The magic i

[John Kenneth Swinswood]

The two transcendental numbers, pi and e, are derived in totally different ways. Pi is the ratio of circumference to diameter of a circle. There is no reason to believe that it is different in any other part of the universe, or that it changes with time. We understand it to be a universal and eternal constant. Similarly, e, first calculated by Euler, is accepted as a universal constant and assumed to be eternal. Exponential relationships have existed since the origin of the universe; radioactive decay, first order chemical reactions, etc. There appears to be no physical reason to connect pi and e. However, we have Euler's Identity e^ipi+1=0 which can be rewritten e^ipi=-1 or e^ipi=i^2.

This equation relates only e, i and pi. So, from the beginning of time e and pi were related through i. How magic is that?

[DrP]

53 minutes ago, John Kenneth Swinswood said:

The two transcendental numbers, pi and e,

Transcendental? 

 

 

[Prometheus]

Transcendental numbers.

[DrP]

10 minutes ago, Prometheus said:

Transcendental numbers.

Never knew it was a number classification.   You learn something new everyday (or something old in this case).  

[Sensei]

16 minutes ago, DrP said:

Never knew it was a number classification.   You learn something new everyday (or something old in this case).  

"better late, than never"..

 

[studiot]

1 hour ago, John Kenneth Swinswood said:

The two transcendental numbers, pi and e, are derived in totally different ways. Pi is the ratio of circumference to diameter of a circle. There is no reason to believe that it is different in any other part of the universe, or that it changes with time. We understand it to be a universal and eternal constant. Similarly, e, first calculated by Euler, is accepted as a universal constant and assumed to be eternal. Exponential relationships have existed since the origin of the universe; radioactive decay, first order chemical reactions, etc. There appears to be no physical reason to connect pi and e. However, we have Euler's Identity e^ipi+1=0 which can be rewritten e^ipi=-1 or e^ipi=i^2.

This equation relates only e, i and pi. So, from the beginning of time e and pi were related through i. How magic is that?

Not at all magical.

You offered a geometric interpretation of pi, but not one for e.

Geometry connects both.

ae^x is the curve that has its slope proportional to x at all points.

 

See the geometrical proof here (comes after the analytical proof)

https://en.wikipedia.org/wiki/Euler's_identity

[Prometheus]

8 minutes ago, studiot said:

Not at all magical.

It's certainly magical in the sense of being beautiful (not sure if that is what the OP is referring to).

I wish maths was actually taught with at least a little sense of wonder: like climbing a mountain to enjoy the view, it makes a difficult subject worth the work.

[DrP]

19 minutes ago, Sensei said:

"better late, than never"..

 

I was surprised I hadn't heard of it as I did a ChemPhysics degree in the 1990's and they are pretty heavy on the maths. I am surprised I never heard of it doing A-Level maths.

Maybe I did and just forgot it as a useless piece of information  -  I have never used the knowledge so could have forgotten the definition I suppose... a few decades have passed. I suspect it is something I have just never come across.

 

[John Kenneth Swinswood]

"plus.maths.org" describes e and pi as transcendental numbers, meaning that they are not roots of any algebraic equation with integer coefficients.

The value of pi is fixed by the geometry of the circle. The value of e is fixed by the fact that the function e^x is proportional to its gradient or in other words the differential of e^x is e^x. This means that when the physical universe started, complete with first order reactions, the value of e was determined.

The geometry of the circle and the physics of first order reactions have no apparent connection yet they determined the values of pi and e from the beginning, before mankind and mathematics. Thanks to Euler we now have a mathematical connection and it only involves one other quantity - i. That is a source of wonder.

[studiot]

22 minutes ago, John Kenneth Swinswood said:

That is a source of wonder.

Maybe.

But it is not magic.

And why is it more interesting that to note than say

4 = 2x2 =2+2

or 6 = 2x3 = 2² + 2 = 1+2+3

or that 13 is not (interestingly) related to any other number but 1 and itself.

or that Pi = 2 x Wallis product?

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

 

Edited May 30, 2018 by studiot

[mathematic]

The connection between pi and e comes about when geometrizing complex numbers.

[John Kenneth Swinswood]

Applying the MacLaurin series you can derive infinite series for e^x, sinx, and cosx. It then follows that e^ix = cosx + isinx, and putting        x = pi yields Euler's identity. No geometry involved. Granted, the complex number e^ix can be represented on an Argand diagram and the locus of values (x varying from 0 to 2pi) is a circle of unit radius, but this was not used to derive Euler's equation.

Arguing about how the identity equation is derived misses my point, which is:- (a)pi is solely determined by the geometry of the circle.       (b) e is derived from first order reactions. (c)These two have no physical connection, but they are connected mathematically by just i.

If that is not magical then it is at least beautiful.

