Why is the square root of -1 useful in mathematics?

[Borogoves]

i am abslotuely not a platonist, you cannot deduce that from what i said. i said i and e are equally "real". onotologically they are practically equivalent. complex numbers, being a divisoin structure on R^2 are equally as "real" as the real numbers since they onlyu require the reals, and the basic properties of sets for their definition. and i is incredibly useful in "the real world". point out something in the "real" world that is e in a way that i cannot be expressed. labellign the reals real and the vomplexes imaginary speaks only of our psychology not our mathematics.

Yes I agree that the labelling of reals/complex is a misnomer.
It all depends on the definition of real or imaginary.

Exponential growth or decay is ubiquitous in the real world, whereas i is far from intuitive. The irony is that i, pi, and e are all captured beautifully in the Euler formula.

 

[matt grime]

but that has nothing to do with the mathematics of it.

 

[Borogoves]

but that has nothing to do with the mathematics of it.

What are you talking about ?

What do you personally mean by real or imaginary ?

Point out an application of i then ? What matters is that a mathematical model, describes the world even though it may not be perfect.

Do you study math to make discoveries in your chosen field or because you enjoy it as a subject in its own right ?

 

[toocool_sashi]

Lets go back to some electricity. V=iR Ohm's Law. i denotes current. Why isn't current usually denoted by "c" for current or "A" for ampere?? well "mathematicians" believe that current is denoted by i so as 2 show the importance of complex numbers in electrical engineering ;) u can find out from any elec engg abt the validity of this point...i wudnt know...im starting mechanical engineering course in abt 2 weeks :) cheers!

 

[lurflurf]

Lets go back to some electricity. V=iR Ohm's Law. i denotes current. Why isn't current usually denoted by "c" for current or "A" for ampere?? well "mathematicians" believe that current is denoted by i so as 2 show the importance of complex numbers in electrical engineering ;) u can find out from any elec engg abt the validity of this point...i wudnt know...im starting mechanical engineering course in abt 2 weeks :) cheers!

Is this a joke(there was a ;))? The symbol used to represent something is not important. The "i" in Ohm's law has nothing to do with complex numbers. To avoid confusion scientist and engineers commonly use j^2=-1 when confusion might arise with another symbol, such as when i is being used to represent current.

 

[meckano]

I've read somewhere that the square root of -1 is handy when it is divided by itself, leaving ofcourse, 1.
I think it was used to show that a ?muon? leaves the opposite side of a mountain as soon as it enters the mountain.

I prefer, at times, to see the square root of -1 and division by zero as meaning that it has left our 'real' world / ceases to exist... whatever.
In the muon case above, I would see it as leaving 'reality' as it hit the mountain, and another was created at the same time on the opposite side.
Similarly, an electron in a copper conductor does not flow, rather it bumps one which bumps the next. ~~~And the little one said, roll-over, roll-over...
- solong as that is still the believed scenario for an electron.

 

[ChrisHarvey]

I've just started reading "Visual Complex Analysis" by Tristan Needham. It is an amazing book that explains complex numbers in such an clear way from a geometric point of view. I covered complex numbers in a basic way in high school and was left asking myself the same questions being asked here. What exactly are they? What do they mean? They just seemed to be a mathematical curiosity. Having only just read the first few chapters I can honestly say I feel 'happy' with them and am seeing uses for them where I woudn't usually think of them. I would recommend the book to anyone. One thing that sticks in my mind so far is the geometric derivation of Euler's formula in chapter 1. Who would have thought that e^(i*theta) = cos(theta) + i*sin(theta) could seem obvious when thought of through the eyes of geometry (and when shown!)?

Visual Complex Analysis