Hint 'parochialism' is the sole obstacle to solution of this exercise

Parochialism is the state of mind, whereby one focuses on small sections of an issue rather than considering its wider context.

 

Ok... Now what does √(-3) mean?

 

Parochialism is the state of mind, whereby one focuses on small sections of an issue rather than considering its wider context.

...or clinging to familiar/comfortable ideas...

 

Ok... Now what does √(-3) mean?

I have studied but forget

 

I have studied but forget

Can you write it as an approximate decimal expansion?

e.g. the approximate value of √3 may be written as √3≈1.73205 So how would write the approximate value of √(-3)

 

my calculator is showing math error.

 

Guess you'll have to use your head

 

-1.73205

 

Hello,

Ever thought on imaginairy numbers?
https://en.wikipedia.org/wiki/Imaginary_number
http://www.mathsisfun.com/numbers/imaginary-numbers.html

Bertus

 

Bertus, that mean you have eye on thread

 

@RRITESH KAKKAR I probably wont be available tomorrow – So I leave you with something to think about suggested by my very good friend @Aleph(0) :

Prove that: √(-1)^√(-1) = e^(-∏/2)

Where 'e' is Euler's number (i.e. the base of the natural logarithm ≈2.71828)

Don't worry if you can't manage the proof! -- It requires some trigonometry --- Just think about it... OK?

Best regards and I'll chat with you later in the week!
HP

 

For your amusement listen to the linked video -- pretend he's saying 'numbers' every time he says 'lovers'

 

@RRITESH KAKKAR

Gotta run -- again, I won't have much time tomorrow -- chat with you in ≈ 42 hours -- Congratulations on your good work today!!!!

Best regards
HP

 

WHY there is holiday

 

first i want a answer from you, what is the importance of study or research?
I mean what will have if i will become fully trained engineer or anything else?

This question has many answers and they are tied to the individual. I studied (and continue to study) because I love to learn new things -- and it can be just about any topic. Others study with purely pragmatic motives, meaning that they recognize that, to achieve their goals, they have to learn certain things. The goals themselves are highly varied. For one person it might be to make enough money to buy the things they've always wanted. For another it might be to be able to help their community or communities half a world away. For some it might be so that they are able to do things such as open their own business or hike across the mountains and survive off the land.

Narrowing the discussion to engineering and such, it still depends on your motives. I love engineering and am fortunate that I can make a decent living at it. But I could make a lot more money if I were to focus on different things, but I prefer to focus on areas that give great personal satisfaction but limit my income significantly. It's a choice I've made and, most days, I'm happy with it. But whatever your goals are in engineering, you can't escape the fact that, at the heart of it, engineering is about problem solving and you have to study and practice to become good at it. There are lots of engineers that think if they just do the minimum to get a piece of paper that says they have a degree in engineering that some company is just going to throw lots of money at them. Well, it seldom works that way. In order to be successful you have to be good and you won't get good unless you put in a lot of time and effort. Above all, one thing you need to always remember is that even if you don't love engineering, you will always be competing against people that do.

 

But I could make a lot more money if I were to focus on different things, but I prefer to focus on areas that give great personal satisfaction but limit my income significantly.

you must be rich man as you are pilot, my ear want to know how much you earn/salary.
as you said you got full marks.

 

Via the the stipulation that R1 ≤ R2 and the implicit lower limit of R1 = 0 (owing to the fact that it is a resistor)

So:
Req=(0*R2)/(0+R2) Indeed yields a lower limit of 0 for Req

So... what's the upper limit or Req?

Keep in mind that the goal is to relate the limits on Req in relation to the values of R1 and R2. We aren't looking for the limits on Req for all values of R1 and R2, but for the specific values they happen to be.

So we can't just say that R1 could be zero ohms. It can't. R1 is whatever R1 is.

Let's use an example to illustrate the practical point.

You have two resistors, R1 = 1800 Ω and R2 = 2200 Ω. Without batting any eye or doing any more work than dividing by two, you know that the parallel combination of these two resistors has to be more than 900 Ω (half the smaller one) but can't be more than 1100 Ω (half the larger). The question that I posed is aimed at establishing this relationship. Picking another two values, R1 = 1800 Ω and R2 = 8200 Ω. As before, we know that the value has to be more than 900 Ω and less than 4100 Ω, but the other part of the question allows us to know that the largest it can be is 1800 Ω (the smaller of the two). Again, the point of the question is to show that this is the case, instead of just accepting it on faith.

 

you must be rich man as you are pilot, my ear want to know how much you earn/salary.
as you said you got full marks.

Flying was a hobby for me -- though I always wanted to get my instructor's ticket so that at least others could pay for most of my flight time. I'm definitely not a rich man and flying is sufficiently expensive that I never was able to do a lot of it. I only have a little over 300 hours pilot-in-command time. I haven't flown in over a decade and, due to health issues, there's a good chance I will never fly pilot-in-command again.

 

see you tomorrow.

 

There's an even tighter upper bound: (R1)/2 ≤ Req ≤ (R1+R2)/4

Yes, if you are willing to use both values in the bound, you can come up with a tighter bound.

This one might be a good one to use to show RRITESH the idea here.

Regardless of whether R1 is bigger than R2, we know that, since anything squared is non-negative, that

0 < (R1-R2)²

If we multiply this out, we get

0 < R1² - 2R1R2 + R2²

If we add 4R1R2 to both sides we get

4R1R2 < R1² + 2R1R2 + R2²

The right hand side is simply

4R1R2 < (R1 + R2)²

Dividing both sides by (R1 + R2) we get

(4R1R2)/(R1 + R2) < (R1 + R2)

and dividing both sides by 4 we get

(R1R2)/(R1 + R2) < (R1 + R2)/4

The left hand side we now recognize as being Req, so

Req < (R1 + R2)/4

Of course, Req < R1 becomes the tighter bound as soon as R2 > 3R1