Understanding Euler's Formula and Complex Numbers

Introduction

• Complex numbers expand the idea that the square of a number does not have to be positive.
• Euler's formula: relates exponential function with trigonometric ones.
• Aim: derive Euler's formula using physics principles.

Concepts of Complex Numbers

• A complex number has a real part (x) and an imaginary part (y).
• Represented on a coordinate plane: x-axis for real part, y-axis for imaginary part.
• Each complex number can be considered as a radius vector from the origin to the point (x, y).

Trajectory and Motion in the Complex Plane

• Particle's position in complex plane changes as per z = e^(t) with t as real time.
• At t = 0, particle is at z = 1 and accelerates along the real axis.
• The velocity (dz/dt) remains real, directed to the right, and equal to the coordinate, which grows over time.
• Changing the trajectory by adding an imaginary unit next to time shifts the initial velocity to i (upward direction).

Imaginary Multiplication

• Velocity of the particle is the coordinate multiplied by the imaginary unit i.
• Multiplying by i rotates the radius vector 90 degrees counterclockwise.
• At z = e^(it), velocity is perpendicular to the radius, causing circular motion with constant radius (1 unit).

Motion Analysis Using Newton's Second Law

• Applying F = ma where acceleration is the derivative of velocity with respect to time, directed perpendicular to velocity vector.
• Perpendicular force doesn’t change kinetic energy, only direction, leading to uniform circular motion.

Conclusion

• Uniform travel around the circle with constant angular speed of 1 radian per second.
• Angle fi is equal to time t.
• Validates the identity e^(i*pi) = -1.

Key Points

• Complex numbers: real + imaginary parts.
• e^(t) gives real exponent growth; e^(it) gives circular motion in the complex plane.
• Multiplication by i results in 90-degree rotation.
• Newton's second law explains uniform circular motion.
• e^(i*pi) = -1 verified: no violation of energy conservation laws.