 

[YaDinghus]

On 5/30/2018 at 4:00 PM, DrP said:

On 5/30/2018 at 3:36 PM, Sensei said:

"better late, than never"..

 

I was surprised I hadn't heard of it as I did a ChemPhysics degree in the 1990's and they are pretty heavy on the maths. I am surprised I never heard of it doing A-Level maths.

Our college professors and TA's didn't mention this classification, either. I learned about transcendental numbers from Numberphile on youtube. I guess classifications of numbers just aren't relevant to what physicists and chemists do. 

@John Kenneth Swinswood Math is full of wonderful relationships. It's really too bad that PBS Studios isn't producing any more of the Infinite Series. 

[mathematic]

4 hours ago, John Kenneth Swinswood said:

Applying the MacLaurin series you can derive infinite series for e^x, sinx, and cosx. It then follows that e^ix = cosx + isinx, and putting        x = pi yields Euler's identity. No geometry involved. Granted, the complex number e^ix can be represented on an Argand diagram and the locus of values (x varying from 0 to 2pi) is a circle of unit radius, but this was not used to derive Euler's equation.

Arguing about how the identity equation is derived misses my point, which is:- (a)pi is solely determined by the geometry of the circle.       (b) e is derived from first order reactions. (c)These two have no physical connection, but they are connected mathematically by just i.

If that is not magical then it is at least beautiful.

 

The point I was trying to make is that the Argand diagram clearly shows the connection between e and pi, not magic.

[John Kenneth Swinswood]

Pi = circumference divided by diameter. Simple geometry. 

e^x is a function equal to its own derivative. In the physical realm e^kt represents exponential growth if k is positive, or exponential decay if k is negative. No connection with geometry or the circle. Therefore no connection with pi.

Now introduce i.

e^ix is a complex number which leads to a circle on the Argand diagram, and thus to pi.

Without i, no connection between pi and e. Only with i can the connection be made. Magic.

 

[Andri Lopez]

The  transcendent number pi related to the cos(90º/pi).

Cos(90º/pi) = 2/ sqrt{5}

It is built will ruler and compass; out of the circle.

[uncool]

Um. No?

 

cos^2(90 deg/pi) = cos^2(1/2) ~.77 is not 4/5 = (2/sqrt(5))^2.

[Andri Lopez]

The  cos(90º/pi) if is 2/sqrt{5}.

See in: International Journal of Mathematics and Computation Vol 28 (2) 2017

Author: Enfer Diez

[Strange]

1 hour ago, Andri Lopez said:

The  cos(90º/pi) if is 2/sqrt{5}.

Nope: http://www.wolframalpha.com/input/?i=cos(90º%2F+pi)+%3D+2%2Fsqrt(5)

cos(90º/pi) = 0.877582 http://www.wolframalpha.com/input/?i=cos(90º%2F+pi)

2 / sqrt(5) = 0.894427 http://www.wolframalpha.com/input/?i=2%2Fsqrt(5)

1 hour ago, Andri Lopez said:

See in: International Journal of Mathematics and Computation Vol 28 (2) 2017

Author: Enfer Diez

That is behind a paywall: http://www.ceser.in/ceserp/index.php/ijmc/article/view/4674/0

But a calculator will show that you are wrong.

 

[taeto]

On 6/4/2018 at 6:57 PM, John Kenneth Swinswood said:

Pi = circumference divided by diameter. Simple geometry. 

e^x is a function equal to its own derivative. In the physical realm e^kt represents exponential growth if k is positive, or exponential decay if k is negative. No connection with geometry or the circle. Therefore no connection with pi.

Now introduce i.

e^ix is a complex number which leads to a circle on the Argand diagram, and thus to pi.

Without i, no connection between pi and e. Only with i can the connection be made. Magic.

 

In the study of physical oscillating systems, with possible dampening, you naturally obtain second order linear differential equations for their movement, as a consequence of Newton's second law F=ma, where F is force, m is mass and a is acceleration (the second derivative of the position as a function of time). The basic example is a ball suspended on a spring while subject to a viscose environment such as a surrounding air or liquid.

The solutions are products of trigonometric functions like sin(at+b) and cos(at+b) at time t, that provide the oscillations, with exponential functions e^-wt , that provide the dampening effect. This is not magical, just consequences of the fact that the trigonometric functions are very closely related to the complex exponential function. The exponential function and the  trigonometric functions are also the basic functions for which their second derivatives are proportional to themselves. Which is why they appear naturally together in descriptions of how physical systems behave. 

The precise identity e^it = cos(t) + i sin(t) does the rest to establish a connection between e and pi.

[John Kenneth Swinswood]

Undamped harmonic motion gives a displacement formula Asin(at+b). Where A is the amplitude of the oscillation? When the motion is damped the amplitude decreases exponentially but the periodicity is unchanged. There is no physical connection between the exponential and trigonometric components.