Can you factorial a negative number

The factorials for real negative numbers may be defined by the integral equation,. The factorials of negative real numbers are complex numbers. At negative integers the imaginary part of complex factorials is zero, and the factorials for -1, -2, -3, -4 are -1, 2, -6, 24 respectively. The moduli of negative real number factorials and imaginary number factorials are equal to the factorials of respective real positive numbers. The present paper also provides a general definition of fractional factorials and multifactorials.

The factorials follow recurrence relations. Beta function on the real negative axis has also been redefined in the context of new concept. Draw Function Graphs — Recheronline. Competing interests. The author conceived the idea, made the calculations and written and approved the manuscript. National Center for Biotechnology Information , U. Published online Nov 6.

Ashwani K Thukral. Author information Article notes Copyright and License information Disclaimer. Ashwani K Thukral, Email: moc. Corresponding author. Received Oct 8; Accepted Oct This article is published under license to BioMed Central Ltd. Keywords: Factorials of negative numbers, Factorials of imaginary numbers, Pi function, Fractional factorials, Multifactorials, Gamma function, Beta function.

Background The factorial of a positive integer, n , is defined as,. Open in a separate window. Figure 1. Table 1 Roman factorials. Therefore, n! Table 2 Factorials of some integers as per present concept. Figure 2. Table 3 Complex factorials of some real negative numbers. Figure 3. Figure 4. Figure 5.

Factorials of imaginary numbers Similar to the factorials of real positive and real negative integers as defined in Eqn. Table 4 Complex factorials of some imaginary numbers. Figure 6. Graph of complex factorials of imaginary numbers. Figure 7. Figure 8. Table 5 Periodicity of factorials of imaginary numbers. Multifactorials and fractional factorials Let c n be a sequence defined by Eqn.

Table 6 Fractional factorials and multifactorials. Fractional factorials and multifactorials of real positive numbers Fractional factorials Multifactorials z z! Table 7 Complex gamma of real negative and imaginary numbers.

Figure 9. Figure Conclusions The present paper examines the Eularian concept of factorials from basic principles and gives a new concept, based on the Eularian concept for factorials of real negative and imaginary numbers.

Footnotes Competing interests The author declares that he has no competing interests. References Anglani R, Barlie M. Factorials as Sums. HO] The Factorial Function and Generalizations. Factorial, Gamma and Beta Functions. University of Waterloo, Waterloo. Correspondance Between Leonhard Euler and Chr.

Goldbach, — In Memorium: Milton Abramowitz. Amer Math Monthly. The early history of factorial function. Arch Hist Exact Sci. Leonhard Euler: His Life, the man, and his works. SIAM Rev. The only extension of the definition of factorial that I have encountered that does have values for negative integers is the Roman Factorial:. How do you find the factorial of negative numbers? Calculus Limits Continuous Functions. Steve M. Feb 5, The gamma function is not the same as the factorial function, however it does have the property that for positive numbers that: n!

George C. The modulus of the fractional factorials and multifactorials of real and imaginary numbers as proposed above follow recurrence relations:. Gamma values of real negative numbers are given in Table 7. Figure 9 represents the gamma of factorials of real negative numbers as per the present concept Eqn. A plot of the gamma function for real numbers as per the proposed concept. Similar to the gamma of real negative numbers Eqn.

Figures 10 and 11 represent the gamma for imaginary numbers. A plot of the gamma function for imaginary numbers as per the proposed concept. Polar graph for real X-axis vs. Beta may be defined as Culham ; Weistein c ; Wikipedia c :. Therefore, the beta function of negative numbers, as per the present concept may therefore be defined as. The graph of beta function is given in Figure Graph for B x ,0.

Factorial function was first defined for the positive real axis. Later its argument was shifted down by 1, and the factorial function was extended to negative real axis and imaginary numbers. Recently, the author Thukral and Parkash ; Thukral gave a new concept on the logarithms of real negative and imaginary numbers. Earlier the logarithms of real negative numbers were defined on the basis of hyperbola defined for the first quadrant and extended to the negative real axis, but the author defined the logarithms for the real negative axis on the basis of hyperbola located in the third quadrant.

Similarly, the author in this paper has defined the factorial function for the real negative axis. The factorials of real and imaginary numbers thus defined show uniformity in magnitude and satisfy the basic factorial equation c n n! Another lacuna in the existing Eularian concept of factorials is that although the factorials of negative integers are not defined, the double factorial of any negative odd integer may be defined, e.

Wikipedia b. Another strange behaviour of double factorials is that as an empty product, 0!! The present concept will remove anomalies in factorials and double factorials. The present concept generalizes factorials as applicable to real and imaginary numbers, and fractional and mutifactorials.

The present paper examines the Eularian concept of factorials from basic principles and gives a new concept, based on the Eularian concept for factorials of real negative and imaginary numbers.

The factorials of negative real numbers are complex numbers. At negative integers the imaginary part of complex factorials is zero, and the factorials for -1, -2, -3, -4 are -1, 2, -6, 24 respectively. The moduli of negative real number factorials and imaginary number factorials are equal to the factorials of respective real positive numbers. The present paper also provides a general definition of fractional factorials and multifactorials. The factorials follow recurrence relations.

Beta function on the real negative axis has also been redefined in the context of new concept. Draw Function Graphs — Recheronline. Google Scholar. University of Waterloo, Waterloo. Goldbach, — In Memorium: Milton Abramowitz. Amer Math Monthly. Dutka J: The early history of factorial function. Arch Hist Exact Sci , 43 3 Article Google Scholar. SIAM Rev , 50 1 Gronau D: Why is the gamma function so as it is? Teach Math Comput Sci , 1: Ibrahim AM: Extension of factorial concept to negative numbers.

Notes Theory Discrete Math , Lefort X: History of the logarithms: an example of the development of a concept in mathematics. Cornell University. Book Google Scholar. Roman S: The logarithmic binomial formula. Amer Math Month , Srinivasan GK: The gamma function: An eclectic tour. Thukral AK: Logarithms of imaginary numbers in rectangular form: A new technique.

Can J Pure Appl Sc , 8 3 Can J Pure Appl Sc , 8 2 Weistein EW: Factorial. Weistein EW: Double factorial. Weistein EW: Beta function. Wikipedia: Factorial.