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.

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.

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.

"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.

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 only neat, but beautiful. Thanks. The point I'm trying to get at is this... Looking at the line in the animation we can see that pi is between 3 and 4. Using a ruler we could find that it is between 3.1 and 3.2; with a vernier scale or micrometer between 3.14 and 3.15, etc. If we could improve the accuracy of measurement by a factor of ten every day for a thousand years we would still not have the exact measurement. In fact it would never be accurate. Is this another "uncertainty principle"? Ken.

What is pi exactly? Ken.

Never mind about the definition of the meter. If the diameter of the circle is one unit then the circumference is pi units. If the circumference is one unit then the diameter is 1/pi units. This means that at least one of the two measurements is an irrational number. How can this be reconciled with the fact that the diameter and the circumference are real physical quantities? And what does it tell us about accuracy of measurements? Ken.