why zero factorial is equal to one factorial?

Question

I will prove 0=1. we know that, From Factorial definition zero factorial is equal to one and one factorial is equal to one. so,0!=1!. factorial get cancelled both sides, we get 0=1. Is this right..?

Answer 1

What gives you the right to cancel out the factorial? By the same token I can claim that if $x^2 = y^2$ then $x=y$, which implies that since $1^2 = 1 = (-1)^2$, we must have $1 = -1$, just as absurd as your conclusion...

Answer 2

Factorial ("!") is not an expression - it is an operator.

For example, $\sin(0) = \sin(\pi) = 0$ but you can't cancel "sin" to get $0 = \pi$.

Answer 3

The symbol "!" denotes a function that takes as input nonnegative integers, and has an output defined by a recurrence relation. Perhaps it might be easier to replace a "!" on the right with something that looks more like traditional functional notation: $$ \operatorname{fact} : \mathbb{Z}_{\ge 0} \to \mathbb{R} \qquad \text{defined by} \qquad \operatorname{fact}(n) := \begin{cases} 1 & \text{if $n=0$, and}\\ n\cdot \operatorname{fact}(n-1) & \text{otherwise.}\end{cases} $$ What you are asserting is that $$ \operatorname{fact}(m) = \operatorname{fact}(n) \iff m = n. $$ This is (more or less) the definition of an injective (or one-to-one) function. If a function is one-to-one, then every output corresponds to exactly one input. When this happens, we can "cancel" the function on two sides of an equation. However, the factorial function is not one-to-one. As you have already noted, $$ \operatorname{fact}(0) = \operatorname{fact}(1) = 1. $$ Since this function is not one-to-one, we can't "cancel" it as you want.

That being said, note that $\operatorname{fact}$ can be restricted to a domain where it is one-to-one. Indeed, it is sufficient to throw away zero. If you define $$ \operatorname{fact} : \mathbb{Z}_{> 0} \to \mathbb{R} \qquad \text{defined by} \qquad \operatorname{fact}(n) := \begin{cases} 1 & \text{if $n=1$, and}\\ n\cdot \operatorname{fact}(n-1) & \text{otherwise,}\end{cases} $$ then you get a one-to-one function that can be "canceled".

Answer 4

Two things:

1. "!" is not a variable rather it is an operator.
2. Factorial function (domain is whole number) is not injective.