Mathematical Curiosities. (1 Viewer)

[nightweaver066]

Yeah so just whenever you integrate stick in a +C, its more natural to have 2 constants, but usually we only bother ourselves with one!

Thanks again.
Btw have fun at the party tmrw if you're going

 

[seanieg89]

This is just from my course notes summarised:






which is what we wanted. So the method is justified - but we skip a lot of theory!

Ignoring potential division by zero problems of course.

 

[asianese]

Thanks again.
Btw have fun at the party tmrw if you're going

What party...?

 

[asianese]

Ignoring potential division by zero problems of course.

Most people ignore division by zero

 

[iStudent]

Can someone explain why in:

a + sqrt(b) = c + sqrt(d)

a = c and
b = d

Teacher introduced it but never explained it when proving that if P(a+sqrtb) = 0, then P(a-sqrtb) = 0, where P(x) is a polynomial with real coefficients

 

[asianese]

The left hand side made up of a rational and irrational part. Same as the right hand side. If you know that both sides are equal, then since you've split up the rational and irrational parts completely, then they must be equal. Basically like if you know 'a' balls plus 'b' cubes equals 3 balls and 5 cubes, then a=3 and b=5. (Balls and cubes are different and so you can be sure you've split them up completely.)

 

[seanieg89]

Most people ignore division by zero

Well, that's cool if people state that these things are nonzero in the hypotheses of the working but it completely changes the situation. One solution can become infinitely many.

 

[asianese]

Well, that's cool if people state that these things are nonzero in the hypotheses of the working but it completely changes the situation. One solution can become infinitely many.

Yeah I understand. No doubt division by zero leads to proof that 0=1 etc.

 

[seanieg89]

The left hand side made up of a rational and irrational part. Same as the right hand side. If you know that both sides are equal, then since you've split up the rational and irrational parts completely, then they must be equal. Basically like if you know 'a' balls plus 'b' cubes equals 3 balls and 5 cubes, then a=3 and b=5. (Balls and cubes are different and so you can be sure you've split them up completely.)

are two ways of writing the same number that have different "rational and irrational parts". How would you define the rational part of a real number unambiguously?

 

[seanieg89]

Yeah I understand. No doubt division by zero leads to proof that 0=1 etc.

Well I don't mean doing illegal things, I just mean that if h,g can be zero in your problem, then you will get several solutions. (Or something like that.)

 

[seanieg89]

Can someone explain why in:

a + sqrt(b) = c + sqrt(d)

a = c and
b = d

Teacher introduced it but never explained it when proving that if P(a+sqrtb) = 0, then P(a-sqrtb) = 0, where P(x) is a polynomial with real coefficients

This fact is true whenever a,b,c,d are rational numbers and b,d have irrational square roots. (So b,d > 0. If b or d were negative, then this fact is still true but for different reasons.)

Assume a is not equal to c, which implies that b must also be unequal to d. Then:



This implies that is rational, and since is irrational, we have that is irrational.

But this is a contradiction, as:



Hence when such an equality holds we must have a=c,b=d.

 

[rolpsy]

Can someone explain why in:

a + sqrt(b) = c + sqrt(d)

a = c and
b = d

this isn't true as stated…

2 + √1 = 1 + √4
so 2 = 1 and 1 = 4, right?

you need a, b, c, d to be rational and one of b and d to not be a perfect square

edit: take a look at the post above

 

[asianese]

are two ways of writing the same number that have different "rational and irrational parts". How would you define the rational part of a real number unambiguously?

Ah yeah true. Mah bad.

 

[rolpsy]

interesting thread!

something that used to annoy me:

• If dy/dx isn't a fraction, then why is it that
  
  
  (inverse function theorem and chain rule)
  
  What are precise statements and how do you prove them?

 

[Carrotsticks]

interesting thread!

something that used to annoy me:

• If dy/dx isn't a fraction, then why is it that
  
  
  (inverse function theorem and chain rule)
  
  What are precise statements and how do you prove them?

On phone atm, but somewhere you can start playing around with.

Consider dy/dx as the limit of deltaX/deltaY (in this form it is now a fraction so they have multiplicative inverses etc ) as say x approaches zero, and use limit laws.

 

[seanieg89]

interesting thread!

something that used to annoy me:

• If dy/dx isn't a fraction, then why is it that
  
  
  (inverse function theorem and chain rule)
  
  What are precise statements and how do you prove them?

Well the inverse function theorem is more the statement that if a function is continuously differentiable in a region and has nonzero derivative at a point (or nonsingular Jacobian if we are dealing with higher dimensions), then the function with a small domain containing this point has an inverse, and this inverse is differentiable. Once we have this fact, the chain rule tells us that the derivative of this inverse must be the reciprocal of the derivative of our original function. (Or in higher dimensions, the Jacobians are inverse linear maps.)

