Why 0 Factorial is 1? | Everything you need to Know about 0!

0 factorial is 1
0 factorial is 1

No wonder math is one of the most frustrating and confusing subject and the number 0 makes it more complicated. For instant, 0 factorial is 1

    \[ \boxed{ 0! =1 } \]

0 multiplied by any number is 0

    \[ \boxed{0 * N = 0} \]

and any number divided by 0 is undefined

    \[ \boxed{\frac{N}{0}\ = 1} \]

Thats enough we don’t want to make our readers bored by showing every mathematical calculations where 0 is involved. So, let’s talk about 0 factorial only.

Before we start talking about the 0 factorial you should know about the concept of factorial first. So, let’s start with factorial.

What is Factorial?

I don’t want to make this boring like in your Maths class in school. So, I will try to explain everything in simple term. Factorial is the number of possible ways in which anything can be arranged or ordered.

Suppose there are three empty boxes in front of you and you have 3 cards of different cards Red, Black and Blue. Now you have you have to put three cards in each boxes.

The possible ways you can do this are:

  1. Red in first box, Black in second and Blue in third.
  2. Black in first box, Red in second and Blue in third.
  3. Black in first box, Blue in second and Red in third.
  4. Red in first box, Blue in second and Black in third.
  5. Blue in first box, Black in second and Red in third.
  6. Blue in first box, Red in second and Black in third.

From the above example we can see that we can put three cards in three boxes in 6 ways which is equal to 3 factorial (3!).

Its easy with the different number of cards like if we have to put 4 cards in 4 different boxes we just do 4 factorial. But the problem arises when we decrease the number to 0.

By the definition that we just mentioned, its like putting 0 cards in 0 boxes. But that doesn’t make any sense. So, how does 0 factorial is 1? We will discuss it today.

In mathematical term:

We are student of mathematics so let’s discuss about factorial in mathematical term.

Factorial of a number n, denoted by n! is defined by the product of all the positive number less than or equal to n.

    \[ \boxed{n! = n * (n-1)*(n-2)*.......*3*2*1} \]

Hmmm…. again this definition doesn’t make sense with 0 factorial. What is the positive number less than 0?

So, following the definition of factorial doesn’t makes any sense with the number 0 but still we say 0 factorial is 1 and there is no mistake in it.

Why 0 factorial is 1? (0! =1 )

There are different ways to prove 0 factorial is 1. Lets use the definition we just explained above to prove it.

From the definition above:

    \[ \boxed{n! = n * (n-1)*(n-2)*.......*3*2*1} \]

We can also write it as:

    \[ \boxed{n! = n * (n-1)!} \]

If you don’t understand what we just did, check the mathematical definition of factorial and tally with the statement we just mentioned.

The above equation holds true for any natural number n. Let’s suppose n = 1 ,

    \[ \boxed{1! = 1 * (1-1)! } \]

It’s equal to

    \[ \boxed{1! = 1 * (0)! } \]

As we know that 1! = 1, So, for the above equation to be true the value of 0! should be equal to 1 else 1! = 1 which is against the definition of factorial.

If you are struggling to understand the proof of 0 factorial then don’t worry we are here to make it easy for you to understand so let’s do an easy proof.

When n = 6,

    \[ \boxed{6! = 6 * 5 * 4 * 3 * 2* 1 } \]

When n = 5,

    \[ \boxed{5! =  5 * 4 * 3 * 2* 1 } \]

When n = 4,

    \[ \boxed{4! =  4 * 3 * 2* 1 } \]

Let’s wait for a moment here and compare the sequence we see in 6!, 5!, and 4!. Did you see they are following certain pattern?

When we move from 6! to 5!, we divide the right hand side of the equation by 6.

When we move from 5! to 4!, we divide the right hand side of the equation by 5.

Let’s follow this pattern for 2! , 1! and 0!,

When n = 2,

    \[ \boxed{2! =  2* 1 } \]

Divide the right hand side of above equation by 2 to find 1!.

When n = 1,

    \[ \boxed{1! =   1 } \]

Similarly to find the 0! divide the right hand side of above equation by 1.

    \[ \boxed{0! =   1 } \]

Boom! this shows 0 factorial is 1. It was easy right?

What if we go beyond 0 to negative numbers?

Lets follow the pattern we just mentioned above and try to find for -1.

For this we have to divide the right hand side of above equation by 0.

Dividing by 0……..

    \[ \boxed{-1! =   Undefined  } \]

Yeah its undefined but Do you know why the number divided by 0 is undefined?

If no then stay keep connected it may be the next topic in Math is Meth.

 

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