Subtraction and Division of Neutrosophic Numbers
Florentin Smarandache 11 UniversityofNewMexico,Gallup,NM,USA smarand@unm.edu
Abstract
Inthispaper,wedefinethesubtractionandthedivisionofneutrosophicsingle valued numbers.Therestrictionsfortheseoperationsarepresentedforneutrosophicsingle valued numbers and neutrosophic single valued overnumbers / undernumbers / offnumbers.Afterwards,severalnumeralexamplesarepresented.

Keywords
neutrosophic calculus, neutrosophic numbers, neutrosophic summation, neutrosophicmultiplication,neutrosophicscalarmultiplication,neutrosophicpower, neutrosophicsubtraction,neutrosophicdivision
1 Introduction

Let��=(��1,��1,��1) and��=(��2,��2,��2) be two single-valued neutrosophic numbers,where��1,��1,��1,��2,��2,��2∈[0,1], and0≤��1+��1+��1≤3 and0≤��2 +��2+��2≤3
Thefollowingoperationalrelationshavebeendefinedandmostlyusedinthe neutrosophicscientificliterature:
1.3 Neutrosophic Scalar Multiplication ⋋��=(1 (1 ��1)⋋,��1 ⋋,��1 ⋋ ), (3) where⋋∈ℝ,and⋋>0.
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Subtraction and Division of Neutrosophic Numbers
1.4 Neutrosophic Power
��⋋ =(��1 ⋋,1 (1 ��1)⋋,1 (1 ��1)⋋), (4) where⋋∈ℝ,and⋋>0.
2 Remarks
Actually, the neutrosophic scalar multiplication is an extension of neutrosophicsummation;inthelast,onehas⋋=2. Similarly, the neutrosophic power is an extension of neutrosophic multiplication;inthelast,onehas⋋=2. Neutrosophic summation of numbers is equivalent to neutrosophic union of sets,andneutrosophicmultiplicationofnumbersisequivalenttoneutrosophic intersectionofsets. That's why, both the neutrosophic summation and neutrosophic multiplication (and implicitly their extensions neutrosophic scalar multiplication and neutrosophic power) can be defined in many ways, i.e. equivalently to their neutrosophic union operators and respectively neutrosophicintersectionoperators. Ingeneral: ��⊕�� =(��1 ∨��2,��1 ∧��2,��1 ∧��2), (5) or ��⊕�� =(��1 ∨��2,��1 ∨��2,��1 ∨��2), (6) andanalogously: ��⊗�� =(��1 ∧��2,��1 ∨��2,��1 ∨��2) (7) or ��⊗�� =(��1 ∧��2,��1 ∧��2,��1 ∨��2), (8) where"∨"isthefuzzyOR(fuzzyunion)operator,defined,for��,�� ∈[0,1],in threedifferentways,as: �� 1 ∨�� =��+�� ����, (9) or �� 2 ∨�� =������{��,��}, (10) or �� 3 ∨�� =������{��+��,1}, (11) etc.
Critical Review. Volume XIII, 2016
Subtraction and Division of Neutrosophic Numbers
for all��1,��1,��1,��2,��2,��2 ∈[0,1], with the restrictions that:��2 ≠1, ��2 ≠0, and ��2 ≠0
So,theneutrosophicsubtractiononlypartiallyworks,i.e.when��2 ≠1,��2 ≠0, and��2 ≠0
Therestrictionthat
(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2)∈([0,1],[0,1],[0,1]) (22)
is set when the classical case when the neutrosophic number components ��,��,��areintheinterval[0,1].
But,forthegeneralcase,whendealingwithneutrosophicoverset/underset /offset[1],ortheneutrosophicnumbercomponentsareintheinterval[Ψ,Ω], whereΨiscalled underlimit andΩiscalled overlimit,withΨ≤0<1≤Ω,i.e. one has neutrosophic overnumbers / undernumbers / offnumbers, then the restriction(22)becomes:
(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2)∈([Ψ,Ω],[Ψ,Ω],[Ψ,Ω]). (23)
3.1 Proof
The formula for the subtraction was obtained from the attempt to be consistentwiththeneutrosophicaddition.
