ऑयलर सूत्र संवादात्मक
Euler's formula links exponentials, complex numbers and trigonometry in a single line. This simulation shows why: as the angle advances, the point e^(iθ) travels around the unit circle, with its shadow on each axis tracing a cosine and a sine.
इस सिमुलेशन का उपयोग कैसे करें
- Advance the angle θ and watch the point travel
- Project onto the real and imaginary axes
- Overlay the resulting sine and cosine curves
क्या देखें
- The point never leaves the unit circle, because the modulus is always 1
- Its horizontal shadow traces a cosine, the vertical one a sine
- At θ = π the point lands exactly on −1
इसके पीछे की भौतिकी
Euler's formula states e^(iθ) = cosθ + i·sinθ. Multiplying by e^(iθ) is a rotation through θ in the complex plane, which is why the point moves on the unit circle. Setting θ = π gives the celebrated identity e^(iπ) + 1 = 0, tying together five of the most important constants in mathematics.