- #1
Natasha1
- 493
- 9
I was asked to prove 0! = 1?
Where do I start?
Is this possible? Any hints welcome...
Where do I start?
Is this possible? Any hints welcome...
Natasha1 said:Well 3! is 3x2x1 = 6
2! is 2x1= 2
1! is 1x1 = 1 but 0?? haven't got a clue
Physics Monkey said:You can easily check (using integration by parts), that the following
[tex]
n! = \int^\infty_0 dt\, t^n e^{-t} \equiv \Gamma (n+1),
[/tex]
is true whenever n is a positive integer 1, 2, 3, etc. This integral is the definition of the so called gamma function [tex] \Gamma (z) [/tex], and I would encourage you to check what I have said thus far. Now, since the gamma function reproduces the standard factorial for integer n, we could equivalently define the factorial to be this integral for integer n. However, this definition is nice because it immediately allows us to define the "factorial" of many more numbers, including zero. Using the definition above, what would [tex] 0! [/tex] be?
You can also compute [tex] 0! [/tex] hueristically by using [tex] n! = n(n-1)! [/tex], but this way still requires you to somehow extend your definition of factorial in order to be rigorous.
It boils down to a definition any way you slice it.
Arman_AAT_Khos said:Factorials represent combinations of a set of objects. so 2 objects can be ordered 2 different ways, 3 objects can be ordered 6 different ways, etc... Matt is right 0! is defined as 1.
It is mathematically impossible for 0 to equal 1. This is because 0 is a number that represents nothing or the absence of quantity, while 1 is a number that represents a single unit or quantity. Therefore, the two values cannot be equal.
Proving that 0=1 has no real-world significance. It is only used as an example in mathematical proofs to demonstrate the importance of logical reasoning and the consequences of incorrect assumptions or statements.
No, it is not possible to prove 0=1. It is a false statement that goes against the basic principles of mathematics. Any proof claiming to show that 0=1 is inherently flawed or incorrect.
When attempting to prove 0=1, you must first understand the basic properties of numbers and how they relate to each other. This includes the concept of addition, subtraction, and multiplication, as well as the properties of zero and one. From there, you can use logical reasoning and mathematical equations to show why 0 cannot equal 1.
Understanding the difference between 0 and 1 is crucial in mathematics because these numbers have unique properties and play a significant role in many mathematical concepts and operations. Confusing or equating 0 with 1 can lead to incorrect solutions and flawed reasoning in mathematical problems.