# A neutrosophic binomial factorial theorem with their refrains.

1 Introduction (Important questions)

Q 1 What are the types of indeterminacy? There exist two types of indeterminacy

a. Literal indeterminacy (I).

As example:

2 + 3I (1)

b. Numerical indeterminacy. As example:

x(0.6,0.3,0.4) [member of] A, (2)

meaning that the indeterminacy membership = 0.3.

Other examples for the indeterminacy component can be seen in functions: f(0) = 7 or 9 or f( 0 or 1) = 5 or fix) = [0.2, 0.3] [x.sup.2] ... etc.

Q 2 What is the values of I to the rational power?

1. Let

[square root of I] = x + y I 0 + I = [x.sup.2] + (2xy + [y.sup.2]) I x = 0, y = [+ or -] 1. (3)

In general,

[2k root of I] = [+ or -] I (4)

where k [member of] [z.sup.+] = {1,2,3, ...}.

2. Let

[cube root of I] = x + y I 0 + I = [x.sup.3] + 3[x.sup.2]yI + 3x[y.sup.2][I.sup.2] + [y.sup.3][I.sup.3] 0 + I = [x.sup.3] + 3[x.sup.2]y + 3x[y.sup.2] + [y.sup.3])I X = 0, y = 1 [right arrow] [cube root of I] = I. (5)

In general,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where k [member of] [z.sup.+] = {1,2,3, ...}.

Basic Notes

1. A component I to the zero power is undefined value, (i.e. [I.sup.0] is undefined), since [I.sup.0] = [I.sup.1+(-1)] = [I.sup.1] x [I.sup.-1] = I/I which is impossible case (avoid to divide by I).

2. The value of I to the negative power is undefined value (i.e. [I.sup.-n], n > 0 is undefined).

Q 3 What are the indeterminacy forms in neutrosophic calculus?

In classical calculus, the indeterminate forms are [4]:

0/0, [infinity]/[infinity], 0 x [infinity], [[infinity].sup.0], [0.sup.0], [1.sup.[infinity]], [infinity] - [infinity]. (7)

The form 0 to the power I (i.e. [0.sup.I]) is an indeterminate form in Neutrosophic calculus; it is tempting to argue that an indeterminate form of type [0.sup.I] has zero value since "zero to any power is zero". However, this is fallacious, since [0.sup.I] is not a power of number, but rather a statement about limits.

Q 4 What about the form [1.sup.I]?

The base "one" pushes the form [1.sup.I] to one while the power I pushes the form [1.sup.I] to I, so [1.sup.I] is an indeterminate form in neutrosophic calculus. Indeed, the form [a.sup.I], a [member of] R is always an indeterminate form.

Q 5 What is the value of [a.sup.I], where a [member of] R?

Let [y.sub.1] = [2.sup.x], x [member of] R, [y.sub.2] = [2.sup.I]; it is obvious that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; while we cannot determine if [2.sup.I] [right arrow] [infinity] or 0 or 1, therefore we can say that [y.sub.2] = [2.sup.I] indeterminate form in Neutrosophic calculus. The same for [a.sup.I], where a [member of] R [2],

2 Indeterminate forms in Neutrosophic Logic

It is obvious that there are seven types of indeterminate forms in classical calculus [4],

0/0, [infinity]/[infinity], 0. [infinity], [0.sup.0], [[infinity].sup.0], [1.sup.[infinity]], [infinity] - [infinity].

As a conjecture, we can say that there are ten forms of the indeterminate forms in Neutrosophic calculus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that:

I/0 = I x 1/0 = I x [infinity] = [infinity] x I.

3 Various Examples

Numerical examples on neutrosophic limits would be necessary to demonstrate the aims of this work.

Example (3.1) [1], [3]

The neutrosophic (numerical indeterminate) values can be seen in the following function: Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence y = [e.sup.-[infinity]] = 0

OR it can be solved briefly by

y = [x.sup.[2.1,2.5]] = [[0.sup.2.1], [0.sup.2.5]] = [0,0] = 0.

