iAxB ∴N(αβ) = (xy + A.B)2 - (xB + yA + iAxB)2 = x2y2 + 2xyA.B +(A.B)2 - (xB + yA)2 - 2(xB + yA).i(AxB) - i2(AxB)2 x2y2 + 2xyA.B +(A.B)2 - (xB + yA)2 + (AxB)2 = x2y2 + 2xyA.B +(A.B)2 - x2B2 - 2xyA.B - y2A2 + (AxB)2 = x2y2 + 2xyA.B +(A.B)2 - x2B2 - 2xyA.B - y2A2 - (AxB)2 = x2y2 + (A.B)2 - x2B2 - y2A2 - (AxB)2 = x2y2 - x2B2 - y2A2 + (A.B)2 - (AxB)2 = x2y2 - x2B2 - y2A2 + A2B2 = [(A.B)2 - (AxB)2 = A2B2 ] =(x2 – A2)(y2 – B2) = N(α)N(β) ∴N(αβ) = N(α)N(β) Definition: The product of Mixed numbers are associative, i.e. (αβ)γ = α(βγ) . 3 References: [1] Edwin Hewitt & Karl Stomberg, Real and abstract Analysis (Springer – Verlag – 1965). [2] R. Courant & D. Hilbert, Methods of Mathematical Physics, Volume I (Interscience Publishers, Inc., New York – 1953). [3] Md. Shah Alam, “Study of Mixed Number”, The Proceedings of the Pakistan Academy of Sciences, 37(1) 119-122 (2000). [4] I.N. Herstein, Topics in Algebra, Blaisdell Publishing Company – 1964. [5] Thomas W. Hungerford, Abstract Algebra An Introduction (Saunders College Publishing – 1990). [6] Louis H. Rowen, Ring Theory (Academic Press, Inc. – 1991) Journal Home Page © Journal of Theoretics, Inc. 2001 4