JAMES GRIME: We at Numberphile often receive this question Is today's topic Brady, what do they want to ask? BRADY HARAN: They asked: "Why is the factorial of 0 1?" JAMES GRIME: Yes Why is the factorial of 0 equal to 1? We must first understand what factorial is Let the natural number n The factorial of n is like this Add an exclamation mark after n This number is equal to You multiply all the whole numbers less Natural number equal to n n times (n-1) times (n-2) Multiply by– Then repeat until 3 by 2 by 1 Give an example Just the factorial of 5 5 times 4 times 3 times 2 times 1 The result is 120 OK The question is what is the factorial of 0 One way to answer is Look at the law of factorial This law is For example, the factorial of 4 is equal to 4 factorial, is equal to 5 The factorial of 5 divided by 5 If you divide this number by 5 You can take the 5 here The result is the factorial of 4.
The factorial of 5 divided by 5 or 120 divided by 5 Quotient is 24 Which is the factorial of 4. The factorial of 3 is the factorial of 4 divided by 4 24 divided by 4 The answer is 6 Which is the factorial of 3 The factorial of 2 is the factorial of 3 divided by 3. That is 6 divided by 3, the answer is 2. Factorial of 1 Same as the former The factorial of 2 divided by 2 2 divided by 2 Divide 2 by 2 The answer is 1 Then come to play jade Are you looking forward to it? Here comes the factorial of 0 We have to follow the law The factorial of 0 is the factorial of 1 divided by 1 The factorial of 1 is 1 1 divided by 1 equals 1 So the factorial of 0 is equal to 1.
Just follow the rules BRADY HARAN: Who said that laws must be followed? Who made this rule? JAMES GRIME: Actually the factorial doesn’t have to be Complete law But this is really the complete law Allow me to explain in another way BRADY HARAN: Why not let me continue to complete the pattern Isn't negative 1 the next number? JAMES GRIME: Just see what happens I'm not sure Try together Negative 1 factorial The answer is The factorial of 0 divided by 0 1 divided by 0 BRADY HARAN: Oh divide by 0 JAMES GRIME: Brady, stop breaking the math Another way to explain the factorial of 0 is The factorial of n is a combination of permutations of n objects Let me elaborate I want to get some objects Just take out my wallet Find some coins Who told you that mathematicians make very little money? Here is 50 pounds I took out this silver one and this 5 pounds There are 3 objects to arrange these 3 objects How many methods are there in total? There are 6 methods in total Is the factorial of 3 See if i am right Combine the first with the second or do this Third combination fourth Can also be arranged like this Put this coin on top That's five or six combinations Take one coin and two left How many combinations are there for 2 objects? One combination two combinations Take one again How many combinations are there? Yes there is only one There is only 1 way to arrange 1 object Why don't you take the last coin? A bit of philosophy There are 0 objects here How many combinations are there to arrange 0 objects? Only one combination that's it Want me to do it again? that's it A bit philosophical but 0 objects should be arranged There is really only one way Then prove the law is right The factorial of 0 is equal to 1 Actually there is a third method Just draw a chart 1 2 3 4 5 The factorial of 1 is 1.
This is 1. The factorial of 2 is 2 which is about here The factorial of 3 is 6 I don't know it is probably here The factorial of 4 is 24, so it’s quite high in this chart. Then the factorial of 5 is higher I mentioned that the factorial of 0 is 1. So this is the chart In theory, the numbers between integers are also factorial For example, 1 and 2 copies of 1 1 and 2 parts of the factorial of 1 What is the factorial of 1 and 2 of 1? The mathematician has already solved They generalized the concept This is the concept of 1, 2 and 1 factorial Called gamma Is the Greek letter gamma The gamma we call "~" Write it like this The next one will be more complicated The gamma of n is equal to the integral from 0 to infinity Just count t to the minus 1 power multiplied by e to the minus t power dn Not everyone is familiar with this concept Some people are familiar with Other will not This mathematical concept is more complicated But it has something to do with factorial Use this to get the factorial between integers Is the point on this line But I want to emphasize This is a little unexpected, but if I take an integer gamma of n and n is an integer The answer to this function will be the factorial of (n-1).
Pay attention to this. May mislead you A little difficult Since you cannot arrange 1 and 2 copies of 1 object So what is the purpose of this function for calculating non-integer factorials? This is just to generalize that this function is actually useful in different categories E.g. probability Can you apply this function to formulas related to probability For example, continuous time is different from arranging objects This function can be used for continuous events Time is the best example When you generalize different concepts Then we need to summarize the concept of factorial BRADY HARAN: 9 6 and 3 20 44.