That's nice. I didn't say there was no value in intuition. I just said it was unreliable and I demonstrated an example of why its unreliable.
Its better to focus on solutions that don't rely on intuition because intuition is based solely on experience and often can't be applied to everything. This is why there's a strive to find general solutions to things. Intuition can help you, but you should not base your understanding off it. Intuition can lead you astray easily as in my example. People just assumed it was correct for many years until Galileo actually proved it through experimentation.
In fact, I found this explanation way more complicated than just stating that you are performing trigonometry in two different domains. Your sine is scaled by an imaginary value and your cosine is scaled by a real value. The radius equates to an exponential. When you move it pi units you get sin = 0, cosine = -1 which you use algebra to get Euler's identity.
You don't need to introduce complex intuition in order to get it. Intuition is often a crutch people use to try and think they understand something and just blame others when they don't get it much like this post. If intuition was so important, why not submit it as a legitimate mathematical way of proving things?
Its better to focus on solutions that don't rely on intuition because intuition is based solely on experience and often can't be applied to everything. This is why there's a strive to find general solutions to things. Intuition can help you, but you should not base your understanding off it. Intuition can lead you astray easily as in my example. People just assumed it was correct for many years until Galileo actually proved it through experimentation.
In fact, I found this explanation way more complicated than just stating that you are performing trigonometry in two different domains. Your sine is scaled by an imaginary value and your cosine is scaled by a real value. The radius equates to an exponential. When you move it pi units you get sin = 0, cosine = -1 which you use algebra to get Euler's identity.
You don't need to introduce complex intuition in order to get it. Intuition is often a crutch people use to try and think they understand something and just blame others when they don't get it much like this post. If intuition was so important, why not submit it as a legitimate mathematical way of proving things?