Oneconsidersthemostusedneutrosophicaddition:
(��1,��1,��1)⊕(��2,��2,��2)=(��1 +��2 ��1��2,��1��2,��1��2), (24)
We consider the ⊖ neutrosophic operation the opposite of the ⊕ neutrosophic operation, as in the set of real numbers the classical subtraction is theoppositeoftheclassicaladdition+. Therefore,let'sconsider:
(��1,��1,��1)⊖(��2,��2,��2)=(��,��,��), (25) ⊕(��2,��2,��2) ⊕(��2,��2,��2) where��,��,�� ∈ℝ. Weneutrosophicallyadd⊕(��2,��2,��2)onbothsidesoftheequation.Weget: (��1,��1,��1)=(��,��,��)⊕(��2,��2,��2)=(��+��2 ����2,����2,����2). (26) Or, {
��1 =��+��2 ����2,whence�� = ��1 ��1 1 ��2 ; ��1 =����2,whence��= ��1 ��2 ; ��1 =����2,whence�� = ��1 ��2 .
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(27)
Subtraction and Division of Neutrosophic Numbers
3.2 Checking the Subtraction
With��=(��1,��1,��1),�� =(��2,��2,��2),and�� =(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2), where��1,��1,��1,��2,��2,��2 ∈[0,1],and��2 ≠1,��2 ≠0,and��2 ≠0,wehave: ��⊖�� =��. (28) Then: ��⊕�� =(��2,��2,��2)⊕(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2)=(��2 + ��1 ��2 1 ��2 ��2 ⋅ ��1 ��2 1 ��2 ,��2, ��1 ��2 ,��2, ��1 ��2)=(��2 ��2 2+��1 ��2 ��1��2+��2 1 ��2 ,��1,��1)= (��1(1 ��2) 1 ��2 ,��1,��1)=(��1,��1,��1). (29) ��⊖�� =(��1,��1,��1)⊖(��1 ��2 1 ��2 , ��1 ��2 , ��1 ��2)=(��1 ��1 ��2 1 ��2 1 ��1 ��2 1 ��2 , ��1 ��1 ��2 , ��1 ��1 ��2)= (��1 ��1��2 ��1+��2 1 ��2 1 ��2 ��1+��2 1 ��2 ,��2,��2)=( ��1��2+��2 1 ��2 ,��2,��2)= (��2( ��1+1) 1 ��2 ,��2,��2)=(��2,��2,��2) (30)
4 Division of Neutrosophic Numbers
Wedefineforthefirsttimethedivisionofneutrosophicnumbers: ��⊘�� =(��1,��1,��1)⊘(��2,��2,��2)=(��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2)=��, (31) where ��1,��1,��1,��2,��2,��2 ∈[0,1], with the restriction that ��2 ≠0, ��2 ≠1, and ��2 ≠1. Similarly,thedivisionofneutrosophicnumbersonlypartiallyworks,i.e.when ��2 ≠0,��2 ≠1,and��2 ≠1. Inthesameway,therestrictionthat (��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2)∈([0,1],[0,1],[0,1]) (32) issetwhenthetraditionalcaseoccurs,whentheneutrosophicnumber components t,i,fareintheinterval[0,1]. But,forthecasewhendealingwithneutrosophicoverset/underset/offset [1],whentheneutrosophicnumbercomponentsareintheinterval[Ψ,Ω], whereΨiscalled underlimit andΩiscalled overlimit,withΨ≤0<1≤Ω,i.e. onehas neutrosophic overnumbers / undernumbers / offnumbers,thenthe restriction(31)becomes:
Subtraction and Division of Neutrosophic Numbers
(��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2)∈([Ψ,Ω],[Ψ,Ω],[Ψ,Ω]). (33)
4.1 Proof
In the same way, the formula for division⊘of neutrosophic numbers was obtained from the attempt to be consistent with the neutrosophic multiplication.
