Does imaginary part of complex number represents the meaning of down payment or stealing in real life??

Question

I am new to complex numbers and trying to understand them and find real life conditions which could be interpreted by them "like positive one apple is gained apple and negative one apple is lost apple....so what is +1i apple and -1iapple" ....

I know that electromagnetism ,ac, wave analysis and quantum mechanics are explained by complex numbers but i can not hold them in my hand like apples...
So i had first thought that +1i may represent the situation when two persons A,B are making bets "gambling" over the result of match
And from point of view of person A : person A put -1i dollar with intermediate person ,person B put +1i dollar with the intermediate person .....
Here for person A, I consider the dollar he gives for the intermediary person -1i because if he loses, he transfers this -1i dollar to real -1 dollar ....
and the dollar given by B to the intermediary is +1i relative to A because if A wins he converts this +1i into real +1 dollar...
I do not know if my understanding is right but i continue to study complex numbers and i solve the following problem :
p(x)=-0.3x^2 +50 x-170 where p is the profit and x is the lamps produced per week.
"note here that i do not know the price of lamp or the cost of manufacturing and that the profit is nonlinear"
then I try to find number of lamps which make profit of 3000 dollars.
the solution is complex number with real part 83.3333 and imaginary part +-60.1849i.
Now i try to get profit for real part alone "1913.3333"and imaginary part alone"916.6667" then add them together but the result is 2830 not 3000.how this is possible????
Note that 2830 +170 =3000????!!!

During this thinking i tried to calculate the imaginary part after conversion to real 60.1849 ,then i thought of this positive imaginary part as taking full price in advance "down payment of 1752.5783" for future production "to get beyond the maximum profit per week of 1913.3333" and I thought of the negative imaginary part as stealing this 60 lamps from the deal and reselling them so we also get beyond barrier of weekly maximum profit...
but calculations for both cases were not equal to 3000 "1913.3333 + 1752.5783 = 3665.9117" does my guessings about meanings of imaginary part as price in advance or stealing true???

I know that my question may be vague ... this is because i am confused about meaning of complex numbers and i will accept all edits to make it more clear...

Answer 1

Actually no number really exists. The natural numbers are just a convention for counting things. Like, an apple is represent by 1, and see an apple together with another apple is represent by 2, and so on. But for example, you can't say what does means having -1 apple, or $\pi$ apples.

Basically math is about making some "arbitrary" definitions that makes things good. You can define whatever you want (if it is not self contradictory, like let $x=1$ and $x\neq 1)$, but you try to define things in such way that you get good results.

For example, think about analytic geometry. Its about describing plane geometric things. The usually $xy-$plane are useful for, for example, describing straight lines, because the equations are simple, like $y=2x$. But describing rotations on the $xy-$plane, usually, leads to really complicated equations, involing $\sin$ and $\cos$. How can we avoid this?

Think you want to describe plane geometric things, but you are more interested in some rotations. Instead of thinking about the $xy$ plane, why don't you introduce a "vector component" $w$ that rotate things? For example, if you have some vector $v$ and multiply this component, you would have the same vector $v$, but rotated by $90º$ ($\pi/2$ radians) counter-clockwise. Now see that if you have the number 1 (which, in the $xy-$ plane would be the vector $(1,0)$) and apply this component $w$, since multiplying things by 1 intuitively keeps it fixed, you have $1w=w.$ Multiply $w$ again, and you have $1ww=ww=w^{2}.$ But applying $w$ twice means rotating $180º$ ($\pi$ radians). The vector $(1,0)$ in the $xy-$plane rotated $180º$ is the vector $(-1,0),$ namely $-1$. So, you have $w^{2}=-1$.

So, the component $w$ satisfies $w^{2}=-1.$ Well, the mathematicians use to denote this component $w$ by $i$, so $i^{2}=-1.$

That is a way to understand the complex number, and my favorite one.