Example (3.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example (3.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example (3.4)

Find the following limit using more than one technique [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solution:

The above limit will be solved firstly by using the L'Hopital's rule and secondly by using the rationalizing technique.

Using L'Hopital's rule

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Rationalizing technique [3]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Multiply with the conjugate of the numerator:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Identical results.

Example (3.5)

Find the value of the following neutrosophic limit [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] using more than one technique.

Analytical technique [1], [3] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By substituting x= -3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which has undefined operation 0/0, since 0 [member of] [-3,3]. Then we factor out the numerator, and simplify:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Again, Solving by using L'Hopital's rule

The above two methods are identical in results.

4 New Theorems in Neutrosophic Limits

Theorem (4.1) (Binomial Factorial)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; I is the literal indeterminacy, c = 2.7182828

Proof

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is clear that 1/x [right arrow] 0 as x [right arrow] [infinity]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where e = 1 + [[summation].sup.[infinity].sub.n=1] 1/n!, I is the literal indetenninacy.

Corollary (4.1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof:-

Put y = 1/x

It is obvious that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (usingTh. 4.1 )

Corollary (4.1.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], I is the literal indeterminacy.

Proof

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Corollary (4.1.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof

The immediate substitution of the value of x in the above limit gives indeterminate form [I.sup.[infinity]],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So we need to treat this value as follow:-

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using corollary (4.1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem (4.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where a > 0, a [not equal to] 1

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Corollary (4.2.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof

Corollary (4.2.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

using corollary (4.1.1)

1/ln (Ie) = 1/[lnI + lne] = 1/[lnI + 1]

Corollary (4.2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Corollary (4.2.2) to get

= k. (1/1 + lnI) = k/1 + lnI

Theorem (4.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

using corollary (4.2.2) to get the result

= k/1/1 + lnI = k(1 + lnI)

Theorem (4.4)

Prove that, for any two real numbers a, b

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof

The direct substitution of the value x in the above limit conclude that 0/0, so we need to treat it as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(using Th.(4.2) twice (first in numerator second in denominator))

= [lna/[1+lnI]]/[[lnb/[1+lnI]] * lnI/lnI * lnb/lna = 1.

5 Numerical Examples

Example (5.1)

Evaluate the limit [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (using corollary 4. 2.1)

Example (5.2)

Evaluate the limit [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Solution

(using corollary (4.2.3) on numerator & corollary (4.2.1) on denominator)

= [4/[1 + lnI]/2ln3/[1 + lnI]] *[ lnI/4/lnI/ln[3.sup.2]] = 1.

5 Conclusion

In this article, we introduced for the first time a new version of binomial factorial theorem containing the literal indeterminacy (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. Finally, various examples had been solved.

References

[1] F. Smarandache. Neutrosophic Precalculus and Neutrosophic Calculus. EuropaNova Brussels, 2015.

[2] F. Smarandache. Introduction to Neutrosophic Statistics. Sitech and Education Publisher, Craiova, 2014.

[3] H. E. Khalid & A. K. Essa. Neutrosophic Pre calculus and Neutrosophic Calculus. Arabic version of the book. Pons asbl 5, Quai du Batelage, Brussells, Belgium, European Union 2016.

[4] H. Anton, I. Bivens & S. Davis, Calculus, 7th Edition, John Wiley & Sons, Inc. 2002.

Received: November 7, 2016. Accepted: November 14, 2016

Huda E. Khalid (1) Florentin Smarandache (2) Ahmed K. Essa (3)

(1) University of Telafer, Head of Math. Depart., College of Basic Education, Telafer, Mosul, Iraq. E-mail: hodaesmail@yahoo.com

(2) University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA. E-mail: smarand@unm.edu

(3) University of Telafer, Math. Depart., College of Basic Education, Telafer, Mosul, Iraq. E-mail: ahmed.ahhu@gmail.com
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