Do you believe in √(-1)?

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No.  Imaginary numbers are, by definition, imaginary, i.e. products of the imagination.  Therefore, they are not real; if they were, they would be real numbers.  Since the square root of -1 is clearly not a real number but an imaginary number, then by its very definition it cannot exist.

The name has little to do with its meaning and more to do with its history. That would be like saying a strange quark has less reality than an electron because of its fanciful name.

Imaginary numbers were therefore called lateral numbers after their discovery.
They are nowadays called "imaginary" because they are too complex for too many people to understand... 🤓

What still blows my mind, though, is the exponentiation of √(-1) with itself. 🤯

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It still blows my mind, though, is the exponentiation of √(-1) with itself. 🤯

Why would the result seem so unusual? Is it because there is more than one value in the answer?

No, because the result is a real number. The proof of that is very easy to understand, but it's hard to visualize why the result comes into being.

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It still blows my mind, though, is the exponentiation of √(-1) with itself. 🤯

Why would the result seem so unusual? Is it because there is more than one value in the answer?

No, because the result is a real number. The proof of that is very easy to understand, but it's hard to visualize why the result comes into being.

But it is defined such that exponentiation results in a real number - square it and you get -1, fourth power gives +1. Why would other powers, including i, also being real be a surprise?

I was talking about i ⁱ.

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It still blows my mind, though, is the exponentiation of √(-1) with itself. 🤯

Why would the result seem so unusual? Is it because there is more than one value in the answer?

No, because the result is a real number. The proof of that is very easy to understand, but it's hard to visualize why the result comes into being.

But it is defined such that exponentiation results in a real number - square it and you get -1, fourth power gives +1. Why would other powers, including i, also being real be a surprise?

I was talking about i ⁱ.

As am I. My point is that i ^z is real for a number of values of z, and is defined based on its exponential to a specific real value of z = 2. Why is is surprising that there are many other values of z, including complex values such as i, that also give real results?

If the problem is one of visualization, then I have to start by asking how you visualize i ². If it is only as an algebraic formula, I can see where that is hard to extend. If instead you visualize points in a plane that can shift and rotate then it is easier to see how some can end up on the real axis.

It's just too hard to understand for me what "to the power of i" even means.
Is there even a real number x that, exponentiated with itself, delivers the same result as iⁱ = x^x?