Random math questions

[Jeggz]

Can someone please explain how you find the real part and the imaginary part in an expression, when it's not in the standard form?
My teacher said something about real part = (real x real) - (imaginary x imaginary)? But I'm a bit confused..

[b^3]

What do you mean by not in standard form?
Do you mean something like ?
In that case lets look at what happens when we expand it out.

Now we know that , which brings us to

Which is what you're teacher was getting at with real times real minus imaginary times imaginary.

[Jeggz]

What do you mean by not in standard form?
Do you mean something like ?
In that case lets look at what happens when we expand it out.

Now we know that , which brings us to

Which is what you're teacher was getting at with real times real minus imaginary times imaginary.

Sorry I should have been more specific! What I meant was when we're dealing with things like differentiation and antidifferentiation of exponentials. For examples showing how e^ax sin(bx) = (e^ax / a^2+b^2 ) (asin(bx) - bcos(bx)) ? Im really sorry if this question doesn't make sense.. I don't know how to use latex?!

[b^3]

I'll start with the sine case, (if we are integrating sine we will use Im part and if we are integrating cosine we will use the Re part, you'll see why later).

Now we notice that we can turn our expression into something involving a complex exponent.

Now we know that


Now if we wanted to take the imaginary part of our expression, how would we find it? What multiplied by what will give an imaginary part?
Real Real gives Real
Imaginary Imaginary=Real (the terms comes out to give a negative)
Real Imaginary=Imaginary
Imaginary Real=Imaginary

So to find the imaginary part we need to look at the latter two.


If we wanted to take the Real part, when integrating a cosine, we would need to do the first two, that is the Real Real and Imaginary Imaginary.

Hope that makes sense

[Jeggz]

I'll start with the sine case, (if we are integrating sine we will use Im part and if we are integrating cosine we will use the Re part, you'll see why later).

Now we notice that we can turn our expression into something involving a complex exponent.

Now we know that


Now if we wanted to take the imaginary part of our expression, how would we find it? What multiplied by what will give an imaginary part?
Real Real gives Real
Imaginary Imaginary=Real (the terms comes out to give a negative)
Real Imaginary=Imaginary
Imaginary Real=Imaginary

So to find the imaginary part we need to look at the latter two.


If we wanted to take the Real part, when integrating a cosine, we would need to do the first two, that is the Real Real and Imaginary Imaginary.

Hope that makes sense

Thankyou (x million!)
It totally make sense to me now

[#1procrastinator]

How would you prove the product of a rational number and an irrational number is irrational? And the sum of a rational number and an irrational number is irrational?

Thanks

[Lasercookie]

For the sum of a rational number and an irrational number is irrational

Let a = rational number, b = irrational number
So aiming for a proof by contradiction, suppose that , where p is a rational number

Subtract a, a rational number from both sides


Since a and p are rational numbers, we can express them as ratios of integers and where x, y, m, n are integers (also y and n non-zero)



Since this is still a ratio of integers (product of integers are integers, sum of integers are integers - not sure how to prove this, but it seems obvious that it'd be true) this is still rational.

So that means is rational. Which is a contradiction! (yay it worked - assuming I did everything correctly) since we defined b to be irrational at the start.


Edit: Figured out the other one, it was easier than I thought/fairly similar to the one above.

a = rational, b = irrational
where p is rational



So we can show that is rational (product of integers are integers), which means that b is rational, hence contradiction.

Hopefully I haven't done anything iffy.

[kamil9876]

How would you prove the product of a rational number and an irrational number is irrational?

Thanks

FALSE



That's the only exception btw, gotta make sure that the rational number you are multiplying by is non-zero.

[#1procrastinator]

thanks laser, i got something a bit more complicated for the more multiplication one and it ended up being circular lol

wtf son oO

thanks :p

[#1procrastinator]

Let and be polynomials with coefficients from a field F. Supposed m is less than or equal to n and define . Then g(x) can be written as

The point of that was to then show that the set of polynomials under the operations of addition and scalar multiplication is a vector space.

My question is how does defining let us write g(x) in terms of n's rather than m's?

[#1procrastinator]

How would you prove the sequence converges to 0? I'm supposing you need to use the epsilons and deltas but not really sure how to manipulate that inequality to get find N.

Note that this is an assignment question so I'm only after some hints :p

[kamil9876]

My question is how does defining let us write g(x) in terms of n's rather than m's?

It's a bit too pedantic. I guess what the author must have been worried about is that and have a different number of terms, and so it doesn't really make sense to add a vector from to a vector from

How would you prove the sequence converges to 0? I'm supposing you need to use the epsilons and deltas but not really sure how to manipulate that inequality to get find N.

Note that this is an assignment question so I'm only after some hints :p

Seems quite weak, both terms go to zero so you can use linearity of limits.

[#1procrastinator]

Yeah he makes it way more confusing I think. Like if m=n, then both polynomials would have the same number of terms and I can see how we could use 'n' rather than m. But he defines so there would be one less term for g(x).

---

I think I'm supposed to prove that it goes to zero ('show that the sequence converges to zero by arguing directly from the definition of convergence'). Worth 3 marks 

[Lasercookie]

I think I'm supposed to prove that it goes to zero ('show that the sequence converges to zero by arguing directly from the definition of convergence'). Worth 3 marks 

I'm assuming that you're just trying to make sure your answer was correct, but just to double check but you did submit the assignment yesterday right?

But yeah the way I approached it was to split it into two parts, and then showed that each part separately converges to zero (using the definition of convergence stuff), and since limits are linear then a_n converges to zero.

The assignment solutions (they're on wattle now) start off similar, but then use the triangle inequality to say that which I thought was a pretty interesting approach (definitely not one that came to my head)

[#1procrastinator]

 

I'm assuming that you're just trying to make sure your answer was correct, but just to double check but you did submit the assignment yesterday right?

 

I should stop trying to do the assignments on the day they're due haha.

Thanks...I think I'm gonna spend a bit more time trying to do it in a way I understand first but I've got so much other stuff to do. MUST STOP PROCRASTINATING