O! (that's zero factorial) (1 Viewer)

[m_isk]

Can anyone explain to me how 0! is equal to one?? and also, is there any way to integrate cosine squared and sine squared other than using cos2x?? thanks.

 

[rama_v]

I think it has to do with the fact that a lot of times you will have 0! on the denominator and if it wasnt equal to one, then there's serious porblems. This occurs in nCr especially. Basically, there is only 1 way to chose 0 objects

Here's a much better explanation
http://mathforum.org/dr.math/faq/faq.0factorial.html

 

[KFunk]

It's largely a matter of how it is defined but as rama_v said it's partly to do with the formula for the number of possible selections and the fact that you can arrange zero objects in 1 way.

 

[withoutaface]

Faced between a choice of being able to arrange zero objects one way (ie 0!=1), or not being able to arrange them at all (ie 0! being undefined), convention chose the former.

 

[FinalFantasy]

Can anyone explain to me how 0! is equal to one?? and also, is there any way to integrate cosine squared and sine squared other than using cos2x?? thanks.

Let I=int. sin² x dx
let u=sin²x and dv\dx=1
du\dx=2sinxcosx=sin2x and v=x
I=xsin²x-int. xsin2x dx
let I1=int. xsin2x dx
let u=x and dv\dx=sin2x
du=dx and v=-(1\2)cos2x
I1=-(1\2)xcos2x+(1\2)int.cos2x dx
=(-xcos2x)\2+(1\2)(1\2)sin2x
=(-xcos2x)\2+(1\4)sin2x
.: I=xsin²x+(xcos2x)\2-(1\4)sin2x+C

 

[m_isk]

are we allowd to use parts in a 3u exam??? provided the question is not preceded by prove that cos2x = blah blah blah?

 

[jm1234567890]

are we allowd to use parts in a 3u exam??? provided the question is not preceded by prove that cos2x = blah blah blah?

just use the cos2x method, it will cause you less pain.

why don't you like it? it is 1 step.

 

[m_isk]

hmmmm..i dunno. It's just that there was one question that said "...hence or OTHERWISE integrate cos^2 x. I usually know an "otherwise" method, but this time I didn't until FF came along.

 

[jm1234567890]

hmmmm..i dunno. It's just that there was one question that said "...hence or OTHERWISE integrate cos^2 x. I usually know an "otherwise" method, but this time I didn't until FF came along.

they say otherwise because if you couldn't do the previous part you could still attempt this part.

 

[jarro_2783]

my teacher told me you are allowed to use any legitimate maths in the exam. Even if it isn't part of the syllabus. So if you know some uni maths you can use it in the exam if you want.
As for integral of cos^2 x. My textbook has a formula for it. You can do it from cos2x which gets you there anyway.
It gives the formula as:

I = x/2 + (1/4)sin2x + C

So integral of cos^2 (2x+1) =
x/4 + (1/8)sin(4x+2) + C
edit: that's wrong - it's x/2 + (1/8)sin(4x+2)

FinalFantasy: in your last line I=xsin²x+(xcos2x)\2-(1\4)sin2x+C
xsin²x + (xcos2x)/2 = (2xsin²x + xcos2x)/2
cos2x = cos²x - sin²x, so xcos2x = xcos²x - xsin²x
therefore, (2xsin²x + xcos2x)/2 = (2xsin²x + xcos²x - xsin²x)/2
= xsin²x + xcos²x
= x(sin²x + cos²x)
= x(1)
so the whole integral becomes x/2 - (1/4)sin2x + C

 

[who_loves_maths]

just going back to 0! being equal to one,
the more technical reason is within Set Theory. All sets have subsets, (and all subsets have sub-subset, etc) and any set must have the empty set as a subset (yr. 9 maths) so 0! is the number of empty sets as subsets which is just one for every set.
it's logical since an empty set is itself counted as a set, so if an empty set is the subset of a larger set, then 0! for that set must clearly exist = 1.

 

[underthebridge]

Whether this is a proof or not I'm not sure, since it's assuming the combinations formula. Using the formula, nCr = n!/(n-r)!r! in the case of nCn, r = n, and so nCn = n!/0!n! = 1/0! but we know that there is only one possible combination of choosing n objects out of n, therefore 0! must equal 1.

 

[Weimin]

As 0!=1 and 1!=1
is it that mean 0=1?
My teacher says 0 is not equal to 1,but one can prove 1=2

 

[FinalFantasy]

As 0!=1 and 1!=1
is it that mean 0=1?
My teacher says 0 is not equal to 1,but one can prove 1=2

u can prove 1=2 only if u perform a logic error during ur proof and ignore it

 

[jm1234567890]

As 0!=1 and 1!=1
is it that mean 0=1?
My teacher says 0 is not equal to 1,but one can prove 1=2

no , cause there is no inverse factorial operator

 

[maths > english]

you can integrate cos²x and sin²x by using complex numbers (example of sin²x attached)

ⁿP_n = n!

ⁿP_r = ^n!/_(n-r)!

therefore ⁿP_n = ^n!/_0! = n!

n! = 0! x n!

0! = 1