What is the concrete meaning of the imaginary number i in math?

[Raddakar]

OK, first things first, I am not a university student in math, or anything. I am actually still in high school. I am very interested in math, even, say, too interested for my level :)

Well, yesterday I learned, on the internet, about the imaginary number i (Being \sqrt{-1}) and I read about it having no occurrence in nature. Well, I, in a couple of minutes of thinking, gave it a meaning. Maybe an expert can say how this doesn't make sense, that's why I came here to talk about this.

Say you are a sanity inspector. You have a little contraption that calculates the amount of water in the sewers. There is, say, 100 litres of clean water all the time, and the rest is waste water. Well, you know that a house produces constantly 10 litres of waste water. So that way, you can know how many houses are in your city with (x-100)/10, x being the amount of water in the sewers. Now, you know that in this particular city, houses are placed in a square grid. You are interested in the amount of houses on one side of that grid, so you may use \sqrt{(x-100)/10} to find out that piece of information.

Imagine now that there is a leakage in the sewers. We lose 100 litres of water, and we have, say, 9 houses, so (x-100)/10 is equal to -1 (because the amount of water in the sewers dropped to 90 litres, being the amount of waste water produced by 9 houses). Well, you blindly continue, and get \sqrt{-1} as the amount of houses on one side of the grid. We can replace that with i to make things simpler. Now, you cannot count i houses. You cannot see i houses. So i must be a different quantity. i can actually be defined by 10 houses worth of leakage. So i is actually an answer to another question than the one you asked. And therefore, i is a concrete, naturally occurring object.

 

[tvavanasd]

...Say you are a sanity inspector. ...

I am a sanity inspector (sorry couldn't help it).

Perhaps you are a sanitation or sewer inspector; likely not a very good one, cause I'm pretty sure that there isn't any "clean water" in the sewers.

The number 100 (clean water in the sewer) in your equation is therefore not valid.

If your grid is 100 houses x 100 houses, you have 10000 houses in your city (not houses^2).

If you look at half of the houses, you are looking at 5000 houses, not the sqrt.,,

 

[Raddakar]

Perhaps you are a sanitation or sewer inspector; likely not a very good one, cause I'm pretty sure that there isn't any "clean water" in the sewers.

Well, what I meant is like some kind of water to, say, decrease the concentration of waste water. Maybe 100 litres is too much...

The number 100 (clean water in the sewer) in your equation is therefore not valid.

The number itself can maybe too big, but no matter how you change that number (unless you bring it down to zero) there is always a point where the leakage is too big and gives a negative square root.

If your grid is 100 houses x 100 houses, you have 10000 houses in your city (not houses^2).

x*x=x^{2}

If you look at half of the houses, you are looking at 5000 houses, not the sqrt.,,

Oh, sorry, when I said "On one side of that grid" I meant:

• * *
• * *
• * *

Where bullets are the "side" and asterisks are the rest of the grid.
9 houses, 3 bullets. \sqrt{9}=3

Thanks for telling me where I wasn't precise and/or realist ^^

 

[Hurkyl]

Well, yesterday I learned, on the internet, about the imaginary number i (Being \sqrt{-1}) and I read about it having no occurrence in nature.

Alas you can't believe everything you read on the internet. While most people will use complex numbers to describe the things that are usefully described by complex numbers, there are a handful of people who are averse to anything that isn't a real number. The fact that there are a couple of useful ways to write a complex number as a pair of real numbers makes them an easy victim of such an attitude.

 

[g_edgar]

See the quote from Freeman Dyson in one of the answers here:

http://mathoverflow.net/questions/30156/demystifying-complex-numbers

 

[Pythagorean]

You can instead think of i as an operator that transforms a number to the orthogonal axis

 

[yossell]

You can instead think of i as an operator that transforms a number to the orthogonal axis

Not sure this is the kind of concrete meaning the original poster was looking for.