We consider the ⊘ neutrosophic operation the opposite of the ⊗ neutrosophicoperation,asinthesetofrealnumberstheclassicaldivision÷ istheoppositeoftheclassicalmultiplication×
Oneconsidersthemostusedneutrosophicmultiplication: (��1,��1,��1)⊗(��2,��2,��2) =(��1��2,��1 +��2 ��1��2,��1 +��2 ��1��2), (34)
Thus,let'sconsider: (��1,��1,��1)⊘(��2,��2,��2)=(��,��,��), (35) ⨂(��2,��2,��2) ⨂(��2,��2,��2) where��,��,�� ∈ℝ
Weneutrosophicallymultiply⨂bothsidesby(��2,��2,��2).Weget (��1,��1,��1)=(��,��,��)⨂(��2,��2,��2) =(����2,��+��2 ����2,��+��2 ����2). (36) Or, {
��1 =����2,whence�� = ��1 ��2 ;: ��1 =��+��2 ����2,whence��= ��1 ��2 1 ��2 ; ��1 =��+��2 ����2,whence�� = ��1 ��2 1 ��2 .
4 2 Checking the Division
(37)
With��=(��1,��1,��1),�� =(��2,��2,��2),and�� =(��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2), where��1,��1,��1,��2,��2,��2 ∈[0,1],and��2 ≠0,��2 ≠1,and��2 ≠1,onehas: ��∗�� =��. (38)
Then: �� �� =(��2,��2,��2)×(��1 ��2 , ��1 ��2 1 ��2 , ��1 ��2 1 ��2)=(��2 ⋅ ��1 ��2 ,��2 + ��1 ��2 1 ��2 ��2 ⋅ ��1 ��2 1 ��2 ,��2 + ��1 ��2 1 ��2 ��2 ⋅ ��1 ��2 1 ��2)=
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5 Conclusion
Wehaveobtainedtheformulaforthesubtractionofneutrosophicnumbers⊖ goingbackwordsfromtheformulaofadditionofneutrosophicnumbers ⊕. Similarly,wehavedefinedtheformulafordivisionofneutrosophicnumbers ⊘andweobtaineditbackwordsfromtheneutrosophicmultiplication⨂. We also have taken into account the case when one deals with classical neutrosophicnumbers(i.e.theneutrosophiccomponentst,i,fbelongto[0,1]) aswellasthegeneralcasewhen��,��,��belongto[��,��],wheretheunderlimit �� ≤ 0andtheoverlimit�� ≥ 1
6 References
[1] Florentin Smarandache, Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset. Similarly for Neutrosophic Over /Under /Off Logic, Probability, and Statistics, 168 p., Pons Editions, Bruxelles, Belgique, 2016; https://hal.archives ouvertes.fr/hal 01340830 ; https://arxiv.org/ftp/arxiv/papers/1607/1607.00234.pdf
[2] Florentin Smarandache, Neutrosophic Precalculus and Neutrosophic Calculus, EuropaNova, Brussels, Belgium, 154 p., 2015; https://arxiv.org/ftp/arxiv/papers/1509/1509.07723.pdf يكفوسورتوينلا لماكتلاو لضافتلا باسح و يكفوسورتوينلا لماكتلاو لضافتلا ئدابم (Arabic translation) by Huda E. Khalid and Ahmed K. Essa, Pons Editions, Brussels, 112 p., 2016.
[3] Ye Jun, Bidirectional projection method for multiple attribute group decision making with neutrosophic numbers, Neural Computing and Applications, 2015, DOI: 10.1007/s00521 015 2123 5.
[4] Ye Jun, Multiple attribute group decision making method under a neutrosophic number environment, Journal of Intelligent Systems, 2016, 25(3): 377 386