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 2pi 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^(ipi) = -1 because no conservation energy law is violated.