A Neutrosophic Binomial Factorial Theorem. with their Refrains

Transcription

1 Neutrosophic Sets and Sstems, Vol. 4, 6 7 Universit of New Meico A Neutrosophic Binomial Factorial Theorem Huda E. Khalid Florentin Smarandache Ahmed K. Essa Universit of Telafer, Head of Math. Depart., College of Basic Education, Telafer, Mosul, Iraq. hodaesmail@ahoo.com Universit of New Meico, 75 Gurle Ave., Gallup, NM 87, USA. smarand@unm.edu Universit of Telafer, Math. Depart., College of Basic Education, Telafer, Mosul, Iraq. ahmed.ahhu@gmail.com Abstract. The Neutrosophic Precalculus and the Neutrosophic Calculus can be developed in man was, depending on the tpes of indeterminac one has and on the method used to deal with such indeterminac. This article is innovative since the form of neutrosophic binomial factorial theorem was constructed in addition to its refrains. Two other important theorems were proven with their corollaries, and numerical eamples as well. As a conjecture, we use ten (indeterminate) forms in neutrosophic calculus taing an important role in its. To serve article's aim, some important questions had been answered. Keword: Neutrosophic Calculus, Binomial Factorial Theorem, Neutrosophic Limits, Indeterminate forms in Neutrosophic Logic, Indeterminate forms in Classical Logic. Introduction (Important questions) Q What are the tpes of indeterminac? There eist two tpes of indeterminac a. Literal indeterminac (I). As eample: + I () b. Numerical indeterminac. As eample: (.6,.,.4) A, () meaning that the indeterminac membership.. Other eamples for the indeterminac component can be seen in functions: f() 7 or 9 or f( or ) 5 or f() [.,.] etc. Q What is the values of I to the rational power?. Let I + I + I + ( + )I, ±. () In general, I ±I (4) where z + {,,, }.. Let I + I + I + I + I + I + I + ( + + )I, I I. (5) In general, + I I, (6) where z + {,,, }. Basic Notes. A component I to the zero power is undefined value, (i.e. I is undefined), since I I +( ) I I I which is I impossible case (avoid to divide b I).. The value of I to the negative power is undefined value (i.e. I n, n > is undefined). Q What are the indeterminac forms in neutrosophic calculus? In classical calculus, the indeterminate forms are [4]:,,,,,,. (7)

2 8 Neutrosophic Sets and Sstems, Vol. 4, 6 The form to the power I (i.e. I ) is an indeterminate form in Neutrosophic calculus; it is tempting to argue that an indeterminate form of tpe I has zero value since "zero to an power is zero". However, this is fallacious, since I is not a power of number, but rather a statement about its. Q 4 What about the form I? The base "one" pushes the form I to one while the power I pushes the form I to I, so I is an indeterminate form in neutrosophic calculus. Indeed, the form a I, a R is alwas an indeterminate form. Q 5 What is the value of a I, where a R? Let, R, I ; it is obvious that,, ; while we cannot determine if I or or, therefore we can sa that I indeterminate form in Neutrosophic calculus. The same for a I, where a R []. Indeterminate forms in Neutrosophic Logic It is obvious that there are seven tpes of indeterminate forms in classical calculus [4],,,.,,,,. As a conjecture, we can sa that there are ten forms of the indeterminate forms in Neutrosophic calculus I, I, I, I, I, I, I, I I, a I (a R), ± a I. Note that: I I I I. Various Eamples Numerical eamples on neutrosophic its would be necessar to demonstrate the aims of this wor. Eample (.) [], [] The neutrosophic (numerical indeterminate) values can be seen in the following function: Find f(), where f() [.,.5]. Solution: Let [.,.5] ln [.,.5] ln ln [.,.5] ln [.,.5] [. [.,.5] ln [.,.5],.5 ] (, ) Hence e OR it can be solved briefl b [.,.5] [.,.5 ] [,]. Eample (.) [9,] [.5,5.9][,] [.5,5.9] [9,] [,] [.5,5.9] [9, ] [(.5)(9), (5.9)()] [.5,7.9]. Eample (.) [.5,5.9] [,] [.5,5.9] [,] [.5,5.9] [, ] [.5 ( ),5.9 ( )] (, ). Eample (.4) Find the following it using more than one technique [4,5] + Solution: The above it will be solved firstl b using the L'Hôpital's rule and secondl b using the rationalizing technique. Using L'Hôpital's rule ( + ) [4,5] [4,5] ( + ) [4,5] [ 4, 5 ] [,.5] Rationalizing technique [] [4,5] + [4,5] + [4, 5 ] + [, ] + + undefined. Multipl with the conjugate of the numerator:.

