Download mathematical tools of mixed number algebra

MATHEMATICAL TOOLS OF MIXED NUMBER ALGEBRA
Md. Shah Alam
Department of Physics, Shahjalal University of Science and Technology, Sylhet,
Bangladesh.
M. H. Ahsan
Department of Physics, Shahjalal University of Science and Technology, Sylhet,
Bangladesh.
Mushfiq Ahmad
Department of Physics, University of Rajshahi, Rajshahi, Bangladesh.
ABSTRACT
Mixed number is the sum of a scalar and a vector. The quaternion can also be written as
the sum of a scalar and a vector but the product of mixed numbers and the product of
quaternions are different. Mixed product is better than the quaternion product because
Mixed product is directly consistent with Pauli matrix algebra and Dirac equation but the
quaternion product is not directly consistent with Pauli matrix algebra and Dirac
equation. Mixed numbers are successfully applied in Quantum mechanics and special
relativity. We can develop a new most general Lorentz Transformation using mixed
number algebra, which we called Mixed number Lorentz Transformation. Most general
Lorentz Transformation has some limitations but Mixed number Lorentz Transformation
has free from those limitations. In this paper we have represented the complete
mathematical tools of mixed number algebra.
Key Words: Mixed Number, Mixed Product, Quaternion Product
PACS No: 02.90. + p
1. INTRODUCTION
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Definition 1
Mixed number[1,2]  is the sum of a scalar x and a vector A like quaternions[3-5]
i.e.  = x + A
Lemma 1
Two Mixed numbers  and  are equal if x = y and A = B
Definition 2
The addition of two Mixed numbers  and  is
 +  = (x + y) + (A + B)
…………… ……………… (1)
where  = x + A and  = y + B
Definition 3
Mixed number  = x + A = ( x + A1i + A2j + A3k ) where x, A1, A2 and A3 are scalar
values and i, j and k are unique Mixed numbers
Lemma 1: The properties of the unique Mixed numbers are
ii = jj = kk = 1 i.e. i2 = 1, j2 = 1, k2 = 1 and
ij = i k, jk = i i,
ki = i j, ji =  i k, kj =  i i, ik =  i j where i = (1).
Lemma 2: They have the following multiplication table
1
i
j
k
1
1
i
j
k
i
i
1
ik
 ij
j
j
 ik
1
ii
k
k
ij
 ii
1
Definition 4
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The product of two Mixed numbers  and  is defined as
 = (x + A)(y + B) = xy + A.B + xB + yA + iAB
……….….……………… (2)
Lemma 1
The product of Mixed numbers is associative
i.e. () = ()
……….….……………… (3)
where  = z + C is an another Mixed number
Proof:
() = {(x + A)(y + B)}( z + C)
= (xy + A.B + xB + yA + iAB)( z + C)
= (xy + A.B)z + (xB + yA + iAB).C + (xy + A.B)C + z(xB + yA + iAB)
+ i (xB + yA + iAB)  C
= xyz + (A.B)z + xB.C + yA.C + i(AB).C + xyC + (A.B)C + zxB + yzA
+ i z(AB) + ix (B  C) + i y(A  C)  (AB)  C
= xyz + (A.B)z + xB.C + yA.C + i(AB).C + xyC + (A.B)C + zxB + yzA
+ i z(AB) + ix (BC) + i y(AC)  B(A.C) + A(B.C)
= x{yz + B.C + yC + zB + i (BC)} + A {yz + (B.C)} + yA.C + i y(AC)
+ (A.B)z + i z(AB) + iA.(BC) +{ A(BC)}
[ using (A.B)C  B(A.C) =  A(BC)]
= x{yz + B.C + yC + zB + i (BC)} + A {yz + B.C + yC + zB + i (BC)}
[ using yA.C + i y(AC) = y AC, (A.B)z + i z(AB) = z AB,
iA.(BC)  A(BC) = i{A.(BC) + i A(BC)} = i A(BC)]
= (x + A) {yz + B.C + yC + zB + i (BC)}
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= (x + A){(y + B)(z + C)}
= ()
or, () = ()
Definition 5
Taking x = y = 0 we get from equation (2)
A  B = A.B + iA  B
……….….……………… (4)
[The symbol  is chosen for Mixed product.]
This product is called Mixed product of vectors [6]. Mixed product is directly consistent
with Pauli matrix algebra and Dirac equation but the quternion product is not directly
consistent with Pauli matrix algebra and Dirac equation [7]. Therefore, Mixed product is
better than quaternion product [7].