Answer 2

This is just an illustration of what Mateus Rocha has already said, but since it relates the usage of complex numbers to a real world problem, I thought it might help you in coming to grips with the use of complex numbers.

I work for an aircraft manufacturer, and we needed to figure out the location of the landing gear axles on an aircraft as it sits on the ground. For most of our aircraft models, the main landing gear use a trailing link configuration. This consists of a trunnion extending from the aircraft, with a bar (the "trailing link") connected to the trunnion by a pivot extending back to hold the wheel. A shock absorber (the "oleo") also connects to the trunnion to the trailing link. The location of the axle depends on how compressed the oleo is. Here is a simplified view of the geometry:

The location of the pivot and the anchor are known constants, as well as the distances from pivot to oleo attachment and axle. By taking a measurement of the oleo, we can know all sides of the triangle and can calculate the location of the attachment point, and therefore also the axle.

When I first did this calculation, I solved it using vectors, setting the origin at the Pivot. The Anchor was a known vector, and I rotated it down to the axle line by the amount needed to make the Oleo the right length.

The algebraic calculation was rather messy. But I reproduced the calculation in a number of variants of the tools we used, until at some point I noticed that a formula I was looking at was exactly like complex multiplication. Suddenly it occurred to me that I had been doing it the hard way.

If I view this as the complex plane, with $0$ at the pivot, and represent the Anchor, Attachment, and Axle as complex numbers, the rotation of Anchor to Attachment is just one complex multiplication, and converting the Attachment to the Axle is another (which in the actual calculation is not a real number, as the Pivot, Attachment, and Axle are not really in a straight line). The Oleo length is $|\text{Anchor} - \text{Attachment}|$, which made the calculation of their ratio more sensible as well.

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The point is, any real-world calculation involving a complex number can be thought of instead as a calculation involving 2 real numbers. So it is never necessary to bring in complex numbers. But there are times when conceptually it is easier to work with complex numbers than pairs of real numbers (in the case of quantum mechanics, so much easier that it boggles the mind how hard it would be to express otherwise).

Answer 3

My take may be a little bit of a simplistic view, but every time that I have dealt with complex numbers, I have always been able to interpret them as some sort of "rotation". In some sense, this is what complex numbers describe.

To use your analogy of gambling, the way that I would interpret that situation would be something like 1+1i to mean the player is gaining a dollar per round but is also cycling a dollar. You would expect that the gambler will experience his wealth increasing by about a dollar per round, but will sometimes be ahead of that trendline by about a dollar, and sometimes will be below that trendline by about a dollar. You would expect an ordinary sinusoidal pattern, if you detrended the line.

If you understand the equation: $$e^{i\pi}+1=0$$ you will start to understand why complex numbers sort of imply rotations. This formula has deep ties to trigonometric functions, and thus circles.

Paul Sinclar's answer describes how he solved for the rotation of a shock absorbing system, and how he figured out that it was easier to work with complex numbers. The vector math was messy, and it was solved by rotating the system. But then he realized it was cleaned up the math to think about the system in the complex plane (because that is where rotations happen).

Quantum mechanics you are dealing with angular momentum which is just a fancy way of saying that things spin/rotate and that has to be conserved. So it is no wonder that complex numbers help to clean up some gnarly partial differential equations in that domain.

For your lamp example, the closest that you can get is by producing 83 lamps. Now, this answer isn't very helpful since the answer is not going to give you back $3000. In some sense, though, the complex number tells you that you need to "spiral" out from 83 lamps (positive imaginary number) with something like an angular component of 60 lamps. Strange interpretation, but it is a static univariate problem and I'm doing my best.

The imaginary number in this case, is more likely telling you that it is not feasible to get your target profit with current production technologies. One interpretation that I like is a little sci-fi, it is how many lamps you would need to produce and rotate into our universe from a parallel universe to achieve your target. Silly interpretation, I suppose, but it gives you an idea that the target you are hoping to hit is not feasible.