Formally:

if f is continuously differentiable in a neighbourhood of a point p, with nonzero derivative at this point p.

I don't really want to write out a full proof of the inverse function theorem here (there are countless sources online), but in 1-dimension it is fairly easy to visualise. If f'(p) is nonzero we may assume wlog f'(p) > 0. Which by our assumption of continuous differentiability implies that f is increasing in a small neighbourhood of p. Increasing functions are invertible. The proof that this inverse is also differentiable is pretty easy to do manually.

As for the chain rule, it is a rather intuitive result. There are two proofs on the wikipedia page, the first is simpler but I prefer the second http://en.wikipedia.org/wiki/Chain_rule#Second_proof as it gets closer to the heart of what a derivative is and the proof generalises to higher dimensions easily.

Formally:

if g is differentiable at p and f is differentiable at g(p).

 

[rolpsy]

On phone atm, but somewhere you can start playing around with.

Consider dy/dx as the limit of deltaX/deltaY (in this form it is now a fraction so they have multiplicative inverses etc ) as say x approaches zero, and use limit laws.

still sounds quite hand-wavy. perhaps it can be formalised in terms of nonstandard analysis..?

Well the inverse function theorem is more the statement that if a function is continuously differentiable in a region and has nonzero derivative at a point (or nonsingular Jacobian if we are dealing with higher dimensions), then the function with a small domain containing this point has an inverse, and this inverse is differentiable. Once we have this fact, the chain rule tells us that the derivative of this inverse must be the reciprocal of the derivative of our original function. (Or in higher dimensions, the Jacobians are inverse linear maps.)

Formally:

if f is continuously differentiable in a neighbourhood of a point p, with nonzero derivative at this point p.

I don't really want to write out a full proof of the inverse function theorem here (there are countless sources online), but in 1-dimension it is fairly easy to visualise. If f'(p) is nonzero we may assume wlog f'(p) > 0. Which by our assumption of continuous differentiability implies that f is increasing in a small neighbourhood of p. Increasing functions are invertible. The proof that this inverse is also differentiable is pretty easy to do manually.

As for the chain rule, it is a rather intuitive result. There are two proofs on the wikipedia page, the first is simpler but I prefer the second http://en.wikipedia.org/wiki/Chain_rule#Second_proof as it gets closer to the heart of what a derivative is and the proof generalises to higher dimensions easily.

Formally:

if g is differentiable at p and f is differentiable at g(p).

ah this is more like it. though admittedly I've always found that first formula utterly indecipherable (this incarnation looks especially nasty)

[HR][/HR]
a few more qns to hopefully get people interested:

• How can we just define and expect things to behave as normal? What are we sweeping under the rug?

• If there is a formula for solving quadratic equations, are there cubic and quartic 'formulas'? Does a general formula exist for quintics and higher degree polynomial equations?

• is often defined (well, it was when I doing my hsc) as the solution to the differential equation . Is this solution, say , unique? (use whatever definition you want)

• Are π (pi) and e "more irrational" than numbers like √2 or √5?

 

[seanieg89]

still sounds quite hand-wavy. perhaps it can be formalised in terms of nonstandard analysis..?



ah this is more like it. though admittedly I've always found that first formula utterly indecipherable (this incarnation looks especially nasty)

[HR][/HR]
a few more qns to hopefully get people interested:

• How can we just define and expect things to behave as normal? What are we sweeping under the rug?

• If there is a formula for solving quadratic equations, are there cubic and quartic 'formulas'? Does a general formula exist for quintics and higher degree polynomial equations?

• is often defined (well, it was when I doing my hsc) as the solution to the differential equation . Is this solution, say , unique? (use whatever definition you want)

• Are π (pi) and e "more irrational" than numbers like √2 or √5?

It's mostly just that notation for the situation is a little cumbersome I guess. Just keeping the idea "the derivative of the inverse is the derivative of the inverse" is good enough to write down any formula you like for it.

I will leave these questions for a little while to give others an opportunity to answer some.

(Not that either is a less valid reason to post here, but are these questions things you personally are curious about or just you trying to promote interesting discussion? They are good questions .)

 

[rolpsy]

It's mostly just that notation for the situation is a little cumbersome I guess. Just keeping the idea "the derivative of the inverse is the derivative of the inverse" is good enough to write down any formula you like for it.

I will leave these questions for a little while to give others an opportunity to answer some.

(Not that either is a less valid reason to post here, but are these questions things you personally are curious about or just you trying to promote interesting discussion? They are good questions .)

probably another reason why some people prefer Leibniz's notation. it reads very nicely

i just thought of a few questions that i think are interesting. hopefully others find them interesting/surprising(?) too. i'll wait a few days then answer them if no one else has (but hopefully this won't need to happen)

 

[seanieg89]

Cool, I thought the questions were a little too well-crafted to be asked by someone who didn't have a pretty good idea of their answers.