Download Comparative Study of Quaternions and Mixed Numbers

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
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(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* .
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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) .
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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)
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