Facts-3

Calculus Level 1

Find the value of ( 1 ) ! (-1)! .

-1 0 \infty 1 Not defined

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2 solutions

Ashish Menon
Jun 17, 2016

For any negative integer ( n ) (-n) :-
( n ) ! = ( n + 1 ) ! ( n + 1 ) = ( n + 2 ) ! ( n + 1 ) ( n + 2 ) = ( n + 3 ) ! ( n + 1 ) ( n + 2 ) ( n + 3 ) 0 0 0 0 0 0 = ( n + n ) ! ( n + 1 ) ( n + 2 ) ( n + 3 ) ( n + n ) = 0 ! ( n + 1 ) ( n + 2 ) ( n + 3 ) × 0 = 1 0 ( because 0! = 1 ) = undefined \begin{aligned} (-n)! & = \dfrac{(-n + 1)!}{(-n + 1)}\\ \\ & = \dfrac{(-n + 2)!}{(-n + 1)(-n + 2)}\\ \\ & = \dfrac{(-n + 3)!}{(-n + 1)(-n + 2)(-n + 3)}\\ \\ & \phantom{0} \phantom{0} \phantom{0} \phantom{0} \phantom{0} \phantom{0} \, \, \, \, \, \, \, \, \, \vdots\\ & = \dfrac{(-n + n)!}{(-n+1)(-n+2)(-n+3) \cdots (-n + n)}\\ \\ & = \dfrac{0!}{(-n + 1)(-n + 2)(-n + 3) \cdots × 0}\\ \\ & = \dfrac{1}{0} \color{#EC7300}{\left(\text{because 0! = 1}\right)}\\ \\ & = \color{#3D99F6}{\boxed{\text{undefined}}} \end{aligned} .

So, we have just generalized that the factorial function of any negative integer is undefined.

good solution @Ashish Siva .._+1

Ayush G Rai - 4 years, 12 months ago

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Thanks! :)

Ashish Menon - 4 years, 12 months ago

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Lol

Swapnil Das - 4 years, 12 months ago

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@Swapnil Das Lol they must have onterpreted it like 1 ! -1! then it is ( 1 ! ) = 1 -(1!) = -1 :P Nice one dude!

Ashish Menon - 4 years, 12 months ago
Pianate Nate
Jun 17, 2016

This question uses understanding. If 2! can be expressed as 3! / 3 , so we can express (-1) ! is equal to 0! / 0. NOTE : 0! = 1! / 1 = 1. The answer will be Undefined because 1 / 0 = undefined

good observation..+1

Ayush G Rai - 4 years, 12 months ago

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