Math and physics

 i, i what ?

What does i = SQRT (-1)  represent in real life. 

I try to find some real life examples in which i plays a role and try to dig out the secrets of i that is the real life meaning of i.

To start I came up with a simple but nice example of it. 

 

 

example 1:

 

suppose I stand on a balcony at 8 meters high, a friend of mine stands on the ground floor. My hand is at 10 meters ready to catch a ball that my friend throws up. we take it for simplification that there is no drag resistance with the air, and even if it would it would not matter with respect to the subjects key problem, the meaning of i. even with drag it would show up but in a very much more mathematical complicated way and in order to avoid that complication, air resistance is excluded.

For more details about object tossing with air resistance watch these:

https://en.wikipedia.org/wiki/Trajectory

and 

https://en.wikipedia.org/wiki/Projectile_motion#Catching_balls

but that is not the subject of interest here.

 

Now as we all know throwing stuff at people goes wrong in the most attempts on average at least once, and so also with my friend. he misses my hand by 1 meter so he just was able to toss the ball 9 meters high with respect to my hand. 

As we also know a falling object (without drag) falls according to a parabolic trajectory AX^2+BX+C. 

 

Now if the level of my hand is taken as the Y = 0 level than a trajectory could look something like:

 

     x      y

     1     -10

     3     -5

     5     -2

     7     -1

     9     -2

    11     -5

    13    -10

 

so at position 7 my friend misses my hand by 1 meter; It is obvious that this has a formula of Y =  -0,25*X^2 + 3,5*X - 13,25

 

This results in the next solution via the ABC formula:

 

             - B +/- SQRT [ B*B - 4* A* C ]

X1,2  = ----------------------------------

                                 2* A 

 

A = -0,25,   B = 3,5 and  C = -13.25  so 

 

             - 3,5 +/- SQRT [ 3,5*3,5 - 4* (-0,25)* (-13,25) ]

X1,2  = -------------------------------------------------

                                             2* (-0,25) 

 

The whole mathematical picture is shwon here below:

 

The solution fo this delivers us an imaginary or complex number  P  + Q * i

 

Question is what is i and what does it represent in the physical reality. 

 

Now as we can see in this case i =  SQRT (B*B - 4*A*C)  which is here   SQRT( 3,5*3,5 - 4* (-0,25)* (-13,25) ) = SQRT(-1). 

and as was stated my friend missed my hand by exactly 1 meter so here it may seem that i = SQRT (-1) that i represents the SQRT of the shortage of my friends tossing ability. 

 

Now lets see if that is correct by downgrading my friends ability even further and lets say he misses my hand by 2 meters in a 2nd attempt. 

What we then get is 

x      y

     1     -11

     3     -6

     5     -3

     7     -2

     9     -3

    11     -6

    13    -11

 

so at position 7 my friend misses my hand  this time by 2 meter; Again it is obvious that this has a formula of Y =  -0,25*X^2 + 3,5*X - 14,25

 

This results in the next solution via the ABC formula:

 

             - B +/- SQRT [ B*B - 4* A* C ]

X1,2  = ----------------------------------

                                 2* A 

 

A = -0,25,   B = 3,5 and  C = -14.25  so 

 

             - 3,5 +/- SQRT [ 3,5*3,5 - 4* (-0,25)* (-14,25) ]

X1,2  = -------------------------------------------------

                                             2* (-0,25) 

 

Now as we can see in this case SQRT (B*B - 4*A*C)  gives us   SQRT( 3,5*3,5 - 4* (-0,25)* (-14,25) ) = SQRT(-2). 

 

And as was stated my friend missed my hand in this attempt by exactly 2 meters

So again it seems that SQRT (-2) = SQRT(2) * i  that i represents the shortage of my friends tossing ability and the 2 in -2 represents the amount of the shortage. 

 

Well today my friend was very exhausted from work and that also shows in a 3rd attempt when he missed my hand by 7 meters as shown in the next graph;

What we then get is 

x      y

     1     -16

     3     -11

     5     -8

     7     -7

     9     -8

    11     -11

    13    -16

So at position 7 my friend misses my hand  this time by 7 meter; Again it is obvious that this has a formula of Y =  -0,25*X^2 + 3,5*X - 19,25

 

             - 3,5 +/- SQRT [ 3,5*3,5 - 4* (-0,25)* (-19,25) ]

X1,2  = -------------------------------------------------

                                             2* (-0,25) 

 

Now as we can see in this case SQRT (B*B - 4*A*C)  gives us  SQRT( 3,5*3,5 - 4* (-0,25)* (-19,25) ) = SQRT(-7). 

 

And as was stated my friend missed my hand in this 3rd attempt by exactly 7 meters. So we see that SQRT (-7) = SQRT(7) * i.  that i represents the shortage of my friends tossing ability and the 7 0f -7 in the B^2 - 4AC represents the amount of the shortage. 

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