leehuan's All-Levels-Of-Maths SOS thread (6 Viewers)

[drsabz101]

Hey guys, can someone please help answer the questions showing full working Thankyou

 

[Nailgun]

Hey guys, can someone please help answer the questions showing full working Thankyou
View attachment 32866

Which questions?
I ceebs doing them atm, but for sanity lol

 

[InteGrand]

Which questions?
I ceebs doing them atm, but for sanity lol

Guessing she meant all of them haha. They're mainly just tedious / dull.

 

[Nailgun]

Guessing she meant all of them haha. They're mainly just tedious / dull.

Ahah they're from an early chapter in the Yr11 pender 3u book, so they're really meant for just-finished-yr10 levelish

 

[InteGrand]

Ahah they're from an early chapter in the Yr11 pender 3u book, so they're really meant for just-finished-yr10 levelish

Haha yeah. Meant tedious like having to solve simultaneous equations etc.

 

[drsabz101]

only the first 2

 

[Nailgun]

For 17, since B is on the y-axis the coordinate is (0,y)
use Pythagoreas theorem to find how high on the y axis it is

Use midpoint formula with A and B to find the midpoint of the ladder
and then use the distance formula from origin to the midpoint to find the radius

 

[InteGrand]

only the first 2

 

[Shuuya]

only the first 2

 

[InteGrand]

 

[Shuuya]

HAHAH so awkward :')

 

[leehuan]

How is your handwriting so on point

 

[Shuuya]

How is your handwriting so on point

You wouldn't say that if you saw my writing for english exams

 

[leehuan]

You wouldn't say that if you saw my writing for english exams

Still a heck load tidier than mine on the integration marathon and etc.

 

[Nailgun]

You wouldn't say that if you saw my writing for english exams

I always assumed you were a guy, but now I'm not sure

 

[Shuuya]

I always assumed you were a guy, but now I'm not sure

Whattt why is my gender being questioned because of my writing >_<

 

[seanieg89]

Can someone explain to me why there are no higher order analogues to the Gaussian Integers and Eisenstein Integers?

And for the sake of normal people, you are allowed to provide geometric intuition.

By this I assume you are asking why the only rings of the form S={a+bw : a,b integers} (equipped with the standard sum and product operations of C) where w is a complex number of unit modulus are the Gaussian integers and the Eisenstein integers.

(If you mean something else please clarify).



For starters, w must be a root of unity, otherwise its powers are dense in the unit circle (exercise). In fact this implies that the ring generated by S must be dense in the complex plane. As S is clearly not dense in the plane (because every nonzero point has magnitude >= min(1,Im(w)) for instance), this implies that S cannot be a ring when w is not a root of unity.

Let us now assume that w=e^(2*pi*i/n) for some n > 2.

We ask for which n will S be a ring.

As we require closure under multiplication and addition, we require:

w+1/w=w+w^(n-1) to be in S. But as this quantity is real this is the same as demanding that it be an integer.

This implies that the real part of w must be a half integer, so it can either be 0 or +-1/2, which yield the Gaussian and Eisenstein integers respectively.

 

[Paradoxica]

By this I assume you are asking why the only rings of the form S={a+bw : a,b integers} (equipped with the standard sum and product operations of C) where w is a complex number of unit modulus are the Gaussian integers and the Eisenstein integers.

(If you mean something else please clarify).



For starters, w must be a root of unity, otherwise its powers are dense in the unit circle (exercise). In fact this implies that the ring generated by S must be dense in the complex plane. As S is clearly not dense in the plane (because every nonzero point has magnitude >= min(1,Im(w)) for instance), this implies that S cannot be a ring when w is not a root of unity.

Let us now assume that w=e^(2*pi*i/n) for some n > 2.

We ask for which n will S be a ring.

As we require closure under multiplication and addition, we require:

w+1/w=w+w^(n-1) to be in S. But as this quantity is real this is the same as demanding that it be an integer.

This implies that the real part of w must be a half integer, so it can either be 0 or +-1/2, which yield the Gaussian and Eisenstein integers respectively.

Yeah, that's all I needed. For the geometric intuition, could that be due to the lack of uniform grid coverage by the unit vectors? The Eisenstein integers form a triangular lattice over the complex plane, and the Gaussian integers form a square lattice over the complex plane. Nothing else could form a lattice structure like those two, so nothing else exists.

 

[seanieg89]

Yeah, that's all I needed. For the geometric intuition, could that be due to the lack of uniform grid coverage by the unit vectors? The Eisenstein integers form a triangular lattice over the complex plane, and the Gaussian integers form a square lattice over the complex plane. Nothing else could form a lattice structure like those two, so nothing else exists.

I don't know how easy/natural it would be to make such a geometric argument rigorous, as this seems to be more of an algebraic/number theoretic fact.

Like, you can tesselate the complex plane by any kind of rhombus you like (with the base one having vertices 0,1,1+w,w where w is an arbitrary non-real number of unit modulus), but the point is you can't pass the multiplicative structure of the complex numbers to this lattice for the reason above.

The restrictive condition is multiplicative closure.

 

[drsabz101]

How to do this question :

A) find the equation of the tangent to y= square root of x, -1. Where x=t
I differentiated this , then sub in t for x .