3 Neutrosophic Sets and Sstems, Vol. 4, ( + ) () ( + + ) + ( + + ) ( + + ) ( + + ) ( + + ) [4, 5 ] [,.5]. Identical results. + Eample (.5) Find the value of the following neutrosophic it + [,] [,6] + technique. Analtical technique [], [] + [,] [,6] + using more than one B substituting -, ( ) + ( ) [, ] ( ) [, 6] [ ( ), ( )] [, 6] [ 6, ] [, 6] [, 6] [,6] [ 6, 6 ] [, ], which has undefined operation, since [, ]. Then we factor out the numerator, and simplif: + [, ] [, 6] + ( [, ]) ( + ) ( [,]) ( + ) [,] [, ] [,] ([,] + [,]) [ 5, 4]. Again, Solving b using L'Hôpital's rule + [, ] [, 6] + + [, ] ( ) + [, ] 6 + [, ] [, ] [, ] [ 5, 4] The above two methods are identical in results. 4 New Theorems in Neutrosophic Limits Theorem (4.) (Binomial Factorial ) (I + ) Ie ; I is the literal indeterminac, e.7888 (I + ) ( ) IX ( ) + ( ) IX ( ) + ( ) IX ( ) + ( ) IX ( ) I +. I. + I! ( ) + ( 4 ) IX 4 ( ) I! ( ) ( ) + I 4! ( ) ( ) ( ) + It is clear that as (I ) I + I + I + I + I + I +!! 4! I n n n! (I + ) Ie, where e + n, I is the n! literal indeterminac. Corollar (4..) (I + ) Ie :- Put It is obvious that, as (I + ) (I + ) Ie ( using Th. 4. ) Corollar (4..) (I + ) Ie, where > &, I is the literal indeterminac.

4 Neutrosophic Sets and Sstems, Vol. 4, 6 (I + ) [(I + ) ] Put Note that as (I + ) [(I + ) ] (using corollar 4.. ). [(I + ) ] Corollar (4..) (Ie) I e Ie (I + ) (Ie) Ie, where & >. The immediate substitution of the value of in the above it gives indeterminate form I, i.e. (I + ) (I + ) I So we need to treat this value as follow:- (I + ) [(I + ) ] [ put As, (I + ) [(I + ) ] [(I + ) ] Using corollar (4..) (Ie) I Ie Theorem (4.) ()[Ia I] + + Where a >, a Note that ()[Ia I] + (I + ) ] Ia I + Let Ia I + I Ia ln( + I) ln I + ln a ln( + I) ln I + ln( + I) (ln a)(ia I) + (Ia I) + ln( + I) +. ln( + I). ln( + I). ln( + I) (ln a)(ia I) + ln(ie). ln ln( + I) ( + I) using corollar (4..) ln I + lne + Corollar (4..) Ia I + + Put as Ia I + using Th. (4.). ( + ) Ia I +. Ia I + Corollar (4..) Ie I + + Let Ie I, as + I Ie ln( + I) + lne ln( + I) Ie I + ln( + I) + ln( + I) ln( + I) Ie I + ln( + I) ln ( + I) using corollar (4..) ln (Ie) + lne +

5 Neutrosophic Sets and Sstems, Vol. 4, 6 Corollar (4..) Ie I + + let as Ie I Ie I +. Ie I + + Corollar (4..) to get. ( + ) + using Theorem (4.) ln (I + ) ( + ) ln (I + ) ln(i + ) + Let ln(i + ) + ln(i + ) e + I + e e I I e I as ln(i + ) + + I e I I e I + ( I e I) + using corollar (4..) to get the result + ( + ) Theorem (4.4) Prove that, for an two real numbers a, b Ia I Ib I, where a, b > & a, b The direct substitution of the value in the above it conclude that,so we need to treat it as follow: [Ia I] + Ia I Ib I + lnb[ib I] lnb + lnb + lnb [Ia I] ( + ) + lnb[ib I] ( lnb + ) lnb lnb + (using Th.(4.) twice (first in numerator second in denominator )) + lnb lnb. + 5 Numerical Eamples Eample (5.) Evaluate the it Solution I5 4 I 4ln5 + ln5 4 + Eample (5.) I54 I + ln5 4 (using corollar 4..) Evaluate the it Ie4 I I I Solution ln[ie 4 I] Ie 4 I I I ( + ln[i I] ( + ( + ln[ie 4 I] ( + 4 ) ln[i I] ( + 4 ) ( + ( + 4 ) 4 ) ( + (using corollar (4..) on numerator & corollar (4..) on denominator ) ln + ln 5 Conclusion In this article, we introduced for the first time a new version of binomial factorial theorem containing the literal indeterminac (I). This theorem enhances three corollaries. As a conjecture for indeterminate forms in classical calculus, ten of new indeterminate forms in Neutrosophic calculus had been constructed. Finall, various eamples had been solved. References [] F. Smarandache. Neutrosophic Precalculus and Neutrosophic Calculus. EuropaNova Brussels, 5. [] F. Smarandache. Introduction to Neutrosophic Statistics. Sitech and Education Publisher, Craiova, 4. [] H. E. Khalid & A. K. Essa. Neutrosophic Precalculus and Neutrosophic Calculus. Arabic version of the boo. Pons asbl 5, Quai du Batelage, Brussells, Belgium, European Union 6. [4] H. Anton, I. Bivens & S. Davis, Calculus, 7th Edition, John Wile & Sons, Inc.. Received: November 7, 6. Accepted: November 4, 6