Definition 6
If  = x + A and  = y + B then, the adjoint of , denoted by *, is defined by
* = x – A
……….….……………… (5)
Lemma 1
The adjoint in  satisfies
(i) ** = 
..……….….…………….(6)
(ii) ( + )* = * + *
……….….……………….. (7)
(iii) ()* = **
………….….……………… (8)
Lemma 2
Scalar() = ( + *)/2
……….….……………… (9)
Vector() = (  *)/2
……….….……………… (10)
Definition 7
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The Norm of a Mixed number  is
N() = * = (x + A)(x – A)
= (x2 – A.A + xA – xA – iAA)
= (x2 – A2)
Where A is the magnitude of the vector A, i.e. A =  A 
or, N() = (x2 – A2)
……….….……………… (11)
It fulfils the requirement that
……….….……………… (12)
N() = N()N()
Definition 8
The inverse of a Mixed number  1 is given by
 1 = (x  A)/(x2  A2)
……….….……………… (13)
Proof:
  1 = 1
or,  1 = 1/ = 1/(x + A)
= (x  A)/ {(x + A) (x  A)}
= (x  A)/(x2  A2)
or,  1 = (x  A)/(x2  A2)
2. PROPERTIES OF MIXED NUMBER
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2.1 Mixed number forms a ring
Mixed number satisfies the following conditions
(i) Additive Associativity: For all , ,   S, ( + ) +  =  + ( + ).
(ii) Additive Commutativity: For all ,   S, ( + ) = ( + ).
(iii) Additive identity: There exists an element 0  S such that For all   S,
0 +  =  + 0.
(iv) Additive inverse: For every   S there exists an element   S such that
 + () = () +  = 0.
(v) Multiplicative associativity: For all , ,   S, ( )  =  ( ).
(vi) Left and right distributivity: For all , ,   S, ( + ) =  +  and
( + ) =   + 
Therefore, Mixed number forms a ring.
2.2 Mixed number forms a division algebra or skew field
Mixed number satisfies the following conditions also
(i) Multiplicative identity: there exists an element 1  S not equals to zero
such that for all   S, 1 = 1 = 
(ii) Multiplicative inverse: For every   S not equals to 0, there exists  1  S
such that   1 =  1 = 1.
Therefore, Mixed number forms a division algebra or skew field.
2.3 Mixed number does not form a field
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The multiplication of Mixed numbers is non-commutative i.e.    .
Therefore, Mixed number does not form a field.
3. APPLICATIONS OF MIXED NUMBER
3.1 Applications of Mixed number in Quantum Mechanics
We could be derived [8] the displacement operator, vector differential operator, Angular
momentum operator and Klein-Gordon equation in terms of Mixed number. Therefore we
can concluded that Mixed number is successfully used in Quantum Mechanics.
3.2 Applications of Mixed number in Special Relativity
Using Mixed number algebra we had found a new most general Lorentz Transformation
which, we called Mixed Number Lorentz Transformation [9]. The Lorentz sum of
velocities for the Mixed Number Lorentz Transformation is associative, the space
generated by the Mixed Number Lorentz Transformation satisfies isotropic property, and
Lorentz sum of velocities for the Mixed Number Lorentz Transformation has group
property without rotation. The Lorentz sum of velocities for the most general Lorentz
Transformation is non-associative, the space generated by it does not satisfy isotropic
property and it does not satisfy group property without rotation. Therefore, it is a very
important application of Mixed number algebra.
4. CONCLUSION:
The mathematical tools of mixed number algebra i.e. the addition, product, adjoint, norm
etc. are clearly explained. We have already been applied the Mixed number in Quantum
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Mechanics and special relativity. The complete mathematical tools are necessary for the
application of Mixed number. Therefore, this paper will be helpful for the further
application of Mixed number.
REFERENCES
[1] Alam Md. Shah, Study of Mixed Number, Proc. Pakistan Acad. Sci. 37(1):119-122.
2000
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[2] Alam Md. Shah, Comparative study of Quaternions and Mixed Number, Journal of
Theoretics, Vol-3, No-6. 2001
[3] A. Kyrala, Theoretical Physics, W.B. Saunders Company. Philagelphia & London,
Toppan Company Limited. Tokyo, Japan-1967
[4] http://mathworld.wolfram.com/Quaternion.html
[5] http://www.cs.appstate.edu/~sjg/class/3110/mathfestalg2000/quaternions.html
[6] Alam Md. Shah, Mixed Product of Vectors, Journal of Theoretics, Vol-3, No-4. 2001
[7] Alam Md. Shah, Comparative study of mixed product and quaternion product, Indian
Journal of Physics – A, Vol. 77, No. 1. 2003
[8] Alam Md. Shah, Applications of Mixed number in Quantum Mechanics, Accepted for
publication, Proc. Pakistan Acad. Sci. 40(2): 2003 (Dec.)
[9] Alam Md. Shah, Ahsan M. H., Mixed number Lorentz Transformation, Accepted for
publication, Physics Essays
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