So (!) Is an operation called Factorial. Factorial is that number multiplied by all positive intgers before itelf. So 1!=1, 2!=2, 3!=6, 4!=24, 5!=120 etc. Factorial can also be expressed (where n is a integer) as n!=n*(n-1)*(n-2)*...*3*2*1, or can be partially expressed as n!=n*(n-1)!. If you set n=1, then 1!=1*(1-1)!=1*0!. Since since we know 1! also know 1!=1, therefore 0!=1. Also anything to the power of 0 equals one. So 0^0=0! !
random bit of explanation so happy frog here doesnt look insane to anyone who understandably forgot math class from 30 years ago.
to "factorial" something you multiple it times all the numbers below it. a factorial is shown in a math as an exclamation mark after the number such as 3 factorial being shown as 3!
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Let's use our example of 3!
3! is 3 times everything below it which is written longways as follows.
3! = 3x2x1
which equals 6.
the principle that allows 0! to equal 1 is from an odd thing with how factorials can be written. You can also write a factorial like this
3! = (4!/4)
to show this is true if we follow up with the answer we get
4 x 3 x 2 x1 = 24
making (4!/4) = (24/4) which equals 6, the same as 3! which we showed above also equals 6.
if we do the same for 0!
0! = (1!/1)
we can rewrite 1! as (2!/2) which is (2x1)/2=1
which leaves us with 0! = (1/1) or
0!=1
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It's like the 0.9999... = 1 trick. It's a neat little toy to play with the expression, but you'll never use it any practical application, as you'll just actually write what you mean.
the same people that will get no practical application out of these "tricks" will get no practical application out of any of this kind of math. those that will get practical applications out of this kind of math will have use for these kinds of tricks.
That... yeah, that's a better point to make.
0! is going to have to be used at some point if you want to justify why a factorial works, program a computer to solve factorials, and shows up in physics (relatively speaking to how broad a subject that is, you could specialize in an entire branch and, in a way, be a part of a human factorial but never use one) o_O.
The point being, even if you are doing something like arranging numbers, when making the arrangement of no numbers, which is still an arrangement, would still be expressed by 1 unless you really just want keep that much consistency (can't really fault anyone on that, as there are some real dicks out there, so if you are submitting a paper or something, someone who thinks you are wrong but can't prove it could attack that).
I'd say perhaps defining "practical" would be a good idea, but as that's subjective, I'd imagine that ends with something resembling a circle... like... idk... 0!
I should explain that entire branch part; that could be confused for fractals, which I'm not talking about. I mean something more like physics over time, where one insight impacts the rest of physics, and it all is... cumulative and compounding; so insights can compound as physics itself expands, like the concept of pressure differences leading to wings, which then someone thought about putting something like that underwater so they worked on the fluid dynamics and added more.
to "factorial" something you multiple it times all the numbers below it. a factorial is shown in a math as an exclamation mark after the number such as 3 factorial being shown as 3!
-
Let's use our example of 3!
3! is 3 times everything below it which is written longways as follows.
3! = 3x2x1
which equals 6.
the principle that allows 0! to equal 1 is from an odd thing with how factorials can be written. You can also write a factorial like this
3! = (4!/4)
to show this is true if we follow up with the answer we get
4 x 3 x 2 x1 = 24
making (4!/4) = (24/4) which equals 6, the same as 3! which we showed above also equals 6.
if we do the same for 0!
0! = (1!/1)
we can rewrite 1! as (2!/2) which is (2x1)/2=1
which leaves us with 0! = (1/1) or
0!=1
-
0! is going to have to be used at some point if you want to justify why a factorial works, program a computer to solve factorials, and shows up in physics (relatively speaking to how broad a subject that is, you could specialize in an entire branch and, in a way, be a part of a human factorial but never use one) o_O.
The point being, even if you are doing something like arranging numbers, when making the arrangement of no numbers, which is still an arrangement, would still be expressed by 1 unless you really just want keep that much consistency (can't really fault anyone on that, as there are some real dicks out there, so if you are submitting a paper or something, someone who thinks you are wrong but can't prove it could attack that).
I'd say perhaps defining "practical" would be a good idea, but as that's subjective, I'd imagine that ends with something resembling a circle... like... idk... 0!