Why 0! = 1

in #math6 years ago

Most of you should be familiar with the factorial function where n factorial (written n!) is the product of the first n natural numbers, e.g. 5! = 1 * 2 * 3 * 4 * 5 = 120. This is a straightforward definition for positive numbers but in maths you will sometimes see zero factorial. It feels like this should come out to zero because zero will appear as a factor but actually the common definition is 0! = 1. Why is that?

If we again look at the definition n! = n * (n-1) * (n-2) * … * 3 * 2 * 1 it becomes clear that n! = n * (n-1)! since (n-1)! = (n-1) * (n-2) * ... * 3 * 2 * 1. The interesting thing about this is that we can rearrange the equation to yield (n-1)! = (n!)/n. Because we divide by n this will not hold true for n=0 but it will do so for n=1. This yields (1-1)! = (1!)/1 which is 0! = 1/1 =1.

There are actually ways to extend the factorial even further so it is also defined for fractions and negative numbers, always keeping the fundamental property x! = x * (x-1)! but they are not as easy to understand as this.

Sort:  

Congratulations @jifka! You have completed the following achievement on the Steem blockchain and have been rewarded with new badge(s) :

You published your First Post
You got a First Vote

Click here to view your Board of Honor
If you no longer want to receive notifications, reply to this comment with the word STOP

Do not miss the last post from @steemitboard:

Meet the Steemians Contest - Special attendees revealed
Meet the Steemians Contest - Intermediate results

Support SteemitBoard's project! Vote for its witness and get one more award!

Congratulations @jifka! You have completed the following achievement on the Steem blockchain and have been rewarded with new badge(s) :

You made your First Vote

Click here to view your Board of Honor
If you no longer want to receive notifications, reply to this comment with the word STOP

Do not miss the last post from @steemitboard:

Meet the Steemians Contest - The results, the winners and the prizes
Meet the Steemians Contest - Special attendees revealed
Meet the Steemians Contest - Intermediate results

Support SteemitBoard's project! Vote for its witness and get one more award!

Congratulations @jifka! You have completed the following achievement on the Steem blockchain and have been rewarded with new badge(s) :

You received more than 10 upvotes. Your next target is to reach 50 upvotes.

Click here to view your Board of Honor
If you no longer want to receive notifications, reply to this comment with the word STOP

Do not miss the last post from @steemitboard:

Meet the Steemians Contest - The results, the winners and the prizes
Meet the Steemians Contest - Special attendees revealed
Meet the Steemians Contest - Intermediate results

Support SteemitBoard's project! Vote for its witness and get one more award!

Congratulations @jifka! You have completed the following achievement on the Steem blockchain and have been rewarded with new badge(s) :

You made more than 10 upvotes. Your next target is to reach 50 upvotes.

Click here to view your Board of Honor
If you no longer want to receive notifications, reply to this comment with the word STOP

Do not miss the last post from @steemitboard:

Meet the Steemians Contest - The results, the winners and the prizes
Meet the Steemians Contest - Special attendees revealed
Meet the Steemians Contest - Intermediate results

Support SteemitBoard's project! Vote for its witness and get one more award!