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[MagmaMcFry]

Only one of the complex roots is called the principal root, it's generally the one with the smallest argument (the argument is the angle to the positive X-axis in the complex plane). So if you take the nth complex roots of any complex number and look at them in the complex plane, they'll form a regular n-gon around the complex origin, and the principal root is the first root you stumble upon when turning the positive real axis counterclockwise.

[Skyrunner]

Question~

₀C₀ != 1, according to my math tutor. In fact, it's not defined. But the equation for nCr is n! /(r!(n-r)!), and since 0! = 1, I can't see any reason why ₀C₀ shouldn't exist :X

Curiously WA seems to say 0C0=1, and I don't have my ol' TI84 with me.

[da_nang]

Question~

₀C₀ != 1, according to my math tutor. In fact, it's not defined. But the equation for nCr is n! /(r!(n-r)!), and since 0! = 1, I can't see any reason why ₀C₀ shouldn't exist :X

Curiously WA seems to say 0C0=1, and I don't have my ol' TI84 with me.

C(0,0) is the same as the number of subsets of the empty set that contain no elements. There is only one such subset, the empty set itself.

[MagmaMcFry]

What kind of definition does your tutor use that does not define 0C0?

[Skyrunner]

The Korean High School Definition, I guess

[Pnx]

Hmmm, so I'm trying to an area between two curves problem.
I've got

2/(1+x²)
and
3-2cos(x)

But I'm having trouble finding where these two graphs intersect. Can anyone help?

[MagmaMcFry]

I'm pretty sure that there isn't actually a way to solve such a graph intersection exactly. Just do a numeric approximation.

[fractalman]

y=2/(1+x²)
and
y=3-2cos(x)

lets see...2/(1+x^2)=3-2cos(x)
2=3+3x^2-2cos(x)-2x^2cos(x)

0=1+3x²-2cos(x)-2x²cos(x)

...lemme guess, that's as far as you got too?

...yeah, just do it numerically, graphing calculators have a pretty decent zoom if you're too lazy to newton's method it. 

[Helgoland]

Hmmm, so I'm trying to an area between two curves problem.
I've got

2/(1+x²)
and
3-2cos(x)

But I'm having trouble finding where these two graphs intersect. Can anyone help?

1/(1+x²) looks a bit like being related to logarithms somehow - through integration, maybe, I haven't done calculus in about a year. I'd just try stuff with analytical methods.

[Vector]

1/(1+x²) looks a bit like being related to logarithms somehow - through integration, maybe, I haven't done calculus in about a year. I'd just try stuff with analytical methods.

It's the derivative of the arctangent.

[Descan]

For math problems I'm stuck on I.E. I don't know what it's SUPPOSED to look like, I use Wolfram Alpha to find out what my end result should look like, and then try and fill in the missing steps.

To that end:

Spoiler (click to show/hide)

They seem to intersect at 1 and -1, with y-intercepts at 2 for 2/(1+x²) and at 1 for 3-2cos(x). How you would derive the intermediate steps, I don't know. I haven't worked on calculus and areas under curves yet.

Edit: Upon further Wolframing, it turns out the intersecting points are (±0.649256, 1.4~~~)

[Helgoland]

It's the derivative of the arctangent.

Okay, great - then integration should do the trick, shouldn't it? There's theorems for that, aren't there?

[MagmaMcFry]

Integration won't really help with anything. As I mentioned earlier, there seems to be no exact method to find roots of nontrivial polynomials in both x and exp(x) (unless you use the Lambert W function, which will still kinda not be useful for anything). And really, if there is such a method, it would be far too advanced for people who are just learning about integration and areas between two curves (which is my impression of the current situation).

[Helgoland]

Idea: Either the solution is a nice round number, then we could just try out some nice round numbers that give nice round cosine values (0 and pi come to mind), or it's not, which would make a numeric approach the best choice.

On the other hand - telescopic sums! Do a taylor expansion of cos x, and bring 1-x² to the other side; that should give an at least tolerable sum.

[Descan]

Well, the intersecting points are -not- round numbers, I can tell you that much. Just look under my spoiler for what the co-ordinates are.