Complex numbers start with an idea that the square of a number
does not have to be positive. This idea gives rise to
famous Euler's formula, which relates exponential
function with trigonometric ones. But why is Euler's formula hold? In this video we will derive it,
based on school physics. A complex number
is an ordinary real number plus an unusual imaginary part.
Complex number z can be represented by a point
on the coordinate plane, plotting the real part of the number, that is x,
along the horizontal axis, and plotting its imaginary part, that is, y,
along the vertical axis. Each point can be considered as a radius vector,
arriving at this point from the origin. Let us imagine that the particle moves
in such a way that its position in the complex plane changes
according to the law z = e^(t), where t is real human time. This is a well-known exponent. Its value remains a real number at any time t. For example, at the very beginning when t is zero, the particle is at point z=1, and it accelerates along the real axis as time goes. Yes, it accelerates. Moreover, the velocity that is the derivative
of the coordinate with respect to time, dz/dt, also remains a real number
on the complex plane. The velocity is always directed
to the right, and its value is equal to the coordinate of the particle,
which, in turn, grows. Now let’s try to imagine how
a particle will move if if we slightly change its trajectory
by adding an imaginary unit next to time. As before, at time t equal to zero, the particle starts from the point equal to one. But the initial velocity of the
particle is now equal to i, that is, it is directed upward
perpendicular to the radius vector. Up to now we don't know where the particle turns out at any other time than the initial one. But the velocity of a particle will always be equal to its coordinate
multiplied by an imaginary unity. What is the meaning of multiplying by i? If we multiply 1 by i than we get i. If next we multiply i by i than we get -1. Each subsequent multiplication by i causes a rotation of the radius vector
by a right angle counterclockwise. But this is also true for any other point
in the complex plane. Multiplying by i will rotate the radius vector
90 degrees counterclockwise. This is indicated by the coordinates
of a point z before and after multiplication. It turns out that wherever the point e^(it) is, the velocity vector at this point
is directed perpendicular to the radius vector. That is, the length of the radius vector
does not increase with time, and this is the motion along a circle, the radius of which is equal to the radius
at the initial moment of time. That is, to unity. In other words, e^(it) equals to cosine of fi plus sine of fi. There is one last question left: how does the angle fi change with time? What if the point moves around the circle unevenly or even very unevenly?..
To address this issue, let's consider Newton's second law: F = ma. It says that
the acceleration of the particle is co-directed with the resulting force. Acceleration is
the derivative of velocity with respect to time. And it equals the product
of the imaginary unity and the velocity. That is, again the direction of the force applied to the particle
is always perpendicular to the velocity vector. Such a force doesn't do work
and cannot change the kinetic energy of the particle,
and hence the velocity itself. Such a force only changes
the direction of velocity. Therefore, the particle moves uniformly
around the circle, and the velocity remains constant from the very beginning
of the movement, that is, it is equal to unity. So, the radius of the circle is equal to one. The speed is equal to one. It turns out that the particle uniformly travels a distance of one radius
per unit time around the circle. And on time 2*pi the particle
will complete a full revolution. This means that the angular speed
of motion is constant and equal to one radian per second. That is, angle fi is equal to time t. This is how physics came in handy where
it wouldn't seem to be expected. Now you can safely write on the fences the famous identity
composed of the main mathematical constants: e^(i*pi) = -1 because no conservation energy law is violated.